Chapter 8 Lie groups and hamiltonian mechanics.

Chapter 8. Lie groups and Hamiltonian mechanics. We shall outlines some interesting applicatione of Lie groups and symmetries to hamiltonian mechanics. Our main emphasis will be on integrable systems and systems that possess large symmetry groups. Although most of the discuasion does not directly involve the representation theory of chapters 1-7 (save for the last section, §8.5), the Lie structure theory of groups and algebras will enter many times. To make our presentation self-contained we included in the first section, 58.1, some basics of the hamiltonian mechanics: Lagrangian formulation, Minimal action principle; EulerLagrange equations; Canonical formalism; sympIectic/F‘oisson structure; conserved integrals, integrability and Darbeaux Theorem.

$8.1. Minimal action principle; Euler-Lagrange equation; Canonical formalism.

1.1. Hamilton’s Minimal Action principle. The state of a classical mechanical system of n degrees of freedom is described by its position vector: q = q ( t ) = ( q l ; ...qn), varying over W”, or more general Riemannian manifold At, called configuration space, and velocity (tangent) vector: q = (ql; ...qn). Its dynamical evolution is determined by the action functional,

s=

li t0

P.(q;q)dt,

whose integrand .t(q;q), called Lagrangian, depends on position and velocity, and has physical dimensionality of energy. In many cases of interest Lagrangian represents the difference of Kinetic and Potential energies, P. = K - P , where h’ = 1 . 2 . 2Q

7

or more generally, is given by a Riemannian metric-tensor { g i j ( q ) }on At,

K = !jC g i , qiqj; while P = V(q)- a potential function. According to the Hamilton’s principle of Minimal Action: a trajectory (evolution) of the classical system must minimize (or give a stationary path) of the actionfunctional. So it satisfies the Euler-Lagrange equation:

a 6q(t) =P q -d(P..e=o. dt .P case of

(1.1)

Equation (1.1) represents a 2-nd order OD system in n variables. In the classical P. = “kinetic” - “potential”, (1.1) turns into the Newton’s equation,

9’’

-aV(q) = F- force.

The canonical formalism reduces the 2-nd order Euler-Lagrange equations to a 1 - st order system of size 2n. We introduce a new set of variables: pi = a. .P.(q;q)- conjugate momenta, ‘11

(1.2)

$8.1. Minimal action principle; Euler- Lagrange equation;

370 and

H ( q ;p ) = p

- L, hamiltonian/energy function.

Solving a system of equations (1.2) for q-variables’, we get q = Q(q;p),and these are substituted in the hamiltonian H ( q ; p ) .Then the Euler-Lagrange equations (1.1) are shown to be equivalent to a hamiltonian system

@ - a PH ;

Ap= - a p .

(1.3)

We shall first demonstrate the canonical formalism in the case of N-particle systems. The corresponding Lagrangians are N

L = 3’ - V ( q ) ,or 3Cmjq: - V, where ( m j } denotes masses of particles. Then the conjugate momenta: p j = r n j q j , and the hamiltonian,

+

+-

H

= c l 2mj p . J 2 V(q);“kinetic” “potential”, a familiar expression from elementary calculus/mechanics. For

more general (“kinetic - potential”) Lagrangians on manifolds A, the Euler-Lagrange (1.1) takes the form & ~ g i j q j ) - a q i v = 0,

while the canonical variables:

..

P; = E g i j Q j ; H = + C g ” P i P j

+ V(q),

( g i j ) denotes the inverse matrix (tensor) to ( g i j ) , and the hamiltonian system

becomes \pi = - a,v. So geometrically, momentum variables { p } can be identified with cotangent vectors on A, and the (position-momentum) phase space becomes a cotangent bundle I*(A).The change of variables (q;q ; L)+(q; p ; H ) , called the Legendre transform, can be interpreted as a map from the velocity phase space, tangent bundle T ( A )= { ( q ; q ) } , to the momentum phase space, cotangent bundle T * ( A )= { ( q ; p ) } , that takes solutions (trajectories) { q ( t ) ; q ( t ) }of the Euler-Lagrange equations (1.1) to those of Hamilton system (1.3) (problem 2). ‘provided it could be solved, i.e. the Hessian of & in q-variables is nonsingular, det(-) a2L # 0. aCr;aqj The latter is always the case with the classical Lagrangians, L = “kinetic” - “potential”, on A, whose Hessian turns into the metric tensor { g j k ( q ) } .

$8.1. Minimal action principle; Euler-Lagrange equation;

371

1.2. Symplectic structure and Poisaon bracket. Phase space 9 = T * ( A ) is equipped with the natural symplectic/Poisson structure, given by a differential (canonical) 2-f0rm2 on 9:

R = C d p i Adqi, (1.4) i in standard (dual) local coordinates { ql...qn;pl...pn}.In other words, we take a basis in the tangent space T q ( A )and the dual basis { d q l ;...dqn} in the cotangent space T z ( A ) , so each point x = ( q ; t ) E T * ( A ) , q E A, ( = C p j d x j E T i , can be represented by a 2n-tuple { q i ; p j } .

{aql;...aqn}

Symplectic structure on a general manifold 9,with local coordinates { x l ;...zm}, is given by a non-degenerate closed %form,

0 = X U j k d X j dXk, (1.5) with the usual proviso that R be independent of a particular choice of { x l ;...xm} (i.e. coefficients { a j k } transform as a tensor on under coordinate changes on 9).Clearly, non-degeneracy of R constraints dim9 to be even. In addition, one requires closedness of R, in the sense that its differential &?= E E i j k a i ( a j k ) d Z i A d x j I\ d X k = 0.

Here Eijk denotes a completely antisymmetric symbol (tensor) in 3 indices, normalized by elZ3 = 1. An alternative way to describe a symplectic structure on 9 is in terms of a skew-symmetric bilinear form 1, on tangent spaces {T,(T):xE T},

(1t17) = z b j k t j 9 k ; (77 E T Z . Two structures are related one to the other by R = j - ' d x A dx, in other words matrix (a.J k ) = ' ( a $ ) - ' . Clearly, the standard 2-form 0 yields the standard symplectic matrix, L

J

Examples of symplectic manifolds include: 1) The standard (flat) phase spaces: W2 = { ( x ; p ) }with R = dx A d p , and W2" with 0 = C d x j A dpi- Those are often convenient to write in the complex form: z = x -ti p E C", then R = A d?i.

3.z

2) Cotangent bundles "*(A) over manifolds A, with R = E d q j A d p j ; 'Let us remark that (1.6) is independent of the choice of local coordinates on A. Indeed, if ,denotes a coordinate change, then differentials { d q j ] are transformed by the Jacobian map f' = 8 9 while differentials of co-vectors by inverse transpose of f'. So the product C d p j pI d q j remain8 invariant. f:q+q'

(r),

58.1. Minimal action principle; Euler-Lagrange equation;

372

3) The 2-sphere S2= {(c$;O)} with R = sinc$dc$A do.

4) Co-adjoint orbits of Lie groups: O C ~ (chapter * 4). The tangent space T , (2 E 0 ) is identified with the quotient 0/6, - Lie algebra modulo stabilizer subalgebra of z OZ = { ( : a d i ( z ) = 0). As a bilinear form on tangent spaces,

I [(;171)

Q(f;rl)= .(

= (ad;(.)

Id; t77l E (5.

5 ) Finally, we shall mention the so called KiiMer manifolds, complex manifolds A with a hermitian metric-form: ds2 =

C b,,dz,dZv;

-

b,,,, = b,,,;

whose imaginary part is a closed 2-form7

R=

b,,dz,,

(1.7)

A dTv.

To check skew-symmetry of R one notes that in real coordinates {z,, = Sz,,; y,, = St,,},

R = Adz A d z + 2Bdz A dy + Cdy A dy,

where A = C and B are real (antisymmetric and symmetric) matrices S(b,,,,) %(b,,,),

i.e. (b,,”)

= B + iA. Hence

and

defines a symplectic structure on A.

Symplectic structure on any phase-space 9 (cotangent bundle, co-adjoint orbit, etc.), allows to assign certain vector fields, called hamiltonian fields, to functions (observables) F on 9, F 4 Z F = j(aF).

In other words we take a gradient vector field of F and “twist” it by a skewsymmetric linear map 1 on tangent spaces {Tz(9’)}.The standard symplectic structure (1.7) on phase-space R” x R”, yields hamiltonian vector field, z, = a p F . - a q F .

aq

ap.

Each vector hamiltonian field generates a hamiltonian flow, {ezp(tE,)}-a fundamental solution of ODS,

5 = B(aF(z)), (1.8) which generalizes the canonical system (1.3). Hamiltonian flows possess many special features, for instance, all of them preserve the canonical (Liouville) phase-volume, d”q

-

d”p on 9,since all respect the canonical/symplectic form 1, or R (problem 3), and “Liouville volume” = R A ... A 0.

Symplectic structure also defines a Lie-Poisson bracket on the vector space of observables { F ( z ) }on 9, namely

{F;G) = (r(aF)I aG),

58.1. Minimal action principle; Euler-Lagrange equation;

373

which in special cases, W2" or T * ( h ) ,turns into,

{ F ; G }= a , F - d , G - a , F . a , F . The reader can verify directly all properties of Lie bracket (skew-symmetry and Jacobi identity) for {F;G}3. In fact, the Poisson bracket of any two observables corresponds to the standard Lie bracket of their hamiltonian vector fields,

-- --

{ & G } ~ E { C G= I [ZF;5,] = = F = G - Z G C F . Thus the space of observables { F ( z ) } on 9 acquires a structure of an m-D Lie algebra, a subalgebra of all vector fields bD(9).The corresponding Lie group consists of all canonical transformations on 9, i.e. diffeomorphisms {q5} that preserve the symplectic structure,

Tq5'%#) = 8.; equivalently coordinate changes, y = d(z), that preserve the canonical 2-form,

We shall list a few examples of canonical transformations: i) symplectic matrices { A E S p ( n ) } in R2" = { ( q ; p ) } ; they clearly preserve the standard 2-form: R = C d p j A dqj. ii) any coordinate change (diffeomorphism) y = @(z), from manifold .A to N, induces a canonical map,

YO);

q5:(x;o+(@(x);TAwhere A denotes the Jacobian matrix of @ at {z}, A = @: (problem 4). The reader has probably noticed some coincidence in terminology: symplectic Lie groups/algebras on the one hand, and symplectic structure/geornetn'es on the other. The relation between two becomes apparent now: Jacobian matrices of canonical (symplectic) maps are symplectic matrices4! 1.3. Conserved integrals; action-angle variables and the harmonic oscillator. The hamiltonian evolution (1.8) of the position and momenta variables { q ; p } gives rise to 3The Jacobi identity on general symplectic manifolds results from closeness of the canonical 2form 0. 41n this regard symplectic groups plays the same role in the symplectic geometry, as orthogonal groups in the Riemannian geometry. There exists, however, a striking difference between two kinds of geometries. The Riemannian geometry is fairly rigid in the sense that isometries of m (even in the best case of symmetric spaces) form only a finite-dimensional Lie group, whereas "symplectic isometries" (canonical maps) are always infinite-dimensional!

374

$8.1. Minimal action principle; Euler-Lagrange equation;

evolution of any other observable (function) F on T * ( A ) ,

P = {F;H}. Functions that remain constant along trajectories of the hamiltonian flow, { F , H } = 0, are called first integrals of (1.8). Indeed, the Legendre “back-transform” (q;p)+(q;q), takes such F into a function F ( q ; p ( q ; q ) ) ,constant along trajectories (solutions) of the E-L equation (1.1) i.e. gives a 1-st integral of (1.1). Hamiltonian H is itself an integral, as {H;H} = 0. In dimension n=l, it is the only integral. So 2-nd order E-L equation, Lq- $L4 = 0, is reduced to a 1-st order ODE: H(q;p( ...;q ) ) = E -const.

In the classical (Newton) case this yields I *+v,I.2 zq +V = E, ZP

whence we get an implicit solution in the form of integral

5 J;i&

= t - to.

(1.9)

More generally, each Poisson integral F of the hamiltonian system (1.3) reduces its total order 2n by 2. The proof is based on the following general result.

Dubeaux Theorem: Any set of functions (obseruables) {F,; ...Fk;G1;...G,} on a phase space (symplectic manifold) 0 that satisfy the canonical commutation (Poisson bracket) relations { F j ; G i }= Jij, can be locally eztended to a canonical coordinate system on 0,{pl;...p,;q,;...q,}, p; = pi (i=l, ...k) and qj=Gj (j=l;...m).

In particular, a Poisson integral F can be made the 1-st momentum variable p , of the new coordinates. Then hamiltonian H ( p ; q ) becomes independent on the corresponding canonically conjugate position ql, and p , is constant along the flow,

p1 =

Q1

= { p , ; H } = 0.

Setting pl= El-const we reduce H to a hamiltonian H ( E , ; p , ;...p,;q,;...q,) in ( n 1) degrees of freedom, so the system becomes:

(4i‘=

p . = d H; ‘li

-dPiH;

i=2; ...n; and p , = El; q1 = ql(0)

t

+ IH(...)ds. n

A hamiltbnian system (1.3) in n degrees of freedom is called completely integrable (in the Liouville sense), if it has n functionally independent, Poisson commuting integrals {Fl;Fz;...F,}, { H ; F i } = 0; and { F i ; F j }= 0, for all i , j . (1.10) Functional independence means gradient-vectors { VF,} are linearly independent

58.1. Minimal action principle; Euler-Lagrange equation;

375

at each point t = ( q ; p ) . In the standard terminology, commuting integrals (1.10) are said to be in involution. Let us remark that any set of n functionally independent integrals {Fl;...Fn} allows to reduce the total order of the system by n, i.e. to bring (1.3) to a first order system in n variables: Fl(G4)= El F,(q;q)'= En' Complete integrability yields, however, more than a mere reduction of order by n, as each subsequent integral F , respects the joint level sets and evolutions of all preceding variables in the canonical reduction. Hence, the system could (in principle) be reduced to order 0, i.e. solved completely! The procedure is best illustrated by the harmonic oscillator

{

H=izpf+wfqionW2".

As first integrals of H one could take all coordinate oscillators,

{ H i = ;(pi

zHi.

+ wiqi): 1 5 i 5

72).

Obviously, H= The joint level sets of { H i = E j } form a n-parameter family of invariant tori in WZn, and the hamiltonian dynamics consists in a uniform motion in some direction along the torus. Introducing polar coordinates (P; 8 ) in the i-th phase plane { q j ; p j } after , rescaling, qi+wiqj, the j-th hamiltonian becomes H 1. = L( ? + q?) = LP?. 2Pc I 2 1 In polar coordinates the flow of each

Hiis given by an OD system

!= O, i.e. P = E -con~t, 8 = 0, + Et. (4 = r The Poisson bracket { r ; e }= i,hence {$';fI} = 1, and the entire set of variables { H,; ...H,; 8,;...en} satisfies the canonical commutation relations: {H..B I' 3.}=6.. 1J' Returning to a general completely integrable hamiltonian system with n commuting integrals { F,; ...F,} (called actions), the Darbeaux Theorem implies the existence of a canonically conjugate set of angle variables: {el;...en}, that satisfy the canonical relations The hamiltonian H in new coordinates { F; 6) becomes a function of actions only, H = f(Fl;...F,J, since aej-- Fj = 0. The joint level sets { F j = Ei} form a foliation of

the phase-space into invariant tori Tn (or products TkxWn-'),

and the dynamics

376

$8.1. Minimal action principle; Euler-Lagrange equation;

resembles the oscillator case. Precisely, if

(P;q)-+(F;e>,and

4j

denotes the canonical coordinate change:

+

F~ = E ~e,(q ; = ej(o) i g 8H F i ( ~... l ;E"), the hamiltonian flow in the action-angle coordinates, then solution in the original coordinates becomes (1.11) ( p ; q ) ( t=) @-I(... E ~...; e,(t) ...). Let us remark, that solution (l.ll), although written explicitly, may be of limited utility unless one is able to compute the canonical map 4j.

Our next goal is to establish integrability, find l-st integrals and, if possible, explicit solutions of different hamiltonian systems. We shall review a number of the classical models, and also discuss some newly discovered examples. Our principal tools, in the study of conserved integrals, will be symmetries of the problem, and the Noether Theorem, to which we turn in now.

$8.1. Minimal action principle; Euler-Lagrange equation;

Problems and Exercises. 1. Check the equivalence of the Euler-Lagrange equations (1.1), and the hamiltonian system (1.3). 2. (i) Demonstrate that the "squareroot-kinetic-energy" functional of a Riemannian metric {gij(z)), L[z] = ( cgij(z)2i2j)1/2, yields geodesics, as extremal curves. So the EulerLagrange equation for L is precisely the equation of geodesics (see Appendix C). The geometric meaning of this result is quite transparent: the Lagrangian density Ldt represents the arc-length element of the metric g! (ii) Show that the Legendre transform kinetic energy:,functional K = + c g i j ( z ) i i 2 j of metric g, becomes the hamiltonian H = i c g r 3 ( z ) p i p j , of the dual metric (gr3) = (gij)-l, on the cotangent space.

3. Check that the Jacobian map ];o :[f-

a hamiltonian vector field

2, = ( a ( p ; q ) ; b ( p ; q ) ) a; = apF; b = -8,F; has determinant 1, hence the hamiltonian flow of F preserves the Liouville volume d"p A d"q. 4. i) Any coordinate change, q+(q),

on configuration space Ab defines a canonical

transformation,

Ad: ( q ; P)+(d(d;=dC YPN; on the phase-space T*(Ab)(Check that A preserves the canonical 2-form dq A dp).

4

ii) Apply part (i) to transform the standard canonical (q; pkvariables in the phase-space R2x W2 to polar { ( r ;O ) } , and spherical {(r;d;fl)}-coordinates. Compute the hamiltonian H = p2 V ( q )on R2 in polar and spherical coordinates, and show

+

2

Hpolor= P,

1 2 + -P r2 + V ; Haher= p j +$p$

+ sin24b2)+ V ;

iii) Do the same for elliptical coordinates: where {rI;-P2) are distances fig.1). Show

of focal points: {( f a;O)} (see

Fig.1: Elliptical coordinates in R2 are made of confocal ellipsi: ( = r1 + -P2 = Const; and hyperbolae: q = -Pl - r2 = Const. Here z = a ch t c o d ; y = a s h t sine; and parameters ( = 2 a c h t ; q = 2acose. Elliptical coordinate change can also be viewed as a conformal (analytic) map, w =t + i b z =z+iy, given by z = $ e W + e-"').

377

378

58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction. 58.2. Noether Theorem, wnservation lam and Marsden- Weinstein reduction. Conserved integrals of hamiltonian systems are often derived from the oneparameter groups of symmetries via the celebrated Noether Theorem. In particular all basic conservation laws of Physics: energy, momentum, angular momentum, arise this way from the translational and rotational symmetries of the system. Going further in this direction, we consider systems with ‘‘large” Lie groups, or algebras of symmetries (not necessarily commuting). Such symmetries allow to reduce the system to fewer variables, via the Weinstein-Marsden reduction process.

Given a configuration space of a classical mechanical system with Lagrangian

L = L ( q ; q ) , we consider a one-parameter group of transformations

CC:(q;t)+(q*;t*) on

A x [O;co)(point transformations along with time reparametrizations),

L*

+

+ ...

= q e$(q;t) = 11,,(q;t) (2.1) = t + q q q ; t ) + = &(q;t)* In other words vector field $J = Sq and scalar field q5 = S t , represent infinitesimal 9*

...

generators of the family Cc. Obviously any group of transformations (2.1) of A generates transformations on the path-space of AL (trajectories of the system):

(40 t ) +uq*(t*); t*)= ( q + 6%t + S t ) = 11,€(Q A).

(2.2)

O

2.1. Noether Theorem: A n y one-parameter group of transformations (2.2) with

generators (11, = Sq;$ = St), that leaves invariant S = L(t;q;q)dt,gives rise to a conserved integral

I

J=p.bq-HSt=p.$J-Hq5

the

action-functional

Const

Here p = a. L, and H = p . q - P. are the canonical variables (conjugate momenta Q and the hamiltonian) of L. The proof follows from the general variational formula for a functional L (problem l), when we allow free motion of the end points as well as all reparametrizations of the timevariable: t+t* = t+6t, ‘1

6s = ( p a 6 9 - H a t ) [

to

+ J(Lq-$Li*469 ‘1

- 4 61)dt.

(2.3)

to

The first factor in the integral represent the E L equation that vanishes along any critical path. If furthermore functional S is invariant under (2.2), then

6s = 0, and we get

J ( t l ) = ( p . 6 9 - H 6 t ) I t , = J ( t , ) - constant along any critical path! QED.

Remark: Conservation of Noether integral J = p a 11, - Hq5 can be recast in the hamiltonian formulation. Indeed, function J = J(q,p; t ) , considered on the phase space

58.2. Noether Theorem and the Marsden- Weinstein reduction.

379

of variables (q; p ) clearly satisfies the equation,

J,+(J;H}=O. Thus each symmetry of the Lagrangian L produces a Poisson commuting integral (symmetry) of the corresponding hamiltonian H. So any Lie group/algebra of symmetries of the Lagrangian gets represented by the Lie group/algebra of hamiltonians. 2.2. Conservation laws. As an application of Noether's Theorem we shall derive the basic conservation laws of classical mechanics.

Energy conservation. If Lagrangian is time independent, L L ( q ; q ) ,then time shifts: t-+t+, form a translational symmetry group of L, whose generators: II, 0, #=I. Hence the Noether integral becomes hamiltonian/energy function, 0

IJ

= H ( q ;;p ) = Const].

.Momentum conservation. We assume now that the space shifts q+q+cu in certain directions u leave Lagrangian invariant. Then the generators 4 = 0; II, = u yield

J, =p . ~ , the u-component of the momentum remains constant. The standard example is a N particle system in R3 with pair interactions,

e = ;xq; - V ,

where potential V = C v(qi - q j ) . Obviously, simultaneous shifts, do not change V , hence system,

(Q1 ... QN)+(Ql+";... nNt"), 21 E R3, L. Thus we get conservation of the total momentum of the

1-1

.Angular momentum conservation. The source of the angular momentum conservation are rotational symmetries of the Lagrangian, like the central potential (Kepler) problem: V = V ( I q I ) in L = ;q2 - V . Let us assume that P. is invariant under rotations in the ij-th coordinate plane. The corresponding symmetry generator is a linear vector field

+

and the Noether integral becomes

.ij=[

-1

]:[I' 0

J i j = Piq j - P jqi

380

58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction.

- the ij-component of the angular momentum. In the N-body system with potential V = x u ( I qi - q j I ), only simultaneous rotations of ( q1...qN) by u E Sq3) leave L invariant. So the total angular momentum is conserved,

Remark. A relativistic particle Lagrangian has the form

so the corresponding symmetry group consists of Lorentz transformations S0(1;3), whose

generators include rotations in spatial directions,

-

I

-

J i j =I:[ 1 5 i;j 5 3, i.e. Lie algebra R = 4 3 ) c so(l;3), as well as relafiuislic boosts:

generators of hyperbolic rotations,

C 4 1 ; 3 ) . The corresponding Noether integrals are JOi

= PoQi + P i 9 0

2.3. The momentum map and Weinstein-Marsden reduction. Let 8 = span{F,; ...Fm)be a Lie algebra of hamiltonians on the phase space A. Then Lie group G of O acts on 311 by canonical transformations { e z p ( t Z F ) F : E (5). Any such algebra O defines a map J from JL into the dual space 0*,by evaluating hamiltonians F E O at a point z E A (“point evaluations” are clearly linear functionals on C ( A ) ! ) ,

( J , I F ) = F ( z ) ,for all F E 0. The resulting momentum map, intertwines two actions of group G: canonical action on A, and the co-adjoint action on 6*:if g = g(t) = ezp(tF), denote denote a Lie group element (of generator F E 0), and #g = e z p ( t Z F ) - the corresponding canonical transformation, then

J(dg(z))= Ad;(J(z)),for all g E G; z E A.

(2.5)

Furthermore, map J preserves the Poisson structures on JL and O*. We remind the reader that the dual space of a Lie algebra has a natural Poisson bracket (chapter 6 and f8.l), defined in terms of the Lie bracket on 0.Namely, for any pair of functions Fl(z); F z ( z ) on @*, gradients { a F .} belong to the Liealgebra itself, so one can 3

Poisson-Lie algebras

(5

often arise as symmetries of hamiltonians

H on A. The

58.2. Noether Theorem and the Marsden- Weinstein reduction.

38 1

momentum and angular momentum operators are obvious examples of the momentum map,

P, = u . p (u-fixed direction), and J = {J. = ( p A z) . 3E 3)

k -pkzj}.

= p .z 3

The former corresponds to translational symmetries (in the u-th direction)

O 21 Wm, the latter to rotational symmetries, 6 N 4 m ) . We shall see that symmetries of hamiltonians { H } play a double role. On the one hand they apply to reduce the number of variables (degrees of freedom). On the other hand some, seemingly complicated, systems could be “lifted” to larger, but “simpler” systems, typically based on Lie groups and symmetric spaces. We shall analyze a few important examples of this sort in the next section. But here we shall concentrate on the “reduction part”. Given a Lie algebra O of symmetries of hamiltonian H , we consider a joint level set of all observables { F E O} (it suffices to pick a basis Fl;F2;...Fm in 0 ) ,at a “level toE O*”,

A. = J-*(to). Subset A, is invariant under the flow of e z p ( t H ) . Furthermore, A, is invariant under the stabilizer of to,subgroup Go = {g E G:Ad,(to) = 0}, an obvious consequence of (2.5). The action of group Go on A, splits it into the union of orbits (a fiber bundle), and the reduced space we are interested in is the quotient of A. modulo Go, the orbit space5 R.

In our setup space R, c R = Jb0/G,, also has a natural symplectic structure: the Poisson bracket on A. restricted to Go-invariant functions. This Poisson bracket turned out to be non-degenerate, hence yields a symplectic structure on Ro (a Theorem due to Weinstein and Marsden). We shall illustrate the reduction procedure by 3 examples. 2.4. Examples.

Radial (spherically symmetric) hamiltonian H = H ( I q I ; I p I ) in W3 x R3, a generalization of the classical Kepler central-force problem: H = i p z V ( I q 1 ). Any such H has an Sq3)(angular momentum) symmetry, the momentum map being,

+

J : (%P)+9 x P

E43).

Here we identify 3-vectors [ = ( a ; b ; c )with antisymmetric matrices, 51t should be mentioned here that orbit-spaces are typically non-smooth manifolds with corners and edges (take, for instance, R”, modulo Sqn)). But there is always a dense open set 0,of ‘generic (maximal-D) orbits” in Q, that possesses a smooth (differentiable) structure!

'-

M -

c -c

-L b -a

-b a

;

-

2

so q x p becomes a element of 4 3 ) . We fix a joint level set

4 = { ( % P ) : P x q = Jol, of the (constant) angular momentum (Noether’s Theorem). Without loss of generality

vector J , could be taken, as (O;O;jo),where j , = I Jo I . Then vectors ( q ; p ) belong in the plane orthogonal to the (constant) angular momentum, which reflects the well-known property of central potential forces: planar motion!

A 3-D manifold 4 = { ( q ; p )E R2x W2; q A p = j,} is foliated into orbits of the stabilizer subgroup GoN Sq2)of J , E d 11 4 3 ) . Clearly, any u E Sq2)takes 21:

k;p)+(qU;pU),

Hence, the reduced dynamics in the phase-space N=W2 is given by the hamiltonian.

where j o = IJoIis the total (fixed) angular momentum. Thus we have reduced he number of variables to 1, which shows complete integrability of rotationally symmetric hamiltonians (problem 2).

Example: Integrable hamiltonians H. A commuting family of integrals { J,} maps phase-space 9 into the Lie algebra R". Here group G N U" (or R"' x Un-"') is generated by the flow {ezp(t,Z, + ...+t,Z,)} of hamiltonian fields {Z,} of {Jk};the coorbits { t } are points in w", stabilizers Gc = U", and inverse images J - ' ( f ) coincide with invariant tori in 9. So the reduced space J-'(()/T" is trivial. 0

0 Example: Left-invariant metrics on Lie groups. Tangent/cotangent spaces of Lie group G at each z E G can be obtained from a single space (Lie algebra), 6 21 T,, and its dual O* N T,*,by left translations,

q : t E T e - - + q - t E T q andpETB-+q*-'*pET;; , qEG. For the sake of presentation we shall consider the matrix group and algebra G

58.2. Noether

Theorem and the Marsden- Weinstein reduction.

383

and 0.Then q .( will be the matrix multiplication. The left-invariaot fields on G are of the form: ( ( q ) = q - 6 (( E O ) , and any left-invariant metric is uniquely determined by its restriction at {e}, a symmetric bilinear form B on 8,

(Bq-'

I q-' v); (;v E T,. L = $(B... I ...), and the *(

*

Thus we get a Lagrangian corresponding hamiltonian H = $(B-'... I ...) on phase-space T*(G), which possess a large symmetry group G of right translations {z+z.q; s;q E G } . Those clearly commute with left-invariant vector fields {( = ( ( q ) = q . ( } . The corresponding family of right-invariant hamiltonians (integrals of H ) consists of {J&; P') = (t * Q I I)'):

t E w' E q , on T*(G).

Identifying co-vector p' at point { q } with a co-vector p = q - * . p' E @* (via left shift with q - ' ) , we can write such functions J's on G x @* N T*(G)as, J&

P) =

I

I

- '& PI ) = (ad,(<) P ) = (t I Qd;;(P)).

So the momentum map becomes IJ(q;p) = ad,*(p)I

To get the reduced space we take the inverse image of p , E @*, J - ' ( P o ) = {(q;P):~dp*(p)= PO} = {(q;$-1(po):q E G}, (2.7) and note that J - ' ( p o ) projects onto a co-adjoint orbit 0 = O(po) (2-nd component in the RHS (2.7)). In fact, space J - ' ( p o ) is foliated over 0 with fibers, isomorphic to Go and its conjugates {G, = q-'Gq}. So the quotient-space J-'(po)/Go can be iden.tified with orbit 0, and the reduced hamiltonian becomes,

H = @?-'(I(),

restricted to 0.

In the next section we shall illustrate the foregoing discussion of symmetries, conserved integrals, integrability and reduction with a few classical examples, including Kepler one- and two-center gravity problem and the Euler rigid body motion.

384

58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction.

Problems and Exercises. 1. Prove the variational formula (2.7). Steps, i) Consider all possible variations of path q ( t ) , q-+q+6q, along with all reparametrizations, t+t* = t+6t. Write the variation of the functional, 6s = JL(t*;q*;$)dt* - J L ( t ; qdq ;z)dt, ii) Change new parameter t* back to the old t , and show that the 1-st integrand becomes

L(t+6t;q+6q; d(9+69))(1 + 6 i ) = L( ...;.,.;‘-)(l+ ...). d(t+6t) 1+6t The dot above any variable/function (q;6q; etc.) indicates its time derivative d. dt iii) Expand the latter in the Taylor series to the 1-st order in variations

-

6S= J L t 6 t + L q * 6 q + L 4 ( 6 4 - 4 6 i ) + L 6 i I

iv) Integrate by parts of all terms containing 64; 6t and get,

v) Observe that the off-integral terms above combine to p * 69 - Hbt, while the &factor inside the integral, after term-wise differentiation and cancellations d reduces to (Lq- ;iiL4). 4 6 4 and completes the proof.

2. Check directly that all 3 components of angular momentum J . .= q.p .- q .p.*

Poisson commute with H = !jp2

I3

3

3

+ V, for any radial V, 8

8’

{ J i j ; H ) = 0. Show that { J i j l i< satisfy the so(n) of antisymmetric matrices,

relations of generators of the Lie algebra rotation in the ij-th coordinate plane.

68.3. Classical examdes

385

38.3. Classical wmples. This section will illustrates the concepts and methods of the preceding parts (§f8.1-8.2), particularly the role of symmetries, integrability and reduction, by a few classical examples: spherical pendulum, Kepler problem, Euler rigid body and 2center gravity problem.

3.1. Spherical pendulum. The configuration space here is the sphere

S2= {s2+y2+z2 = 1) in UP; and Lagrangian: L = $(k2+y2+i2)- V ( z ) .In spherical coordinates (q5$), p. =(;

$2

+ sin24 8’) - V (cos4),

and the corresponding hamiltonian:

H = $ ( p i t sin-’4p;) t V . The system possesses a rotational symmetry about z-axis, whose generator by Noether’s Theorem gives a conserved integral (z-component of the angular momentum) p g = 0 = g - const. One could verify directly the commutation relation system is reduced to a 1-st order ODE in +variable, $($2

{H;p e } = O! As the result

the

+ V(cosq3)= E ; 8 = g t7 g2 s m (p

which is then integrated as in 58.1,

I

2 (dE4-

)

=t

- to; e - eo = gt.

J -

3.2. Kepler 2-body Problem. The motion of a 2-body system about the joint center of mass is described by the Lagrangian, 1 = Tq2- V , with potential V = -2, 191 the corresponding hamiltonian H = &p2 V . In spherical coordinates (r;4; e),

L

+

+(i2 r2(@

+ + sin2482))- v ( T ) ,

and

H = L2(Pr2 t f . -2 (p(p 2 t sin-2p;))t V(r).

(3.1)

The S0(3)-rotational symmetry of central potential V yields conservation of the angular momentum J= q x p , by Noether’s Theorem. In particular, the direction of J remains constant, hence 2-body motion is always planar (in the plane orthogonal to 15). The Kepler problem is thus reduced to a plane motion, where

68.3. Classical examples.

386

e = $(i.' + r'b')

- V ( r ) ;H = f(p:

+ r-'p;) + V ( r ) .

But this L also possesses a rotational symmetry, L 8 + c . Hence the Noether integral p g = g - const, i.e. { p e ; H } = 0. Thus the 2-D system is completely (Liouville) integrable. Moreover, two integrals reduce it to a I-st order ODE, as above :(i.'

+ $1 t V ( r )= E , =+ T=Ak 2(E2

= t; 8 = 8,

+ gt.

So we get a family of solutions r ( t ; E ' g ) depending on 2 parameters (constants of '(e t ; E , g S integration) E, g. To show complete [Liouville) integrability of the 3-D central potential problem does not require, however, the 2-D (plane) reduction. Indeed, 3 commuting integrals can be exhibited directly:

H = f p 2 iV ( r ) ;J , = J,, and J z = J i +

J i + J:.

The commutation relation: { J , ; J 2 } = 0, follows from the 4 3 ) Lie bracket relations. The reader could easily recognize J2, as Casimir (central) element of the enveloping algebra of 4 3 ) . 3.3. Eula rigid body problem. The rigid body in W3 has a density distribution {p(x)}. In the absence of external force its motion consists of the free (linear) motion of the center of mass, f = $zpdx, f = vt

+ b,

(momentum conservation), and rotations about f . We shall restrict our attention to nontrivial rotational components of the dynamics, i.e. study it in an inertial coordinate system6 with origin at the center of mass 0. Fig.2: Rigid body with 2 coordinate frames, the fated frame (dashed) and the moving frame (solid).

Two coordinate frames can be assigned to a rotating body: the fixed rest frame (of the ambient space), and moving body frame, both centered at 0 (fig.2). We shall

'i.e. system whose axis move at a constant velocity u.

387

68.3.Classical exarndes

Q

posit ion:

q = utQ

velocity:

2,

angular velocity:

w = 6u-I = ut( R

angular momentum j =

J

a =q

1

52 = u-19

vxqdp= /(wxg)xqdp

j =4J)

acceleration:

V=nxQ

=q =w x q =Ut(V)

= Lj x q

J= / V x Q = /(RxV)xVdp J=BR

+w x (w x q )

A = u;

* ( a ) = j2 x Q

+ R x R x Q.

The angular velocities { w ; R } here could be understood in two possible ways, as antisymmetric matrices (elements of Lie algebra 4 3 ) ) , or 3-vectors: w,R E W3 CY 4 3 ) . So operation Q-4 x Q, coincides with the adjoint/co-adjoint action, ado(Q), under the standard identification of l@ and ' 43),

-2

Similarly, multiplication with u E &3) ad,(Q) = u-'Qu. A symmetric matrix B = /(ad$dp

1

-2

corresponds to the adjoint action:

= /(&'I - ' Q Q ) d p

,['2;

represents the inertia temor of the body (in the fixed body-frame),

B=

-22

-2y

-2%

z2+r2

-yt

-2y

2'+y2

The inertia tensor relates angular momentum to angular velocity,

pzmi

the same way as mass/metric tensor relates momentum to velocity in the standard, Now the basic Newton's law: i p = F-force, takes on a form:

8.3.Classical examples.

388

We can write the latter as

$ j = ut(J

+ J x 0 )= ...

In the absence of external force the RHS (torque) is 0. So we get a system of equations for unknown (body frame) parameters: J(t);R(t), J -t J x R = 0;

J=BR

or

IBb -t B R x R = 01

(3.2)

To simplify equations (3.2) and write them down explicitly we introduce a special inertial body frame, where matrix/tensor B is diagonalized,

.=[I'

1,

J

the eigenvalues of B are called principal moments of inertia. If R = (wl;w2;w3),then

(3.2) turns into a system of 1-st order equations for { w j } with quadratic nonlinearities,

I

I& -k (Iy- Iz)wzw3 = 0 I y L j 2 + (I,- I , ) w 1 ~ 3= 0 ; I,w3 + ( I , - Iy)w1w2= 0

whose solutions are given in terms of Jacobi cnoidal (elliptic) functions [Erd]. Let us remark that equations (3.2) arise in the usual way from the minimalaction principle with (kinetic) Lagrangian on the phase-space T(G) or T*(G), of Lie group7 G = S0(3),

JP. I

P. = K = i

'JI

w x q I 2dp = T

R x Q I 2dp = i(BR 10)=$(B-'JI J).

(3.3)

has 2 conserved integrals: hamiltonian, H = P. = ;(B-'J I J ) ; and total angular momentum: 3" = = I BR I '. Third integral can be chosen as any (e.g. 2) component of the angular momentum. But more systematic reduction-procedure should Lagrangian

exploit the momentum map. Let us notice that P. is a special case of left-invariant Lagrangians on group S0(3), considered in example 3, the metric being given by the 71t may be somewhat confusing to see Lagrangian (3.3) without velocity (time-derivatives) terms. The reader should keep in mind, however, that R itself represents the "velocity" (tangent) vector on G. So the 1-st order E L equations (3.2) are, in fact, 2-nd order in the position-variable { ~ ( t )E G}. However, (3.2) can solved directly for O(t), whence u(t) is recovered by integration of u - % = 0.

$8.3. Classical examples

389

inertia tensor B ,

e = $tr(tBt);f E (Jj = 4 3 ) ; H = ;tr(pB-'p);

p E @*.

Here Lie algebra 6 and its dual O* are identified via the ad-invariant product

(tld= tr(tv)* The momentum-map reduces H to a co-adjoint orbit 0 = 0 ( p , ) - a sphere in 0 N W3, with the natural (invariant) symplectic structure R = sinddd A dB, and the reduced hamiltonian becomes, $(B-'p 1 p), restricted on 0.So we see once again the integrability to result from the symmetry-reduction! 3.4. Twmxxker gravity problem. We place two centers at points {-a;a} on the

z-axis. The corresponding Lagrangian is given by

It possesses an obvious rotational symmetry about z-axis, so J , = p g is an integral. The existence of another commuting integral is not so obvious. It results from a hidden symmetry of the Coulomb potential: V = L, the so called Lagrange-Laplace191 Runge-Lenz vector,

where J = p x q is the angular momentum. One can verify the commutation relation { Li;H } = 0 , and show that a system of hamiltonians {Jz; J y ; J,; L,; L,; Ly; L,) satisfy the Lie algebra brackets of 4 4 ) or so(1;3) depending on sign of y, with the angular momentum part represented by ro(3)-matrices, while the Runge-Lenz occupying the 'J3-component,

0

0

0

0

O

a

b

e (3.4)

-p

J=

aJ,

-a

*c

0

+ p J y + y J , and L = aL, + bLy -t cL, (problem 2 ) .

In particular, hamiltonians J , and L, commute. Furthermore, the Coulomb potential is the only one (among other central potentials) to possess the Runge-Lenz type symmetry (problem 1). For the two-center gravity problem we modify the definition of the Runge-Lenz

$8.3. Classical examples.

390 vector to be

9-0

= P X J - 71IQ-al-

9+a

Yzlq+al.

The new L is no more constant under the hamiltonian flow, but one has Proposition: Hamiltonians H and L satisfy the Poisson bracket relation

t3

{ J ; H } = a x ((+1

2 )

x P},

where q1 = q - a; q2 = q t a; rl,z = Iql - q21. The proof involves standard calculations with Poisson brackets

{ L ; H } = { p X L - - 7191 r1

...;$+?+...}={p 2

The first bracket {p x L.r-'}= ' '

while the second

xL.I?.+...}-{7+...;T}. 7191

' rl

1

-Q'x L r:

+ p x ( 91T rl

PZ

(3.5)

1

x9);

2 91.p tFil 2

1- r1-3 (Pq: - P .919l),

and similar relations hold for qz; rz -terms. Combining the rl-terms of (3.5) we get b [ P x (91 x 9 ) - 91 x (P x 9 ) + 91 x (P x 9J.

4

(34

Remembering that q = ql+a, and interpreting x as adjoint action of 4 3 ) , a x b CI [a; b]

= oda(b),

we get in (3.6) adqlad,(p)-~d[gl;a](~)=adaadql(p) = a x ( q 1 XP), QED.

It follows from the Proposition, that z-component of L (in the direction of vector a ) , Poisson commutes with H , { L , ; H } = 0. Also J , and L, commute, as in the onegravity-center case (check!). Thus we get a complete set of commuting integrals: H;J,;L,. In cylindrical coordinates (r; B;z ) they become

Substituting p r = i.; p , = i ; p g = r2h = g in (3.7), it is reduced to a 1-st order

from which f and 2 can be found in terms of other constants and variables,

88.3. Classical examples

1

391

i. = f ( T ; e; E , F , g)

.

i = g(r;& ...)

S=

/r’

The latter could in principle be solvecfeexactly. We shall demonstrate it in the case of the planar motion using elliptical coordinates (see problem 4 of $&I), and the Liouville’s method of separating variables. The 2-center planar problem still possesses a Runge-Lenz symmetry, L , = ( J -~ a2p2,) az(T 71 - 5). 72

+

Here we use (z;y)-coordinates in the plane with centers placed along the y-axis (see fig.3). Fig.3. A two gravity-center system with masses at { * a } .

Vectors J is orthogonal to the plane, while L lies in the plane. Passing to elliptical <=+; r +r2 rl=- fl-r2

coordinates:

where {rl;rz} are distances the from the moving point to the gravity centers, we get

The hamiltonian in elliptical coordinates takes the form

The resulting H belongs to a general class of hamiltonian systems, studied by Liouville (1849), that allow separation of variables. Liouville hamiltonians H have the kinetic and potential-energy terms of the form,

where B = C Bj, and all functions {aj; Bj;Vj} depend on a single variable qi’ It is easy

to check that (3.8) has n Poisson-commuting integrals

F 3. = 4 2 3. p32 + V 3.- H b 3’.*

(3.9)

which also commute with H (problem 3). Integrals {Fi}are not independent, since their sum.

eFj=O.

j=1

But H along with any (n-1) of {Fj} form a system of n commuting integrals, so (3.8) is completely integrable. Furthermore, fixing the energy-level: H = E , and the values of

$8.3. Classical examples.

392

conserved integrals: F j = a j , (3.9) is reduced to a system of uncoupled ODE’s, 4 2

2aj

3

+ V3. - Eb3. = a3.; j = 1;2;...

The latter could be solve explicitly (in quadratures) by writing (3.10)

One first introduces ”local time” (3.11)

then solves ODE’s (3.10) with the RHS = d r , to get solutions { q j = q j ( r ) } , and substitutes them in (3.11) to produce “physical time” 1 in terms of the “local time” r , t = j B ( q l ;...q,) ds.

So any Liouville system is integrable completely and ezactly (in quadratures)! Returning to the 2-center problem we find B = t2- q2; a l ( t ) = t2-a2; a2(q)= a2 -q2;

vl(t)= -yt;

~ ~ ( =7-y”). )

Hence system (3.10)-(3.11) becomes (3.12)

with quatric polynomials

+ + a);Q = 2(q2-a2)(Eq2- y’q -a).

R = 2(t2 - a2)(Et2 y t

So ( ( T ) , q(r)can be expressed through the Jacobi elliptical functions ([Erd]).

Problems and Exercises:

1. Show, if a “central-force” hamiltonian H = $p2+ V ( I q I ) Poisson commutes with

then f = y - const, and V =

L = p x J - f ( I q I )L, - Coulomb potentibf.’

IQI 2. Verify the Lie bracket relations (3.4) of 4 4 ) , or so(1;3) for a combined “angular momentum-Runge-Lenz algebra” { J ; L}. 3. Verify that Liouville integrals { F j } (3.9) are in involution and Poisson-commute with the hamiltonian H of (3.8).

58.4, Integrable systems related to classical Lie algebras

393

$8.4. Integrable systems related to d a h l Lie algebras. Many interesting examples of integrable hamiltonians arise as interacting n-particle systems on R, where H = $ C p: V, and potential V is made of all two body interactions: V = Cui,, u. . = u(qi - q,), or nearest neighbor interactions. The foremost those are CalogereMoser systems: u = l / z z (and other special u), and the Toda lattice: a chain of nonlinearly coupled mass-spring systems: u = ezp(qi - qi+l). Both types turned out to be closely related to some classical Lie algebrss, e.g. 4.).

+

fi:

In this section we shall develop the basic Lax-pair formalism for such systems, then outline the so called projection method following Olshanetski-Perelomov (see [Pel). The idea is to view such systems as reductions (projections) of the geodesic flow on “large” symmetry group, or symmetric spaces {X}. Although the number of variables increases, in passing from R” to 36, we gain in simplicity of the ensuing flows. While the Lax-pair formalism proves integrability of the Caloger+Moser and Toda hamiltonians, the projection method yields an effective computational procedure for conserved integrals.

N-body hamiltonians. We shall study hamiltonians of the form H = i C p ; i-V , where potential 4.1.

V=

C

l
- sum of all 2-body interactions, or

v(qj-qj);

(4.1)

V = Cv(qj- qj+1)

(44 i - nearest neighbor interactions. The former are often called Calogero-Moser systems (for special v), the latter are exemplified by the well known Toda lattice. It turns out that for special classes of 2-body potentials v(q) the resulting n-body hamiltonians (4.1) and (4.2) are completely integrable. For (4.1)these functions are

(I) v = 1 . “2’ .1

(11) v = 2. or -

sinh’aq’

2

L

o

cosh’aq’

2 (111) v = L! -or--2. *

sin2aq’

cos’aq’

(IV) v = a’p(aq) - Weierstrass p-function (V) 2, = 4-2 4- ,*q2 (VI) Toda lattice: V =

i

bie

-a(qi-qj+l)

Systems (I-VI) are closely related to certain classical Lie groups and their homogeneous spaces G / K . As we shall see they represent “projections” (via symmetry reductions) of the geodesic flow (free motion) on such spaces.

58.4. Integrable systems related to classical Lie algebras

394

The latter can be illustrated by 1-body potentials of the type (I-VI). Indeed, free motion on the Euclidian 2-plane with hamiltonian

H = g P ; 4becomes upon the radial reduction with fixed angular momentum po = g, (I) H = ; p : t $ .

2

Similarly the W 2 harmonic oscillator H = :pz g2 4- OZ?. (V) H = #ppT.4- 7

+ w2(x2+y2),reduces to

Hamiltonians of type (11)-(111) arise in the radial reduction of the free (geodesic) motion on the hyperboloid { x I x * x = x i - - x; = l}, embedded in the Minkowski 3-space with the indefinite metric: ds2 = dxf d x i - d x i , and the standard Euclidian 2-sphere

+

{ x i t X'1-i- x i = 1) respectively. Let us also remark that potentials (1-111) are special cases of the Weierstrass function (IV). 4.2. The Lax-pair formalism. Most known examples of integrable systems (both in finite and infinite-D) can be recast in the form of so called Laz pair formalism. Namely, each point in the phase-space z = ( q ; p ) is assigned a pair of linear operators (matrices):

L = L ( z ) (typically self-adjoint), and M = M ( z ) , in such a way that the Hamiltonian evolution is equivalent to the Laz equation

Iii = [L;MI = LM - MLI

(4.3)

We shall see that evolution equation (4.3) preserves eigenvalues of operator L(t), so we immediately get a family of conserved integrals: eigenvalues { &(q; p): 1 5 k 5 n}, or some functions of {Ak}, e.g. characteristic coefficients of L. Proposition 1: For any operator function M = M(t), the eigenvalues of L(t), that solves equation (4.3), remain constant {Xk(t)= X k ( 0 ) } . If M were constant, then L ( t ) = e-i'ML(0)eitM would be given by conjugating L ( 0 ) with one-parameter subgroup, generated by M , so the result would obviously hold. In general, the role of the group { e i t M } is played by the propagalor (fundamental solulion) U(l) of the ODs:

58.4. Integrable systems related to classical Lie algebras iUl= M(t ) u

{

U ( 0 )= z

395

*

Then L ( t ) = U(t)-'L(O)U(t) is still isospectral to L(0). Another argument is based on traces of powers of L. Indeed it suffices to check that all traces tr(Lm) = But derivative, -$Zm)

=

C XP- const, for m=l; ...n.

L"'-'-Jt

L j , 80

differentiating the trace of Lm, we get

t),

tr(lr"j = m t t ( ~ m - 1 and the latter is equal to

= 0,by (4.3), QED.

tr(L"'-'[L;M])

Our goal is to construct Lax pairs for the Calogero-Moser and Toda systems (IVI) and to show that the resulting integrals: eigenvdues {Xk(L)}, traces {xm= tr(Lm)}, or the characteristic coefficients {bm(L)} (coefficients of the characteristic polynomial p(X) = det(X - L), are in involution. 4.3. Construction of Lax pairs. We use the following Ansatz for matrices L and

M:

L = P -I-ix,

(4.4) where the diagonal part P = d i a g ( p l ;...p,) depends on the momentum variables, while the off-diagonal part X = (xiJ is determined by an odd function x(q) of positions only, 2 . . = X(Qi

'I

- qj).

Similarly, matrix

M=Z$Y, with the diagonal part 2 = diag(zi), made of functions " i = C'z(qi-qj), I

and the off-diagonal part Y = (yij), determined by an even function y(q), Yij = d q i - qj).

The Lax equation then takes the form

iP - x = [P;Y ]-I-i [ X ;21 + i [ X 21.

(4.5)

Writing down matrix entries of (4.5)yields a system of equations for diagonal entries ip'= 3 ix'(Xjkykj-Yjkxkj) = 2ix'xjkYjk; (we took into account the parity of functions x, y). The off-diagonal entries satisfy

Comparing the RHS of (4.6)and (4.7)with the H&il;onian

(4.6)

system of potential

58.4. Integrable systems related t o classical Lie algebras

396

(4.8)

Im:x j k ( z j - Z k ) = ~ x j m x ~ m - x m k x ~ m .

Introducing variables: [ = q j - qm; 9 = qm - q k , and remembering the form of matrix entries { z j } of 2, formulae (4.8) yield a functional-differential equation for an odd function x and and even z,

The analysis of (4.9) is outlined in problem 3 (see [Ca];[Pe]). One can show that

and function takes on one of the following expressions,

P;

a cot h(a(); a sinh-'(at) a eot(a0; asin-'(at) dn a%(aO; am(.€);a sn(a€)

The last line gives x in terms of Jacobi elliptic functions: sn, cn, dn. The corresponding potentials v = x 2 + C are then found to be those of the list (I-IV). Potentials of type (V), u = w2q2

2

+c,require q2

a slight modification of the basic Lax

construction. One takes the evolution equation

~ L = [ c ; M I ~ L~ =LL;* . Evolution (4.10) does not preserve eigenvalues of L, or traces latter can be shown to satisfy, x,(t)

(4.10)

xm = tr(Lm),

but the

= xm(0)ezp( f i w t ) . Now the integrals of motion

can be determined from a pair of auxiliary operators

N , = L+L-; N , = L-L+. T h e latter satisfy the usual Lax equation,

i N i = ";;MI, so their powers and eigenvalues provide conserved quantities. Specifically, we take a pair

of matrices

L* = L f i w Q ,

$8.4. Integrable systems related to classical Lie algebras with the above L and diagonal Q = diag(ql; with

2I.k

... 4,).

397

Then a simple identity

[Q;MI = X, =-q j l q along k , with the Lax equation: i L = [&MI, leads directly to the

modified equation (4.10).

4.4. Complete integrability of Hamiltonian (I-IV). Integrability of systems (I-VI) will be established in two different ways.

1-st argument is fairly general and simple, but it applies to the repulsive, shortrange potentials v(q). The former means that derivative v'(q) 0, so the Newton forces: F i , = -&(qi-qQj), between the i-th and j-th particles, have always the direction of

<

i.e. particles repel each other. "Short range" refers to the effective particle interaction at large distances: Ej potential v(q) decays sujjiciently fasf? at {oo}, then

qi-qj,

hamiltonian trajectories { q ( t ) ; p ( t ) } become asymptotic t o the f r e e motion (lines in the phase-space), qj(t)

-

qoj

+ pit, as t-mo.

So particles do not "effectively feel each other" at large distances.We observe that conserved integrals,

xm = tr(Lm) with

matrix

L of (4.4)represent

polynomials in

the momentum variables { p l ;. . . p n } , whose coefficients depending on { z ( q j - q j ) } i j . One can show that for any repulsive, short-range potential v , and any hamiltonian path { q ( t ) ; p ( t ) } , the distances between pairs increase, - qj(t)-,

as t+m,

hence their motion becomes asymptotically free. Therefore, large-time limits of { xm} turn asymptotically into functions of the momenta variables only. Precisely, if

dt:(Qo;Po)-(q(t);P(t)), denotes the hamiltonian evolution of H in R2n, then

cp7 - mth Newton symmetric polynomial, as b o o .

x m ( q ( t ) ; p ( t ) ) - + x m (= p)

Obviously, the infinite-time limits of

{xm]

Poisson commute. But the

Hamiltonian flow preserves Poisson brackets,

{ F 0 dt; G 0 4tl = { F ;GI 0 dt, for any pair of observables F,G on W2n. Since a commutator of two integrals

{ x j ; x k }is

also an integral, and, it follows that

x j ; x m commute

lim { x3.;xk } o +t = 0,

t-w

for all time

It is often sufficient to require

W

0

t, including t=O, which proves complete

I u(q) I dq < 00.

58.4. Integrable systems related to classical Lie algebras

398

integrability (see problem 4)! 4.2. Remark: Let us briefly discuss the scattering process for type-I hamiltonians. Scattering refers to hamiltonian dynamics with certain asymptotic behavior at large time, typically linear (free) motion:

( 9 ( t ) ; p ( t ) ) - ( q f + t P f ; P f ) , as t-fw. So the =--oo"

asymptotic state (q-;p-) is transformed into "+w" state (qt;pt), by a

canonical (symplectic) map Y, called scattering map. Ordering particles according to their relative position on R (from left to right), we can write

pt

< ... < p;;

p;

> ...> p i .

Comparing conserved integrals at both extremes, xm(pt) = x,(p-), conclude that, p; = p i ; p ; = pi-,; vectors, q; =;:9

q;

= qi-,;

...

... A

we immediately

similar relation holds for relative position

So the scattering transform amounts to reordering

particles.

Fig.4 illustrates the scattering process f o r a train of particles traveling along R asymptotically free at { fa}.At t = - 00 the first particle q1 has the highest momentum, and the last q, the lowest. They interact via pair potentials { u ( q i - q j ) ] and ezchange momenta. A s the result the left particle (qJ becomes the slowest, while the right one (q,,) gets the highest momentum, with the rest of them occupying intermediate states.

The above argument does not apply, however, to the "long-runge", or attractive, potentials (11-VI). So we shall give another (direct) argument to verify vanishing of the Poisson-bracket for eigenvalues { X k ( q ; p ) }of L,which applies to all cases. 'Ind argument. Given a pair of eigenvalues {X;p} and eigenvectors 4 = (4,; ...I$,); $ = ($I;...$,,)

of L,

zf#J= ( P + X ) I $ = Ad; L$ = p$; we differentiate (4.11) in variables q and p ,

(4.11)

58.4. Integrable systems related to classical Lie algebras

399

that results from (4.11), and the functional equation (4.9), to bring the bracket { & p } into the form

p-Ak

#j

($k~jRkj+*kdjRkj)Zjk-~ (~kajRkj+dk*jRkj)zjk. k # j

Since the expressions inside the first and second sum are antisymmetric in (jk),each sum vanishes. Thus we establish integrability of systems (I-VI).

4.5. Integration of the equation of motion. The Projection Method. We have

shown complete integrability of hamiltonians (I-VI) by exhibiting a family of Poisson commuting integrals { x k ( L ) }or {xm(L) = ktr(Lm)}. Integrals {Ak}, however, do not yield an explicit solution of the hamiltonian system. The latter will be achieved via the so called Projection method due to Olshanetsky-Perelomov [OP]; [Per]. The idea is to lift the dynamics from n degrees of freedom ( q l ; ...qn) to a larger space of dimension N = n2-1, related to some Lie group, where the motion becomes free (geodesic). Namely, we take the space 36 of n x n hermitian (complex) matrices of trace 0, and consider a free motion: z ( t )= at b, i.e. H = ip'. The reduction (projection) from 36 to W n consists of diagonalizing matrix z,z-uQu-', where Q = diag(ql;... q,,), is made of eigenvalues { q j ( z ) } , and u is unitary.In case n=2,

+

1

!q and the projection consists of radial reduction on the free is 3-dimensional, Q = motion in W3 for a fixed value of the total angular momentum,

L' = I J

I ' = p i + sin-2dpi; J = p x q.

Indeed, in spherical coordinates hamiltonian

H = x p ; +T L' ) ,turns into the type-I

two-body problem upon reduction of the center of mass coordinLteg q = f(ql - q2). Let us return to the general case. To find the evolution of { q j } and of the conjugate momenta { p3. = q 3.} we differentiate the relation, UQU-' = at b,

nQ -

+

u

-

l

-Q ~ ~ - + ~ u-l W --a

~

and rewrite the latter as 'The center of mass coordinate T = & x m j q j ( M = CrnJ-total mass) moves rectilinearly (at a constant speed), due to the momentum conservation (58.2):

& - = J-p-const. dt M

Hence, the dynamics of the n-body system can always be reduced (constrained) by the number of degrees of freedom of T

.

58.4. Integrable systems related to classical Lie algebras

400

5 = u(t)L(t)u-'(t) = a.

(4.12)

Introducing operators

P = Q, M = - k - ' U , and

L = P+i[M;Q];

(4.13)

and differentiating (4.12) once more yields the Lax equation for the pair L,M

Ji+ i[M;L]= 0. The eigenvalues of L (or its traces) are the familiar conserved integrals. But we we need to compute matrix-function Q(t) explicitly. Let us observe that the type-I Lax pair L = P iX, and M = 2 Y of the previous section (v = q-'), does satisfy (4.13). But the initial data a, b can not be chosen arbitrary in 36 to be able to project to a typeI trajectory in the q-space. Such data must be constraint by the angular momentum oper at or1O ,

+

+

J = i[z;51 = i[a;b]= iu[Q;L]u-'. Operator J can be computed for L and Q of the previous section, and is found to be a matrix with all off-diagonal entries equal to constant g and the diagonal 0,

J = g ( ( @ C ' - I ) ; t = (1;l;...1). Now we can write down the solution-curves of the equations of motion. Initial position variable b = z(0) = Q0 can be assumed (without loss of generality) diagonal (ql;...qn). Initial momentum a is taken to be

a = Lo = Po

+ ix,,

where Po denotes the initial momenta, and matrix X has entries 2. .(O)

t3

=1 . Qj - Qj'

+

(so that J = [a;b] is of the right type!). Then z(t) = Q, tLo describes the free evolution in the extended space 36, whose "eigenvalue-projection" on the "Q-space" W" gives the hamiltonian dynamics of type I. Thus we get a solution '"The angular momentum operator arises from the Noether SU(n)symmetry of the free motion in 36. Indeed, the Lagrangian L = I i I = Ir(i i*), is invariant under conjugations, z+uzu-', u E SU(n), whose infinitesimal generators ( = ((2) = ;[.;(I are identified with elements ( E.).(ts Since the dual space of 36 is identical to itself via pairing: ( p ; z ) - ( p 12) = tr(pz*), we get the = ( p I [ ( ; 2 ] )= (( I [ p ; z ] )= const, for all (. Hence, the conserved Noether integral, J = (p I((.)) .)-valued map on the phase-space 36x36 is angular momentum J = [ z ; p ] , thought of as a 4 constant. Let us remark, that SU(n) is a subgroup of yet larger symmetry-group S q N ) , N = n2-1, so J consists of 4")- components of the total angular momentum of L = I it I '!

38.4. Integrable systems related to classical Lie algebras

+-

Q ( t ) = (...qj(t)...) = "eigenvalues" of (Qo tLo) = it

401

;

(4.14)

where { q i ; p j } are initial values of { q ; p } . In other words, the initial data { q ; p } is lifted from phase-space W" x W" to a point {(Q;L): L = P i X ( q ) } in the extended space 36 x M, then the free evolution in 36 is applied: (Q tL;L), and the result is projected back to W". For n = 2 the eigenvalues of Qo tLo can be computed explicitly (problem

+

+

+

1). Systems of type II and III. The role of the flat space

36 will be played now by the

hyperbolic (symmetric) space A = SL(n;C)/SU(n).The latter can be realized by all hermitian positive-definite matrices of det = 1, via the map, z+z*z = r. The free motion on space A is generated by all G-invariant vector fields, "7.)

= z*toz,

where r = z*z, and toE 36, identified with the tangent space of A, at ro = { I } . The projection consists once again of diagonalization of r E A, r = ueaQu-l. Differentiating the latter we can recast it into the Lax form: ii; = [ M ;L], with matrices

L =p

+ i(e-'aQMe'aQ 4a - e'aQMe-'aQ); M

= -iu -1(t)c(t).

(4.15)

Solution of (4.15) satisfying natural consistency condition are given by matrix L = P + ZX,with entries x i j = a coth a(qi - q j ) . We can lift up the initial data to a pair of matrices b = e"Q (diagonal), and matrix I , which solve the equation: 2aL0 = b2lb-l b-"Ub.

+

Thence we find the entries of CU, CUjk = apjbjk

+ ia2(1-bjk)sinh-'a(qj

-qk).

4.6. Poisson structure on cuadjoint orbits; the Toda lattice. We remind the definition of Poisson bracket on Lie algebra (5 (86.3). Let F ( z ) ; H ( z be ) two observables (functions) on (5. Then {F;H}(z= ) (z I [aCaH]),or (in local coordinates)CzkCfjaiFajH, (4.16) i jk (5, relative to some basis {el;%;...} c 0,and {zi}-coordinates of z E @* relative to the dual basis {e:;G; ...}. The corresponding hamiltonian dynamics on (5 is given by where { C f j } denote structure constants on algebra

$8.4. Integrable systems related to classical Lie algebras

402

j.= adgF[z].

(4.17)

Let us remark that Poisson bracket (4.12) on 6 is highly degenerate, as all Ginvariant functions, I ( a d i ( z ) )= I(z),for all g E G, equivalently, ~ d ; ~ ( , ) [ z=] 0, for all z E 6* yields trivial flows (4.17). So such { I } form trivial integrals of (4.17), Poisson commuting with all F on @! The degeneracy of (4.12) can be resolved by restricting the dynamics from 6* to co-adjoint orbits of group G ,

0 = { a d i ( z ) : zE G} cz G,\G, where G, denotes the stabilizer subgroup of z. Clearly, stabilizer 6, coincides with the null-space of the Poisson structure:

J,((;dF) = (z I [ ( ; d F ( z ) ]=) 0, for ( E gZ;and all F . So J becomes nondegenerate on the quotient @/GZ N T,(O)-tangent space of 0 at {z}, and furnishes 0 a symplectic structure. It turns out that many interesting examples of hamiltonian systems (classical and newly discovered) arise in this way. To begin we remark that the ad-invariant (Killing) product on 6 identifies Lie algebra 6 with its dual space O* and transforms (4.17) into the Lax form, k = [ M ;21, with M = d F .

There are several ways to construct Poisson-commuting integrals on @*, and on co-orbits 0 c @*. We shall outline 2 of them. 1-st method to produce commuting hamiltonians on the Lie algebra S is to look for a larger algebra @ 3 8. We assume that 6 is decomposed into the direct sum: 8 N B @ R. Our goal is to build commuting integrals on S, starting from @-invariants.

Theorem 2: Any pair of @-invariant functions F , H E I(@), restricted on S, Poisson commute on !2. We denote two projections from 8 to 8;R by IIl;II,, and call F' = F I Q, H' = H 18 the restrictions of F , H on 8. Their dual spaces 8* = R function F on

(9

and St* = 8'.

-

For any

its gradient BF is decomposed into the sum of Q and R-components,

OF = OF, +OF,. We take point z E Q* and observe that for any Binvariant F , (2

I [B,F;B,HI) = 4. I [ 4 F ; 4 ~ 1 ) ;

since (2 I [BF;...I) = 0 . But H i s also @-invariant,hence

98.4. Integrable systems related to classical Lie algebras

403

I [azF;6J1) = -(. I [6,F; 6 , m

(2

and the latter is 0, since the commutator belongs to

[R;R]c R, while z E S* is

orthogonal to R. So the Poisson bracket {F’;H‘} = (zI [ a , F ; O , H ] ) = 0, for all F ; H in

I ( @ ) , QED.

Once again the inner product on 0 allows to recast the hamiltonian dynamics 5 = “d;11(aF)[“17 on an orbit 0 c B* into the Lax form,

L = [L;MI; where L = s;M

= alF(z).

The Toda lattice. As an application of Theorem we shall discuss now the Toda lattice. Here algebra 0 = 4n;R); B = 9- consists of lower-triangular matrices and

$3= d n ) . A @-invariant product is given by

(A;B)+tr(AB);A , B E 4n).

consists of symmetric matrices, Sym(n). We take So the dual space B* = R hamiltonian H = $tr(L2)on 0,and the evolution equation in the Lax form,

L = [ M ;L] = [8,H;L], where 8,H denotes the !&(antisymmetric) part of 8 H ( L ) = L. If

L = L+ + L-

+ D, (L- = TL+),

denotes a decomposition of L into the upper, lower and diagonal parts, then M = L + - L-.

+*;

The Toda lattice corresponds to a special choice of L. Namely, we pick a point

Lo=[

Y.

1 0

(4.18)

and take its co-orbit 0 = { L = ( u - ~ L ~ u ) ~ ~ E ,N, -, }: ~c B*, which consists of all tridiagonal matrices

L=

4 “1

a1 *

..

a,-,

The Lax “mate” of L is found to be

1

: trL =

Cbj = 0

(4.19)

$8.4. Integrable systems related to classical Lie algebras

404

The reader can verify the Poisson bracket relation for coordinate functions { a i ; b j }on 0,

(4.20)

]

Those are, indeed, the Lie bracket relations for &-matrices, representing { u j ; b j } : u p [ ’

b j * diag(0;... 1;O;...-1).

Variables { a i ; b j } are not canonical, but one can easily pass to a canonical set, via the coordinate change,

(4.21) The hamiltonian H then takes the form

(4.22) It is precisely form (4.22) that the Toda lattice was first discovered, namely as a chain/ lattice of n particles” on interactions. Functions

W with the nearest-neighbor exponential (nonlinear!)

{Ik = tr(Lk):k = 2;3;...n} make up

a complete set of commuting

integrals. The corresponding flows are given by the Lax operators,

Mk = (Lk)+- (Lk)-; “upper triangular” - “lower triangular” parts of Lk. This result was discovered by P. van Moerbeke. Another general method to cook up commuting integrals involves shifts in vector space 0 of certain G-invariants functions (polynomials). One starts with any F in the algebra of ad-invariant functions, I ( @ ) , and forms a shifted family

= F(z+Xa):all X E R}. The shifts turn out to form a complete, involutive family on 8, i.e. hamiltonians

FA,a; F p , a Poisson commute,

IF.\,

a; Fp, a}

= 0, for P ~ X ,

and any function, Poisson commuting with all

(4.23)

belongs to the algebra, generated

by them. The argument ([Per], chapter 1) exploits the structure theory of semisimple Lie

+

“Notice that coordinates { q j } are determined by (4.21) only modulo constant shift, qj-+q. c. ? Constant c can be identified with the center of mass coordinate = fCqj, that undergoes a uniform constant-speed motion, T = at b, due to momentum conservation: P = C pi = const, for hamiltonian H. The removal of the center of mass (conserved momentum) reduces the number of degrees of freedom from n to n-1.

+

$8.4. Integrable systems related to classical l i e algebras

405

algebras ($5.1-2).

The are many other modification of the shift construction, we shall skip further details, and just remark, that both methods, as well as other known constructions of commuting (integrable) hamiltonians turn out to be special cases of a general R-matriz (recursion operator) method (see [Per]; [Olv] for details). Let us mention that some classical problems (like “rigid body in an ideal fluid”) can also be brought into such framework.

$8.4. Integrable systems related t o classical Lie algebras

406

Problems and Exercises: 1. Show that the first 3 of Lax traceintegrals x1 = C p j - momentum;

{xm} of (4.19) are

+ 2 c v j k = 2H - Hamiltonian (Noether integrals), x 3 = c P! + C ( Pj - P k ) jk' x2 =

pg

i l k

2. Show that the only steady state (time independent) rational solutions of KdV are sums Of{*]

3. Solutions of Functiondaerential equation: [.(C+r]){Z(O

- z(r])l = z(Oz'(r])- z(r])z'(€)l

(4.24)

We are interested in odd solutions {z(()}, and even { z ( ( ) } . i) Show that z can not be regular at {0}, unless z(() = 0 (take hence z=O);

r]

= 0, and

z

= const,

ii) derive an asymptotic expansion of z(() at small (, +(I) a(-1 P€ ...). Hint: changing r]+-r], the equation is traniformed into,

-

+ +

4(-r]){z(O-z(r])} = z(€)z'(r]) + z(r])z'(O. Let (-q, and expand both sides in powers of small 6 = (-7.

Derive small-r] asymptotics of z(r]),

+ P);

4 7 ) a(r]-2

iii) Expand both sides of (4.24) a t a fixed ( in powers of small and show that the coefficients, r] -2: -az = -az; r] - 1: -az' = -ad go r]l

:(z-.P)z :(z-P).'

r]

(starting from v - ~ ) ,

Lz" == -ayz. ayz --pz"' 12

Here z = z(<);z = z((). The 3-rd equation (q0) yields z

= "z+ .(P+r), 2 '

constant term can always be made 0, i.e. P = -7, since z is determined up to constant, So we get z=a(4.25)

I;sl.

The last equation along with (4.25) gives a 3-rd order ODE for

+

2,

+

(all y)z' - %"' cry2 = 0.

22 6 iv) Multiply (4.26) by z-3 and integrate it to get, Z-~Z''

-

+ 6yz-' + c = 0;

From the boundary condition at {O}, z(()

a(-', it follows that c = - 2 c ~ - ~ .

v) Multiply (4.27) by z3z' and integrate once again to bring it into the form (z')2 = a - 2 2 4 - 2 / 4 2

Now the inverse function

((2)

+ A.

becomes an elliptic integral

(4.26) (4.27)

$8.4. Integrable systems related to classical Lie algebras

2) p = fb2; X = ( r 2 b 4 1 3

407

z(() = abcothbt; abctgb(;

In all other cases integral (4.28) is expressed In terms ot elliptic functions. bxplicit formulae depend on roots of the quadratic equation: w2 - 2pu2w Xu2 = 0.

+

We skip the rest of the analysis (see [Per]), and just remark that all cases the 2-body potential has the form, u ( ( ) = a2p(b() const.

+

4. Lax inkgrala of (I-IV), can be chosen either as traces of {L'"},

characteristic coefficients { P j } of L. Show that

Find

eigenvalue

{g1;g2}

of

system

(4.18)

for

n = 2,

= tr(Lm), or the

and

show

that

P = ! j ( p l + p 2 ) denotes the center of mass coordinates, that undergo a free motion: Q ( t )= Qo + P t , while the relative motion in q1,2

= Q ( t ) f q(t),where Q = !j(q1+q2);

x,

the %enter of mass frame",

9 = I/(PO+tPO),

+t2/g2.

Check that co-orbit of any tridiagonal Lo (e.g. (4.18)) under conjugations with lowertriangular subgroup N - consists of all tridiagonal { L } (4.19). Show the Poisson bracket relation (4.28). i) Split variable

z = -(zP

P-A

+~

-

a )J-(z

P-A

ii) Denote by F,; F, functions FA,a; F p , a; by

+ pa) = z1+ z2.

a,; a,

- gradients 8, ;Oz., and derive 1

= q ( z l I V,F,; aF,1) - q z 2 I [OF,; a,F,I). iii) Use G-invariance of F , , , to show ad* a q z l ) ( ~ l=) "d5F2(z2)("2)= 0. tF1;F , } ( Z )

408

$8.5. The Kepkr problem and the Hydrogen atom

58.5. The Kepler problem and the Hydrogen atom. In the last section we reexamine the classical Kepler 2-body problem and its quantum counterpart: the hydrogen atom, from the standpoint of hamiltonian dynamies, symmetry and quantization. Following Moser, we show that Kepler problem is equivalent to the geodesic flow on 3-sphere. This explains the source of the hidden (Runge-Lenz) S0(4)-symmetry, mentioned in $8.2, and also the high degeneracy of eigenvalues (energy levels) of the hydrogen atom. Then we suggest a direct approach to the hydrogen-spectrum problem by realizing it as a Laplacian on the 3-sphere. Going further along these lines one can discover even larger S q 2 ; 4)-symmetry of the Kepler problem. The resulting “Kepler manifold” then appears as a Weinstein-Marsden reduction of the minimal ceorbit of 00(2;4). The latter can be quantized according to the Dirac prescription: namely, one first canonically quantizes the extended system (ceorbit), then imposes symmetry-constraints on the quantum/ operator level. The net result is once again the exact hydrogen spectrum!

The Kepler problem in W3 describes the motion of a body about the fixed gravity center at the origin. Its hamiltonian,

H=’2-(*. ZP rt

(5.1)

has an obvious SO(3)-symmetry, whose Noether generators make up 3 components of the angular momentum J = p x q . Furthermore, we have shown in $8.3 (two-center gravity problem) that H possesses an additional (hidden) symmetry called the RungeLent vector,

L = p x J-%. In fact, the J and Gvectors combine together to form Lie algebra 4 4 ) (or 4 1 ; 3 ) ) , where J sits in the upper-diagonal 4 3 ) block, while L fills in the complementary row and column. Precisely, for any 3-vectors t ,E ~ R3 we take t and 71 components of both vectors, J ( J ) = < * J L; ( q ) = v * L . These scalar hamiltonians on the phase-space w6 = { ( q ; p ) ) satisfy the following Poisson bracket relations,

I

14thJ(71)) = J(tx 71); (40;L(71))= Yt x II); {YO;Y71)) = -2HL(t x v),

(5.2)

where H is the hamiltonian (5.1). Relations (5.2) imply that on any “energy shell” (level

58.5. The Kepler problem and the Hydrogen atom

409

surface) H = X, functions { J ( ( ) ; q y ) }form a 6-dimensional Poisson-Lie algebra (3 of one of the following types: orthogonal, pseudo-orthogonal, or Euclidian-motion, depending on the sign of A. Namely,

4 4 ) ; if X > 0; so(3;l); if X < 0; E, = @Ds0(3), if X = 0 The proofs of all Lese statements axe outlined in prc,.-m 1. Of course, both J and L commute with H. Furthermore, Newton potential V = 8 is the only one among central potentials {V = f(r)}, that makes Runge-Len2 L = p x J f ( r ) $ , a symmetry of H (problem 2).

+

In 58.3 we applied the J and Gsymmetries to establish complete integrability of both the Kepler problem and a 2-center gravity problem. Next we would like to explore further their implications the corresponding quantum problem: the hydrogen atom. This will bring us back to the subject of Quantization (§6.3), and its connections to the group representations.

5.2. Quantization of classical hamiltonian systems. A classical mechanical system is determined by its phase-space, a symplectic manifold T (e.g. cotangent bundle), with a canonical/Poisson structure j, or equivalently, canonical 2-form 0 = j - ' d x A dx. Points { x E T} describe classical states of the system, while functions {f} on Ep represent classical okervables. One particular observable h ( z ) represents the energy (hamiltonian) of the system. Any observable f defines a one-parameter flow on 9, {exptZ!}, generated by its hamiltonian vector-field Zj = j(af). The flow takes any initial state zo into the state z ( t ) at time t. Quantum system is usually described by the quantum (Hilbert) phase-space 36, whose (unit) vectors {4 E J6:ll$112 = 1) give quantum states, while operators { A } on 36 represent quantum observables. The act of observation consists in evaluating observable A at a state 4,

A+(A4I$). Often Hilbert space 36 consists of L2-functions { $ ( q ) } on the classical configuration space A. One requires J hI $ I 'dq = 1, and interprets J D 111, I 'dq, for a region D c A, as a "probability to find the quantum system in region D".There are two standard views of the quantum evolution. The Schriidinger picture considers evolution of states, generated by a one-parameter group of the quantum hamiltonian H,

410

$8.5. The Kepler problem and the Hydrogen atom

= eitR[$Ol, while the Heisenberg picture takes the corresponding evolution of observables $ O + W

Bo+B(t) = e-itHBoeifH.

(5.3)

Quantization, from a geometric standpoint, amounts to constructing a Hilbert space 36 = X ( 9 ) , and assigning quantum observables (operators) {F}, to classical observables {f} on 9, f+F. Typically, one would like to preserve all basic symmetries (conservation laws) of the classical system in the new quantum system. So the correspondence, f +F, should maintain the basic commutator (bracket) relation between observables. Precisely, given a classical hamiltonian h we denote by G its symmetry-group (of canonical transformations on T), and by (5 its Poisson-Lie symmetry-algebra, made of observables {f} on 9,that commute with h. We want to construct a representation of algebra (5 (or group G) by operators on 36, that would take all Poisson-Lie brackets on 9, into the commutator brackets of operators {A,: f E 8 } ,

{f;h}+ih[Af; A,]; f,h E (5. Plank constant h in front of the commutator (a small parameter depending on the choice of physical units) will be assumed 1. The standard example of quantization discussed in $6.1 was the Heisenberg canonical commutation relations, i.e. the Poisson-Lie algebra spanned by all position and momentum variables {qi;pj} in W’”, with brackets, {Pi;qj} = bij. Quantization of CCR led, via Stone-von Neumann Theorem, to essentially unique” quantum phase-space 36 = L2(W”), where the basic observables {qi; pj} were represented by the multiplications and differentiations, qi+Qi[$l=

qi$;

~j+Pj[$l=iaq,$; $ E L2*

(5.4)

Furthermore, we have shown that representation (5.4) can be extended from the Heisenberg algebra, 1-st degree (linear) operators in { q;p},

Wl= { C a j q j + b j ~ j + c } , to the Weyl algebra W = W(Rn), generated by all differentiations and multiplications. In particular, the 2-nd degree component

+

+

+

W2= {f = Caiq; bijqipj cjp; ...}, gave the metaplectic (oscillator) representation of the semidirect product

W,

D

Mp,.

lzPrecisely, Stone-von Neumann Theorem proves uniqueness of irreducible representations of

CCR, so the only source of non-uniqueness are passible multiplicities of {TA }.

$8.5. The Kepler problem and the Hydrogen atom

411

There are many different ways to extend the Heisenberg-CCR through a representation of W. One of them is the standard Weyl convention, which assigns each (ordered) monomial f = q i p j a symmetrized operator,

f+Af = 8QiPj

+ PjQi).

This convention maintains the Poisson brackets, so operators: A f = f ( Q ; P ) , Ah = h(Q;P) obey the relation,

[A!;Ah] = iA{f; h]’

(5.5)

The Weyl quantization rule also extends to higher degree polynomials, for instance, q2p2+ Q(Q’P’

+ QPQP + QP’Q + PQ’P + PQPQ + P’Q’).

But now it fails to maintain the Poisson bracket relation (5.5) (the Poisson bracket for operators gets replaced by a more complicated Moel bracket, which involves higher derivatives of ‘symbols” f , h to all orders (see 52.3). In fact, a general result of Griinwald-van Howe claims that there is no consistent quantization rule that would work for the entire Weyl algebra, and would extend the Heisenberg CCR for { q ; p } (see

[Chi, [GS21). Of course, any classical hamiltonian h = ip’ + V, is still consistently quantized to a Schrodinger operator, H = - 3v’ t V(q). With this in mind we turn now to the hydrogen atom. 5.3.

The hydrogen atom. The real (physical) hydrogen atom consists of single

proton in the nucleus and a single electron. The proton is about 2000 times heavier that electron, and densely packed at the center. So the “classical prototype” of the hydrogen atom” is the standard 2-body (Kepler) system, the role of the Newton’s gravitational potential being played by the electrostatic Coulomb force (both happen to be the same v=

h!).

So the quantum model of an electron orbiting nucleus13 is the Schrodinger operator l30ur model is clearly a hybrid of the quantum and classical principles, the nucleus being treated as a fixed classical point-charge, while the electron represented by a quantum +function. Yet such model makes a good first approximation to the real physical system. It accurately predicts the energy levels of hydrogen, as measure through the radiation (emiasion/absorption) spectra, explains chemical bonds, etc.

58.5. The Kepler problem and the Hydrogen atom

412

H

= -LA - 1 ; in 36 = ~ 2 ( d ) .

(5.6)

IZI

The evolution of quantum states { $ ( z ) E L 2 } is governed by the Schrodinger equation,

ili, = 11141; +(o) = +o-initial state, whose formal solution

$(t)= e ' y l l , o ] . As always we want to analyze $ ( t ) via spectral decomposition of Operator

H (6) is one of

H

(chapter 2).

the best studied "model examples" in quantum mechanics. Its

spectrum is well known to consists of an absolutely continuous part: [O;m) (of infinite

-$:

k = 1;2;3;...}, multiplicity), and a discrete sequence of negative eigenvdues {A, = accumulating to (0). In other words, space L2(W3)is decomposed into the direct sum, L2 = gc

W

gk;

of eigensubspaces { g k } , and an absolutely continuous (spectral) subspace gc.

Eigenfunctions { $ k } of

H are called bound states,

since the quantum evolution takes on

a particularly simple form for such 4,

$(t)= a time periodic motion of frequency A,.

itx

k$k;

In contrast, states

11, E gC scatter in the sense

that probability to find a state in any finite region D c Wn diminishes in time,

I,4(z;

t ) I ~ d 3 ~ + 0as, t+oo.

I

The eigenvalue problem for operator (5.6) can be solved explicitly (see any Quantum mechanics text, e.g. [Bo]; [LL]). For the sake of completness we shall briefly outline the solution. Spherical symmetry of the Coulomb potential allows one to separate variables in polar coordinates (r;d;8) in

@,

i.e. space L2(R3) is decomposed into the tensor product

L2(R+;r2dr)8 36, 36 = L2(S2),and operator

H=

-L(a2+%

where

A, =

2

r

r r

+LA 1-1. r 2 S ri

1 a2 a; + cotea8+ -

sin% 4' denotes the spherical Laplacian on S2. In chapter 4 (34.4) we examined spectral theory of

spherical Laplacian A,, and established a decomposition 36 =

ex,

W

0

into the sum of

spherical harmonics

36, = Span{Yy: -m 5 j 5 m}; dim%, = 2m+l. Each 36,

constitutes an irreducible subspace of the angular momentum algebra

58.5. The Kepler problem and the Hydrogen atom

413

4 3 ) = { J = z x iV}14, and the Laplacian (central Casimir element) becomes scalar on ’rn,

As136, = m(m+l). Accordingly, the entire quantum space L 2 ( d ) breaks into the direct sum of “7?“‘isotropic components of J“, respectively, eigenspacea of A,,

L, = {t/J(r;d,O):As[t/J] = m(m+l)$} N L2(Rt;r2dr)@36,, and the reduced operators H, = H I L, turn into an ordinary differential operators on R+,

H, = - (8: + @r

+7

A S ) - jz. 1

The eigenvalue problem H [ $ ] = E+ is reduced now to an ODE for the radial part R(r),

R”+ :R‘

m(mtl) + [2(E +f) --]R 2

= 0.

= -and r-p = $, brings it to the form

&z

Change of parameters: E-n

R”

+ $R‘ + [(; - i) - Tm ( m]t l ) r

R = 0.

(5.7)

Further simplification of (5.7) results from the substitution,

R = prne-P/’w(p). Exponential p/2 comes here from the “leading part” of ODE (5.7), a t p = OD,

R “ - i R = 0, while power

pm are due

to the homogeneous Euler-type ODE

a2 + $8 - m(m+l) (see problem 3). Then function w solves an ODE, P2

pw”+ (2m+2 - p)w’+ (n - m - 1)w = 0,

(5.8)

known as confluent hypergeometric, or generalized Caguerre equation. Solution of (5.8) is

a generalized Laguerre polynomial, L$mA1)(p), of degree n + m . In order to get a Uregular” solution of the “singular” ODE (5.8) (points p = O;OD are singular!), n must be integer and m 5 (n - 1). So the radial components of an eigenfunction ~t is ~,,(p)

= pme-P/2L$2’)(p).

For a fixed eigenvalue parameter n, the angular momentum number m takes on values m = 0,1, ...( n-1), tJ

and we have 1

= Rn,(p)Yr(q5;6’), { Y r E 36,)

+ 3 + ... + (2n-1)

= n2 “hydrogen eigenfunctions”

of eigenvalue (energy-level)

We immediately notice that the multiplicity of the n-th eigenspace, dime, = n2, is much higher than could be expected on the basis of the apparent so(3)-symmetry! Indeed, a generic rotationally symmetric hamiltonian, I4Let us remark that the angular momentum observables: J k , = P k z , - P , z k , belong to the 2-nd degree (symplectic) component W z of the Weyl algebra. Hence, the angular momentum can be consistently quantized: J k , = xkBm- Zrnak; so J = z x V = V x z.

414

$8.5. The Kepler problem and the Hydrogen atom

H = -~A+V((ZI), should have only (2k+l)-degeneracy of the "spherical-harmonic decomposition": J6= %X,. The abnormal degeneracy of specH indicates some other (hidden) 0 symmetries of the hydrogen problem. The previous discussion should prepare the reader to make a correct guess: the hidden symmetry of H is related to the "Runge-Lenr vector". However, the quantization of L is not straightforward, as L involves cubic terms:

P x J = I P I 2Q - (P.Q)P, which can not be canonically quantized in general. It can be shown, however, (the details are left to the reader) that the correct choice, the one that maintains the classical Poisson-Lie brackets (5.2), is given by the symmetrized Weyl rule

L = ;(Px J - J x P)++s where J = is x V = iV x x (problem 2).

= $(V x J - J x V) ++s,

(5.9)

If we now fix an 'energy leveln of H, eigensubspace = { $ : H $ = A$}, the commutator brackets (5.2) would yield a representation of Lie algebra15 4 4 ) in 8. But irreducible representations of 4 4 ) N 4 2 ) x su(2), are products {am8 a,: k, m = 1;2; ...} (chapters 4; l), of degree mk. In particular, 4 4 ) contains a series of representations { a k @ a a "of } degree k2, which should correspond to eigenspaces of H. So Runge-Lenz suggest a possible explanation of spectral degeneracies of H. However, to establish the precise correspondence between the eigenspaces { G k } of

H, and irreducible representations { a k @ a k }we , need to revisit the classical Kepler problem. The analysis will reveal the true nature of the S0(4)-symmetry of H. We shall see that H (more precisely, its inverse H - ' ) can be realized, as the Laplacian on the 3sphere S3! But S3 is the quotient of the orthogonal group SO(4) N SU(2) x SU(2)/{ fI}, whose representations were analyzed in ch.4-5. In particular, we know that the regular representation R on L2(S3)has multiplicity-free "spectrum" (true for any S"), made of the tensor products a k @ a k ,where { a k } are standard spin-k representations of SU(2) (see problems 7,8 of 55.1). So the Laplacian A,, has spectrum {A, = k'} of multiplicities d, = k2, and we get spec(H) at once. Thus the 'Runge-Lenz" explains the mystery of hydrogen spectral degeneracies.

To convert the hydrogen problem H into the S3-Laplacian, we shall follow a 15assuming negative energy value A = H 5 0, as otherwise one should take Lorentz algebra 4 3 ; 1). The assumption is fully justified, since all eigenvalues of H are indeed, negative.

58.5. The Kepler problem and the Hydrogen atom

415

geometric approach of J. Moser [Mos], based on the stereographic projection. 5.4. Kepler problem and the geodesic flow on

sphere

S" = {zi -t 12 I

S3.A

stereographic map takes

= 1) in R", with the north pole removed, onto the horizontal

hyperplane, @:(zo;z)+w = L

(5.10)

1-x0-

In polar coordinates {(p;4)} on R" and {(6;4)} on S" (fig.5), p = cot;; and

4+4. Fs5. Stereographic projection

@ takes a sphere punched at ihe north pole onto the plane (hyperplane) in R"+'.

Nt

Since @ is clearly a clearly a diffeomorphism (smooth 1-1 map), whose inverse @-l:W-t(zo;2); zo

=*- lw12-1

I w I 2+1'

2=

A. 1 UI I 2+1'

N

the associated canonical map @ = ( @ ; T @ f - l ) takes the natural symplectic structure (1and 2-forms) from T*(S")to T*(W"). Fig.6 Polar angle 8 on sphere S" i s taken into the polar radius p = on R".

COG

Furthermore, map @ is conformal with respect to natural Riemannian metrics on

S"

and W", induced from the ambient space

W"+'. So

its Jacobian

@'

is a conformal

matrix (scalar multiple of the orthogonal one), T@I. @'

= $1,

(5.11)

where p denotes the conformal factor. The matrix transposition in (5.11) refers, of course, to a choice of (Riemannian) metrics in tangent spaces of S" and

W".

The

58.5. The Kepler problem and the Hydrogen atom

416

conformal factor can be easily computed (problem 3), and is found to be

P = W - Iw12)ItI, at each point {(w;t)} on the tangent bundle T(R") of the hyperplane. To check conformality and compute factor p , one just write the Riemannian metric on W" and S" in polar coordinates: ds2 = dB2 t sin%d@ (on

S") + +{dp2

due to p = cot8/2.

(P + I )

t p2d&} (on R");

Then substitution of @ in the kinetic energy form (Riemannian metric) on the bundle T*(S")yields a new kinetic energy function on T*(R"), K = i ( 1 t I w I 2 ) 2 I ( I 2. Here we used a general fact: any diffeomorphisms (coordinate changes) 9 from manifold

Jb to X,

induces a canonical transformation on the phase-spaces,

@:z-y,

N

9 :T*(AJ)-+T*(N);

9 :(2; 0

N

4~ 1' 1 = ; (@(z);TA - *(Oh

(5.12)

where A in the second term of the RHS of (5.12) means the Jacobian matrix 9=' of 9 at 2". In

particular, any hamiltonian f(z;() on T*(AJ)is transformed into N

f

(Y;'I)= f 0 :

= f(@ - '(y);*@'('I)).

In the study of hamiltonian dynamics ' h can be replaced with any function F = u(K), the corresponding hamiltonian vector fields being related by a constant (on any energy level) factor, z, = U'(K)ZK.

In particular, we can take

u ( K )=

a-1 = i ( l tI

w 12)

I ( 1 - 1.

The crucial observation of Moser was to notice, that the hamiltonian flow of Ii' (the geodesic flow on the sphere!), restricted on 0-th level surface of u(K), turns into our old friend, the Kepler flow. Precisely,

u(K)/ItI = H t & where

H(ru;t) = ;I w I 2 -1. l€l'

(yo.

is the Kepler hamiltonian with the reversed position and momentum variables The interchange between {w} and {(} could be implemented by a canonical transformation (involution), fx(w;[)-+(-(;w). (5.13)

58.5. The Kepler problem and the Hydrogen atom

417

The composition of 2 maps, ao3:T*(Sn)+T*(Rn),takes the level-1 set of the kinetic energy (metric) form ((z;q):2K= I q I = 1) - a unit cosphere bundle over S", into the level set {Hfw;<) = - f} of the Kepler hamiltonian,

'

H = H,= f I ( 1

2 - h .

Here subscript 1 refers to constant in the numerator of the Newton/Coulomb potential of H. More generally, we call H, = 4 I f I and observe, using simple ")! homogeneity properties of H, with respect to symplectic dilations: (w; +Ow;&), that level sets, {Ha= E } = {HAu = X'E}.

' p,

In particular, cosphere bundles of different radii in T*(Sn), are taken by 1 = D o

St(S") = {(Z;q):2K(...) = I q I = a'}, into the level set,

{(w;<): H,= -T}.1 2u

Clearly, hamiltonians K and H are inverse one to the other, via map 1, '=H 2 K 0 1. Following [Sou], we call the union of all nonzero covectors over S", T+(S") = { ( z ; q ) : 9# 0}, the Kepler manifold. Let us summarize the foregoing discussion in the following

Thwrem 1: The stereographic map (5.10) composed with symplectic involution D (5.13), 4 = Q o @, takes the Kepler manifold, Ts(S"), onto the negative energy shell 9-= {(w;(): H I< 0) of the Kepler hamiltonian, and t r a w f o m the kinetic (metric) f o r m K = 3 I q I on T+(S") into the inverse 1 / H on T-.

'

Finally, it remains to quantize both problems and to compare their spectra. The geodesic flow of hamiltonian, K = I q I on S" (in fact, on any manifold A) yields the Laplacianl' A.

'

A straightforward approach to relate the "negative part of Schrodinger operator u H" to AS' would be to quantize the canonical map 1 = g o @ , i.e. assign a unitary O -n symmetric spaces A, like the n-sphere, the "quantized Laplacian" can be derived from symmetry considerations. Lie algebra elements must be represented by invariant vector fields on A, generators of point-transformations g:z-+z, g E G. Classically, invariant vector fields {X} on A, are in 1-1 correspondence with invariant hamiltonians { F x ( z ; ( )= X -(; E Tr)on the phase-space T * ( A ) . Quantizing algebra { F x } we get a family of 1-order differential operators { i a X } ,acting on Lz(A). But the hamiltonian K IS the sum of squares of invariant vector fields, which upon quantization yields the invariant Laplacian.

418

58.5. The Kepler problem and the Hydrogen atom

operators U:L2(S")+L2(W"), to the stereographic (symplectic) map 5,and to quantize involution u (the latter is clearly given by the Fourier transform 9:L2(W")-+L2(W")). Then one has to verify that the product W = 9 U takes L2(S") into the (negative) discrete component of H - ', and intertwines both operators,

Since, spectrum of the Laplacian A on

S3is well known (chapter 5):

Id,];

with multiplicity this would yield the result, as s p e d (its discrete part) is the inverse of specA! Another formal "derivation" would be i) to take the "stereographed" spherical Laplacian

ii) to show that the eigenvalue problem

K[$I = E2$; is reduced to a "square root problem", a ( 1t I w

1 ')[$I

= E$;

iii) to observe that the latter turns into a hydrogen eigenvalue problem: (w' -&I$ = $; with interchanged position and momentum {w;GI}.

(5.14)

Then (5.14) becomes an eigenvalue (A = 1) problem for the Schriidinger operator H E . Using the obvious scaling properties of the hydrogen hamiltonian: spec(HE) = 1 s ec(H,) E2

would yield the requisite relation between the k-th eigenvalue of H,and eigenvalue E of A,: Once one could get the discrete hydrogen spectrum from spec(Ag)! Neither of 2 direct arguments, has yet been establlished rigorously. The known methods involve an extension of the Kepler problem to a higher dimensional manifold, W4 x W4* (Kepler represents a U(1)-symmetry reduction of the latter), and the Dirac formalism for imposing constraints in quantum systems with symmetries. The Dirac's procedure [Dir] involves quantizing first the extended (large) system, then imposing "quantum constraints" as operators on the extended quantum Hilbert space 36, so that the

419

58.5. The Kepler problem and the Hydrogen atom

“physical states” become null-vectors of quantum constraints. This procedure could be implemented in many different ways (see [Simm]; [Kum]; [Soul-21; [Mla]; [GS2]; [Hurl). We shall outline an approach, due to [Simm]; [Kum] (see also [GSp]), where the Kepler problem is “conformally regularized”, embedded into a system of 4 constraint harmonic oscillators. 5.5. Couformal regdarktion and the Dtac quautktkm. We consider a system of 4

oscillators described by the hamiltonian

& = fcI ajI2 in the

phase-space C4 equipped

with the natural symplectic 2-form: w = C d a j A d 6 j;

(5.15)

and impose the constraint, b1I2

+ b 2 I 2 = 1631’ + b4I2.

(5.16)

Notice that (5.16) defines a twistor-space II, made up of all complex null-lines in C4, quotient modulo the one-parameter subgroup { e i f : a + e i f a } - a flow of h,. So the twistorspace 17 (a 6-D manifold with the canonical form (5.15)) arises from the WeinsteinMarsden reduction of the constraint oscillator system. Let us also remark that the conformal group G = Sy2;2) acts naturally on II (since G preserves the (2;2)-indefinite form (5.16)on C4, hence it takes null-lines of (5.16) into null-lines). In fact, II represents a minimal co-adjoint orbit of SY2;2), and the constraint system can be obtained by

geometrically quantizing orbit II in the sense of 06.3. 2”. Embedding. We shall embed the Kepler problem, {T*(R3\{O});w = d p A dq; h = :p2

- i}

into the constraint oscillator problem. We fix energy-levels h = -E; (E > 0) (for the

*

Kepler hamiltonian), and h, = 1 (for the oscillator). The embedding procedure (of “Kepler” on the energy shell, h, = -E to the “oscillator”), will exploit the familiar conserved integrals of h,

J = q x p - angular momentum; and L = J x p +A. I91

We remind the basic Poisson-bracket relations for { h ; J ; L } ,

{ h ; J } = {h;L} = 0; {Jj*J 9 3’} = - 6 i3k ’ ’ J kr { J , ; L3’} = -6;jkLk; {Lj;Lj} = 6 83k ’ ’ 2hJk

(5.17)

in other words h commutes with J and L, and the latter form a Lie algebra 4 4 ) (for negative energies h < O!).

Alongside (5.17) we shall use 2 other geometric relations

(problem 4):

J . L = 0; I L 1 2 - 2h I J I 2 = 1. We set p = &%

(5.18)

(constant for a fixed energy level), and define a new set of vector-

58.5. The Kepler problem and the Hydrogen atom

420 variables

z=pJ+& y=pJ-L.

Relations

(5.17-18)

show

that

+ ILI

I z I = Iy I = p2 I J I

= 1,

so

map

!F:(q;p)+(z;y), takes a negative (fxed) energy shell qE= { h ( p ; q )= -E} of the Kepler

phase-space T * ( R 3 ) into the product of %-spheres, 3:4pE+s2 x s2 = ( ( 2 ;y)}

N

CP' x CP'.

Furthermore, the symplectic structure (5.15) on the shell 9, goes into the sum of the natural (volume) forms on the spheres (problem 4), (5.19)

One could check that variables {z;y} do, indeed, Poisson-commute (problem 4). The latter could also be written in terms of the complex (stereographic) parametrization of S2:

z+x = ---1-(z+

1t1112

'

I z I - 1); similarly

w+y

= ...; z,w E C.

Then (5.20)

Thus the reduced Kepler problem yields the product of two projective lines CP', with a 1parameter family of symplectic forms { w E } , depending on the energy-level E. Next we remark that the projective space CP' gives the energy-reduced phase-space for a pair of oscillators. Indeed, restricting the hamiltonian h,, = I z1 I

+ I z, I

= 1/-

(fixed

energy), and factoring the energy-shell modulo its flow, z = (zl;zz)+eitz, we do recover space CP' with the 2-form17 (5.20).

3'. Now we turn once again to the constraint 4-oscillator (5.16) and fix its energy-shell, 4

C I z j I 2 = 1Along with 1

&E

(5.16) this relation gives the same pair of projective spaces

and 2-forms, as the above energy-reduced Kepler system, 1z1I2+

IZ,I~=

1z3l2+ I z 4 1 2 = l

&G

So 2 hamiltonian systems: Kepler (on the negative energy region), and the constraint 4oscillator produce the identical (symplectomorphic) family of reduced energy-shells. Hence, both systems are equivalent (better to say the Kepler system is embedded in the constraint-oscillator, as the latter includes the "collision Kepler trajectories", so it gives a conformal regularization of the Kepler manifold). 4'. Finally, the constraint oscillator problem can be quantized according to the Dirac 17By the same pattern projective space CP" gives (n+l) oscillators, energy-constraint oscillators,

~z,,

I t ...+ I zn12 = Const.

s8.5. The Kepler problem and the Hydrogen atom

421

prescription (see [Hurl). Namely, we first quantize free oscillators (see 56.2) and get 4 commuting operators {aj= - 8: :z +

j = 1;2;3;4} in the Fock-space of symmetric

tensors (polynomials) in 4 variables, 36 21

m

ex,, 0

36,

-

spanned by the products of the

i+ j+m+l = k} (p.2). Each of oscillators

Hermite functions {+i(z1)+j(z2)+,(z3)+,(z4):

spectrum of eigenvalue (f6.2), spec[aj] = {k +i:k = 0; 1; ...}.

{ a j } has an $integral

Following the Dirac procedure we impose constraint on the “quantum level”, i.e. consider solution of the joint eigenvalue problem,

+ ...+a, = A; + 01 = a, + a,.

a1

Ql

The latter boils down to a solution of a simple Diophantian system,

+ ...+

k, k4 = 2n-2; k1 + k, = k, k,

+

3

, -

(5.21)

Here 2n represents the n-th eigenvalue of the quantized constraint oscillator. Thus we get

k, + k, = k,

+ k4 = n - 1.

In other words, the 2n-th

eigenspace of the “constraint

oscillator problem (quantized Kepler)” is obtain from the n-th eigenspace 36,,(12) of the 2-D oscillator a l + a 2 , tensored with the n-th eigenspace 36,(34) X,,, = 36,(12)

@ 36,(34).

of a 3 + a 4 :

Spin variable n for a 2-D oscillator (a,+az) takes on all

integral and $integral values. Indeed, in 56.2 we learned that eigenspaces of the n-D oscillator coincide with irreducible representations

{T”}

of Sqn),

correspond to S q 2 ) . Thus we get the dimension of eigenspace 36,,

80

2-D oscillators

t be. n.n = n2, the

“constraint-oscillator-eigenvalue” A, = 2n, and the corresponding “Kepler eigenvalue”,

Additional results and comments: The material of 58.1 is fairly standard and could be found me-

.L1

,>n classical

mechanics [AM]; [Am]. The same applies to $3.2 (the Marsden-Weinstein reduction appeared first in [MW). Classical problem from the standpoint of Lie symmetries are treated in many sources ([Am], [Oh]). The recent book [Per] by A. Perelomov contains a comprehensive survey of the classical results, as well the recent developments in “integrable hamiltonians” (see also [FM]). This book along with the review article [OP] was our main source in 58.4. The “hydrogen quantization problem” goes to the very onset of quantum mechanics. W. Pauli (1926) and V. Fock (1935) first discovered the So(4)symmetry of the hydrogen hamiltonian on the Lie algebra level (see [LL]). J. Souriau

[Sou] and J. Moser [Ma] reviewed the classical Kepler problem, and applied the “stereographic projection” method to it. The higher S q 2 ; 4)-symmetries were analyzed in

88.5. The Kepler problem and the Hydrogen atom

422

[Simm];[Sou];[Kum];[Hur];[GSp], and more recently in [GS2]. The general review of geometric quantization and the references could be found in [Sn], [Hurl.

Problems and Exercises: 1. Verify that commutation relations (5.2) with constant E define one of 6-dimensional Lie algebras: 4 4 ) ; 4 3 ; l ) or e3= R3 D 4 3 ) , depending on sign of H. 2. Check that the "combined angular momentum"-"Runge-LenZ" quantum vectors (5.9) obey the commutation relations (5.2), and both Operators commute with H.

3. Laguerre polpomiab and the hydrogen bamilbnian. Generalized Laguerre polynomial L g ( z ) , of degree n and order a, can be be defined 88 a regular solution of the ODE, zy"

+(a+l-

z)y'

+ ny = 0.

They form an orthogonal family on Rt with weight w(z) = z"e-=, and can be generated by the Rodrigues formula, ~ g ( z= ) Const z-aeZ[za+ne-7(n).

Show that the reduced hydrogen hamiltonian, R"+:R'+[($-;)--]R

= 0.

m(m+l)

could be brought to the Laguerre form. Steps:

(5.22)

r2

1' Verify the following commutation relations for differential operators: L = 0' and M = z2a2 bza c (Euler-type);

+

+

+ + + +

+

= eAz[(a A)2 6(a A) c] = eAZ[L 2A8+ ( 6 1 ( i ) L[e (ii) M[Z....I = Z.[M + 2sza + s2+(6-1)s]

+ 60 + c,

+ A2)]

2' Take the reduced "hydrogen operator" (5.22), and apply (i) to the product R = e-r/2u(r), to get an ODE, m(mt1) )u = 0. M[u] = a%+ ($- 1)au - (++Break M into the sum of 2 Euler-type operators, M , (M,

+ ~ , ) [ r ' v ] = r8{a2+

(v -

1)8 -

r2

+ M,, and show that + s ( s t l ) -r2m ( m t l ) "1-

For the singular term (...)r-2 in the potential to cancel out index s must be equal rn, and the resulting ODE becomes a generalized Laguerre equation! 4. i) Verify relations (5.18) for { h ; J ; L } ; ii) show that the canonical 2-form on 9, c T * ( p )is taken into the form W , (5.19) by map I; iii) check that variables {zi}, and {yj} Poisson-commute, {zi;yj} = 0. Find the commutation relations among {zi}.