Hamiltonian structures for smooth vector fields

Volume 124, number 4,5

PHYSICS LETTERS A

28 September 1987

HAMILTONIAN STRUCTURES FOR SMOOTH VECTOR FIELDS * Henry D.I. ABARBANEL’ Marine Physical Laboratory, Scripps Institution ofOceanography, A-013, and Department ofPhysics, University of California, San Diego, La Jolla, CA 92093, USA

and A. ROUHI’ Department of Physics, B-019, University of California, San Diego, La Jolla, CA 92093, USA Received 1 July 1987; accepted for publication 24 July 1987 Communicated by D.D. HoIm

We give a constructive method to find a hamiltonian and Poisson brackets for any smooth vector field. The construction is local. We exhibit the hamiltonian and Poisson brackets for the damped, one dimensional, simple harmonic oscillator, wherein one can see the global problems. We suggest how one might usefully employ the construction in physical settings.

1. Introduction

,

In the course of investigating a phase space density formulation of inviscid fluid dynamics [1], we have found it necessary to adress the matter of when a given differential equation can be written in hamiltonian form. By this we mean the following: suppose we have a differential equation defined in an even dimensional space with coordinates z= (z’, z2, z2”) and that orbits Z(z, t) in the space arise as solutions to d t)=Y’~(Z(z, t)), a=l, 2,..., 2N, (1) with Z(z, 0) =z. When can we write the right-hand side of (1), the vector field Y’~(z),as

)

~H( ya(Z){za

H(z)}={za,

where { } are Poisson brackets defined in z space and H(z), the hamiltonian, is a scalar function in the space? For a physical context to this question, we refer to ref. [1] and to chapter 11 of Whittaker’s book on dynamics [21. This brief note is devoted to showing how one can construct both Poisson brackets and a hamiltonian H(z) for more or less any vector field Y(z), at least locally in z space. To us this was quite a surprise and at the very least raises a warning flag about conclusions too rapidly made about the hamiltonian nature ofaflow. Since one, especially in contemporary studies of dynamics, takes some care to distinguish between dissipative and conservative (hamiltonian) systems, we thought it worthwhile and even interesting to bring out some of the subtleties of the matter. To illustrate how far one may proceed along the path of

z”}

~

,

(2)

* Work supported by DOE Grant No. DE-ASO3-84ER1 3165, Division of Engineering and Geosciences, Office of Basic Energy Sciences; ONR Contract No. N00014-79-C-l472, Code 1122 P0; and DARPA Applied Computational Mathematics Program under Contract No. 86-227500-000. Institute for Nonlinear Science.

demonstrating a flow to be hamiltonian, we will explicitly exhibit the Poisson brackets and hamiltonian for the one dimensional damped simple harminic oscillator .

.

The construction we will give here is almost entirely found in general geometric language in section 3.3 of the book by Abraham and Marsden [3] and in more

0375-9601187/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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familiar tensor notation most of the argument is possible to infer from Whittaker [2] where he attributes the result to Lie [4] and Koenigs [5]. Neither reference provides the constructive detail given here nor the example we will discuss, so the reader may find our discussion of some value in seeing the issues. The key to understanding how we can claim a hamiltonian structure for something like the damped oscillator comes from the fact that the construction we make (or Lie or Koenigs must make) is only local. More difficult is the issue of under what circumstances one may determine globally when the vector field Y(z) is hamiltionian. As far as we can see this matter remains unresolved. Our note must turn next to the construction of H and the Poisson brackets for a general Y(z). We will give both the geometrical and tensorial descriptions. We then present our illustrative example and then conclude with a few summary comments,

{A (z),

28 September 1987

B(z) } = {Z’~,zt’ ~0A aB

(6)

~

where repeated indices are summed over. pah(={

2a,~} (7) is called the Poisson tensor. We shall assume it is nondegenerate: det P(z) ~ 0. Now our problem is to find a P~”(z)and an H(z) so that the given vector field Y(z) may be written as

ya(Z) _pa/~(Z).~~i

(8)

To begin we invert P’~(z) to give the Lagrange tensor (two-form) Wah(Z) satisfying ö~ (9) (note that Wab is the negative of the inverse matrix of Pab). This means ~-‘ac(Z)P~(Z)

~j~Wha(Z)

,

Y”(z)

(10)

2. Finding H(z) and Poisson brackets for Y(z) z2~) is some coordiWe begin then, with the2,vector field Ya(z), a= 1, 2, 2N, where z= (z’, nate system i~ithe 2N zdimensional phase space. Poisson brackets { } are, as usual [3], an operation on functions of z, with the property of ...,

...,

,

and the problem is then to determine when the righthand side of (10) is the gradient of a scalar. The Jacobi identity given in (5) as a requirement on Pth(z) translates into the demand that Wab(Z) be closed aWh(Z)aWh(Z)

awca(Z)

(11)

0

antisymmetry {A(z), B(z)} =

aza

(3)

ab

—

and linearity

We may now use this, along with the statement that Wl,aYt~is the gradient of a scalar, called H(z) here. First we have

{aA(z) +/JB(z), C(z)}

~

—

{B(z), A(z)}

+

,

=a{A(z), C(z)}+J3{B(z), C(z)}

,

~)

(12)

from (10). Then the closedness of Wah(Z) gives us the linear partial differential equation for tOah(Z) ya(Z)

Wah(Z)

(5)

so that the Lie group property of the transformations generated by the evolution equations is respected. Perhaps sacrificing some generality we will assume that {A(z), B(z)} can always be written in terms of the brackets of the coordinates {Z’~,z”} as 282

~~Wca

(4)

for arbitrary functions A(z), B(z), and C(z) and constants a and fi. The Jacobi identity is also required {A, {B, C}}+{B, {C,A}}+{C, {A,B}}=0

~C)

Yc(z)

wCb(Z) _y’(z)

Wca(Z).

(13)

If one solves this equation, the imposition of the Jacobi identity in the form of (11) is required. We may assure this, however, by introducing the covari-

Volume

124, number

PHYSICS

43

ant vector (one-form) formed by

~,b(Z)

o!,(z) from which o,~(z)

is

=-&dz) -&a(z)

This guarantees ponents of a(z) equations

a

r‘(z)-gl’“(z)

that (11) is satisfied. The 2N commay be taken to satisfy the 2N linear

a

= - Y’(Z),z,~AZ)

.

(15)

As with any “potential”, CX,is ambiguous up to the gradient of a scalar in determining m,b. Apart from this, our eq. (15) for a,(z) is equivalent to eq. (13) for w,,,(z) together with eq. (14) which connects them. Suppose we have established an a(z) and thus an m&(z). The required Poisson tensor is the negative of the matrix inverse of w,b, while H(z) is given by a line integral in z space

Htz)=H(Zo)+ j,,.

(16)

yb(~)o,,(~),

where z. is any point in phase space, and H(z) is locally independent of the path from z. to z by virtue of the construction of w,b; i.e. WbaYh is the gradient of a scalar, locally. Our problem is then to solve eq. (15) for (Y(Z). We do this by introducing the characteristics of Y(z); namely the orbits Z(z, t) indicated in eq. (1). In terms of the matrix operator

LETTERS

28 September

A

and (o!‘(zo))b is initial value data on the hypersurface S(z) = 0 in phase space. This surface is coordinatized by 2N- 1 coordinates denoted by z. here, and is where we begin the integration at t= 0. For these formulas to have meaning the vector field Y must be transverse to the surface S and not vanish along the orbits Z(zo, t). This is the more precise statement of what we mean by “any vector field” in the introduction. To determine a,(z) from which m&(z) is constructed, we must first identify the position Z on the orbits with z, then solve for the 2N- 1 variables z. and the one variable t in terms of the 2N Ps, and finally use these functions in (19). There are only 2N- 1 variables z;, since the initial value surface is given by one condition, S(z) = 0, on 2N variables. t acts as the one additional piece of information required. We will make all this quite concrete when discussing our example in the next section. Now we give the geometric setting for our problem. In the language of differential geometry [ 3,6], a hamiltonian system is characterized by a manifold (phase space), which we take to be even dimensional, a closed, nondegenerate two-form w, a scalar H (the hamiltonian), and a vector field Y(z). These are all connected by dH=i,w

.

(20)

This is precisely our earlier eq. ( 10) in abstract notation. Since d( dH) =O, we conclude from (20) that d(i,o)=Y,w-i,dw=O,

The formal solution %(Z(Zo,

r

t))abtztz,

t)) .

where Yy is the Lie derivative with respect to Y. w was assumed closed which means do = 0, so we arrive at

(18)

T#=O,

=[ev(-j

\-I

WVtzo, T)) dT,)]+t~“tzd)b,

where the time ordered

product

(22)

which in tensor notation is precisely (13). Finally, since dw =0 may be enforced locally by taking w = da. This leads to

to this is

t))

1:

(21)

(17) we have

$u,tz(z, t)) = -M:tZ(z,

1987

(19) [ ] + is introduced,

yp,(da,)=O=d(yPya), since JZ’,d=ddp, as operators. isfy eq. (23) by taking Y,o!=O,

(23) Finally,

we may sat-

(24) 283

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which in tensor form becomes (15). We have already discussed the solution to this equation.

28 September 1987

This is our Lagrange tensor. The Poisson tensor is clearly / 0 l/Q\ P(x~v)=(\1/Q 0 (33)

)~

3. An example: the damped harmonic oscillator as a hamiltonian system The damped, one dimensional, simple harmonic oscillator is described by coordinates x and v satisfying dx/dt=v, dv/dt=—x—2yv, (25) with y >0. So the vector field Y(z) is Y’(x,v)=v,

Y2(x,v)=—x—2yv.

(26)

Finally we construct the hamiltonian H(x, v) by computing fa[ v’dv’ +(x’ +2yv’)dv’]Q(x’, v’) on the path a(s) = (x’(s), y’(s)) = (sx, sy), 0~s~ 1. With the choice H(0, 0) =0 we have 2+2yxv+v2)Q(x,v) (34) H(x, v)=~(x Just as a check, note that Q = 1 for y = 0, and H= ~(x~+ v2) as expected. Finally we may check that .

~ix

1 ~H V

The Lagrange tensor has the single nonzero component Q(x, v) w(x v)—( —Q(x,v) 0

Q(x,v)\ 0

—

(27)

Q— (x+ 2yv)

~-

Q= 2yQ.

(28)

In the case at hand, phase is two dimensional, therefore w is automatically closed (i.e. the Jacobi identity is trivial). Therefore, it is simpler to solve (28) directly rather than introducing the one-form a as in eq. (14). Set x=0, v=v 0 at t=0; that is, the initial value surface is the v axis: S(x, v) = x. The orbits X( v0, and V(v0, t) are then X(v0, t)=(2v0/v)e~’sin(vt) t)

V(v0, t)=2v0e~”[cos(i’t)—(y/v) sin(vt)], in which v= (1 ~~~2)1/2. means finding Q 50 Q(x, v) =2yQ(x, v)

(29)

Solving (28) along orbits

(30)

.

With the choice Q(v0, t)=l at t=0, we find 2~’ (31) Q(X, V) =e Now use the orbits to solve for v 0(X, V) and t( X, V) to learn .

Q(x,v)=exp[~cot(~~)]

.

284

(35)

Furthermore,

which satisfies v

dv l~H ~—={v,H}=~-~—=—x—2yv.

(32)

dHldt={H,H}=0 (36) by construction So we would conclude, by construction, that the damped harmonic oscillator is a hamiltonian systern. Though our discussion has been cast in terms of non-canonical coordinates, we know from Darboux’s theorem [6] that since det w~ 0, we could find canonical coordinates q and p in which /01 w(q,p)=~1

~

(37)

and in which phase space volume is preserved under the flow. Computationally, a direct application of Darboux’s theorem would entail finding new coordinates q=q(x, v), p=p(x, v) such that {q, p}= 1 ={x, v}(~~

~ ~xavaxäi~v)~

(38)

This partial differential equation q(x, v)here, and p(x, is v).aHowever there is clearly somefor freedom we may as well take p(x, v) = v for example, then

aq 1 aq {q,p}=l={q,v}={x,v}—=—— or

ax Qax

(39)

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28 September 1987

that is

x q(x,v)=$Q(x’,v)dx’.

(40)

J=logv.

(47)

In this case, by construction, q and v would constitute a set of canonical coordinates for the damped harmonic oscillator. Using (32) in (40) we are led to the introduction of unfamiliar special functions. It is possible however to find a very simple set of canonical coordinates for the damped harmonic oscillator by the following method. From eq. (29)

In terms of s and J the equations of motion take the simple form

we are motivated to introduce

dt

s( x, v)

t=

yX~\ V cot—’ (v+ \ vx )

(41)

where we have used the symbol s to avoid confusion, since tin eq. (29) just parametrizes the characteristics of (26) and may differ from the physical value of “time” for the damped harmonic oscillator by a constant (it coincides with the physical time only when x= 0 is an initial condition). Now instead of (x, v) we uses and v as coordinates (this is certainly admissible, locally). This leads to the equations of motion, ds ~=l,

d

_____

~=~2yV_1~~.

V

(42)

Denoting by £~2(s,v) the single nonzero component of the Lagrange tensor in the (s, v) coordinates, we have that

an —

(

2yv+

)an v cot(vs)

+ v cot(Vs)

—

dJp.2(

1

5J)2~

P cot(vs)

(48) —y

Here the equations are in “usual” canonical hamiltonian form where Y’ is independent of s and ~2 is independent o~J.The hamiltonian can then be easily written as a sum of “kinetic” and “potential” terms, H(s, J) =J+ys—log[y sin(vs) —v cos(vs)]

.

(49)

One may easily check that th all dJ all

~

~j~’

~.

(50)

Clearly there is still an issue here for the damped harmonic oscillator is not a hamiltonian system. The issue is the global behavior of the solutions we have established for H(x, v) and Q(x, v). We can see these problems first by noting that the integral of iyWrQ(X,

v)[(x+2yv) dx+vdv]

(45)

Now we construct a quantity J=J(s, v) canonically conjugate to s by taking

a.i a~ {s,J}=l={s,v}—=v av ôv’

1

That, of course, is its steady progress into the sink at x= v= 0. The existence of the sink in Y(x, v) is also the clear cause of our woes. The formulation of the

As in eq. (33) we then have {s, v}=v.

~ (s, J) =

(44)

(43)

In this case we find a particularly simple solution by inspection

s, v) = 1/v.

dt

around the unit circle in the (x, v) plane does not vanish. The one-form i~ois then not exact on the whole plane, though it is closed by construction. This means H(x, v) is multiple valued as is Q(x, v). Furthermore, for y> 1, Q and H are complex. Were we to restrict ourselves to motions in the plane which exclude points on the v axis and require y < 1, all would appear well. Unfortunately, we would then fail to capture the essence of the motion in this system.

—

~

~=

(46)

problem in terms of the (s, J) coordinates also suffers from the same difficulties. The multiple-valuedness in w, which is now simply given by (37), is eliminated here but reintroduced into the vector field (eq. (48)) itself. 285

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4. Conclusions Our main conclusion in this note is that any sufficiently smooth vector field ya(Z) may locally be written as the Poisson bracket of the coordinates z” with a scalar H(z): ya (z)

= {z”,

H(z) }

(51)

,

where the Poisson brackets { } satisfy all the usual requirements. Indeed, we have given a constructive procedure, illustrated by the example of the one dimensional damped simple harmonic oscillator, for determining Poisson brackets and a hamiltonian. Locally one concludes that essentially any vector field is hamiltonian. The issue left unresolved is whether the construction yields a dH which not only is closed, which we force on the solution, but is also exact. As far as we know there is not yet an answer to our question globally. Nonetheless we may have a small useful tool here for the investigation of differential equations one suspects ofbeing hamiltonian. The usual signs ofthis would be a conserved quantity which physically qualifies as an energy and the absence of sinks or sources in Y(z). Under these conditions one may correctly suspect that the underlying differential equation is a Poisson structure; this one may find by our construction. Further, motivated by the knowl. edge of this structure, one may produce canonical coordinates in which to examine the problem at hand. The utility of identifying the underlying hamiltonian form of a given vector field lies primarily in the conserved quantities and simple expression of symmetries that we know in that setting. Finally, we return to the investigations of Lie and Koenigs nearly one hundred years ago as reported by Whittaker. There it is noted that the phase space need ,

286

28 September 1987

not be even dimensional to carry out the Poisson structure construction. This opens up the possibility of making our construction even in cases where the Poisson tensor is degenerate. The construction would proceed on the largest even dimensional subspace on which P~th(z)is not degenerate. Since degeneracy is often the signal of conserved quantities, one arena where our construction may prove useful is in establishing nonlinear stability theorems for flows described by appropriate Y(z) by the use of Arnol’d’s method [7] which makes essential use of quantities conserved under the motion.

Acknowledgement Martin Kruskal asked one of us (H.D.I.A) the question that led to this investigation. Mike Freedman, John Greene and Jerry Marsden assisted us in examining the global questions still left open, and we benefited from further conversations with Ted Frankel and Yu-min Yang.

References [1] H.D.I. Abarbanel and A. Rouhi, Phase space density representation of inviscid fluid dynamics, UCSD—INLS preprint (January 1987). [2] E.T. Whittaker, A Treatise on the analytical dynamics ofpartides and rigid bodies, 4th Ed. (Cambridge Univ. Press, Cambridge, 1961). [31 R.A. Abraham and J.E. Marsden, Foundations of mechanics, 2nd Ed. (Benjamin/Cummings, Menlo Park, 1978). [4] S. Lie, Ark. Mat. Naturll (1877) 10. [5] G. Koenings, C. R. Acad. Sci. CXXI (1895) 875. [6] VI. Arnol’d, Mathematical methods of classical mechanics (Springer, 1978). T. Ratiu and A. Weinstein, Phys. [7] D.D. Holm,Berlin, J.E. Marsden, Rep. 123(1985) 1.