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Nambu mechanics and generalized poisson brackets
Volume 114A, number 2
PHYSICS LETTERS
3 February 1986
NAMBU MECHANICS AND GENERALIZED P O I S S O N BRACKETS G.W. K E N T W E L L Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2601, Australia Received 9 September 1985; accepted for publication 20 November 1985
We show that if a dynamical system has a Nambu bracket representation, it is possible to construct a noncanonical hamiltonian description, by expressing the cosymplectic form in terms of Casimir invariants. This new approach is illustrated for the free-rigid body and the relativistic motion of a charged particle in perpendicular constant electric and magnetic fields.
In recent years there has been a renewed interest in the noncanonical or generalized hamiltonian formulations of mechanics proposed by Pauli [1] and Martin [2]. The basic idea behind noncanonical mechanics is to construct a harniltonian system in which the canonically conjugate variables are replaced by noncanonical variables, which are usually the physical variables of the system. The new algebra is now a dynamical Lie algebra where the physical variables are not c-numbers, but belong to some other, possibly noncommutative, ring. The Poisson bracket associated with this new description still preserves the Lie algebra axioms, but the cosymplectic form is no longer constant and depends on the noncanonical variables. At present there are several methods that are used to construct such a noncanonical description. One method is via a direct change of representation from a canonical hamiltonian system to a noncanonical system by a change of variables [3]. Group theoretical methods have been used by Marsden and Weinstein [4] and proceeds by using the moment mapping of coadjoint group actions of the Lie group for the'configuration space for the system. In general, the generalized Poisson bracket does not produce a hamiltonian phase space in the usual sense, that is a symplectic manifold [5]. It instead produces a family of symplectic manifolds which are parametrized by a set of Casimir invariants, whose number is determined by the corank of the cosymplectic form, that is the Poisson tensor. Since some generalized hamlltonian descriptions 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
are singular, it suggests a connection with a form of mechanics proposed by Nambu [6]. Motivated by Liouville's theorem, Nambu proposed a formulation of mechanics in which the phase space may have odd as well as even dimensions. For this n-dimensional phase space, (n - 1) hamiltonians exist which are constants of motion for the system. As in conventional hamiltonian mechanics a bracket, called the Nambu bracket, can be defmed in the same manner as the Poisson bracket. Bayen and Flato [7], Cohen and Kalnay [8], and Rugged [9] have shown that Nambu mechanics is formally related to singular hamiltonian mechanics. Mukunda and Sudarshan [ 10] and Kalnay and Tascon [ 11 ] have shown that it is possible to imbed Nambu mechanics in a higher dimensional phase space with constraints. In Nambu mechanics, the phase space is spanned by an n-tuple of dynamical variables x i, i = 1,2 ..... n. Given (n - 1) hamfltonians, Casimirs or constants of motion, the equations of motion for the x i are determined by dx i dt
a ( x i , H 1 ..... Hn_ 1) a(x I .....
x n)
= ei]k.., l ~ j n 1 . . . ~ l n n _ l ,
(1)
where ell k l is a generalized Levi-Civita symbol, and ~1 - a/axfi'For the function F i = F l (X i ..... Xn) the Poisson bracket is replaced by { F 1 , . . . , F n } =- eiik...laiFla]F2...alFn .
(2) 55
Volume 114A, number 2
PHYSICS LETTERS
3 February 1986
The canonical relation {qi,Pi} = 6i] where { , } is the canonical Poisson bracket, is replaced by
{F,H}*
(X 1 ..... Xn} = 1,
Ruggeri [9] has investigated, in the finite degree of freedom case, the relationship between Nambu mechanics. From eqs. (1), (6) it follows that
and (qi, qi) = {Pi, Pi) = 0, is replaced by (xl,xl,...,x,}=o.
As an example, Nambu considers the case of a canonical triplet
{.4 ,B,C} = O(A ,B,C)/O(x I ,x2,x3),
(3)
with the evolution equation i:= {F, It 1 ,H).
12°32 = (11 - 13)6°16°3' (5)
Here m is tile angular momentum, o~ is the angular velocity and m i = liw i defines the diagonal moment of inertia tensor I. In the noncanonical, or generalized form, the equations of motion are simply [ 1,2,12] OF 0H dFdt - {F,H)* = k,] ~ Old(X) Ox k O~ .
(6)
It must be shown however that O]k, the cosymplectic form, satisfies the Lie-algebra axioms of antisymmetry and the Jacobi identity {A ,B}*
=
-{B,A}*,
(7)
{.4, {B,C}*}* + {B, {C,A)*)* + {C, {A,B}*)* = 0.
(8) As an example, the free rigid body may be expressed in noncanonical form [12] =
{a),H)*,
where H is given by H = ½m . ~ . The generalized Poisson bracket is 56
(TFX
VH).
(10)
OF OH fi'= {F'HI ..... Hn-1) =]~k O]k OX]. OXk'
(11)
where
O/k = i3,~,in (--1)nelki3 ...i n Oi3H 2 . ..OinHn_ 1 .
12)6o26o3,
13~3 = ( I 2 - I i ) w i c o 2 .
-I.
(12)
(4)
Here F = F(x 1 , x 2 , x 3 ) , and the two constants of motion are H 1 and H. For the free rigid body, there are two conserved quantities: H = ½m. co (the total kinetic energy) and H 1 = ½m2 is a Casimir invariant. Eq. (4) becomes the force free rigid body equations [6] IiCOl = (13
=
(9)
Here one of Nambu's hamiltonians, i.e. H = H I , has been selected to play the role o f H i n eq. (I 1). Eq. (12) also satisfies the Lie algebra axioms. Therefore given the Nambu formulation, it is possible via eq. (12) to obtain the generalized hamiltonian form, via eq. (6). Returning to the example of the free rigid body, eq. (12) and eq. (6) imply eq. (10), which is the generalized Poisson bracket. Thus the Nmbu bracket form provides a simple means to construct the generalized Poisson bracket, which has certain advantages over other approaches. As another example, let us consider the relativistic motion of a charged particle in mutually perpendicular constant electric and magnetic fields [ 13,14]. If H and E are in the x 1 and x 3 directions say, the equations of motion are [14]
~)2 = ~tl-Iv3'
~)3 = }tE(ct) - }tl-Iv2,
ct = LEo 3. (13)
Here ~ is a constant, whose exact form is not necessary for our purposes, while the dot denotes the derivative with respect to the proper time. Razavy and Kennedy [13] have shown how to represent eqs. (13) in Nambu form. The Casimir invariant H 1 is given by H I = u(ct -
X~vo),
(14)
where O is another constant. The hamiltonian H 2 is
H 2 = ~ [kl-lv2/(~2E 2 - 1) - XHo2].
(15)
The equations of motion are represented in Nambu form by 0H 1 0H 2
vi = ~],k eijk Or/ Ovk ,
i,],k = 2,3,4,
(16)
Volume 114A, number 2
PHYSICS LETTERS
where 04 =/act. The generalized Poisson bracket is found to be
{~,~]-* = ~
002 (17)
It is easy to show that using the above bracket, we recover the equations of motion. Eq. (17), by construction, satisfies the Lie-algebra axioms. One may argue that the constants of motion need to be found in the first place if a Nambu prescription is to be found. It is true that this is a limitation, however there exist many dynamical systems which may be integrated. Razavy and Kennedy have shown how to obtain a first integrals of various dynamical systems of the form
~7i = Xi(r/1 ..... r/n) , i = 1,2 .... n, where ~i = dr/i[dt and ×i are arbitrary functions of the dynamical variables. Constants of motion, or first integrals, such that {F, Ci}* = 0 are called Casimir functions (where F is also arbitrary) may also be obtained by group-theoretical means [12]. In any case the above approach provides a convenient and somewhat easy approach to the construction of generalized Poisson brackets. It is also possible to introduce "reduced" Nambu brackets for certain dynamical systems, such as the heavy top equations. This system only has three con-
3 February 1986
stants of motion [12], instead of the five necessary to use the approach of this paper. However by using the method of reduction [12] it is possible to partition the full Nambu bracket into three parts, by utilizing the fact that the configuration space for the heavy-top is the semi-direct product of the euclidean and rotation group. This approach can then be used to obtain the generalized Poisson bracket. The:~e results will be forthcoming, and in a sense the approach of this paper is a special ease of this general theory.
References [1] W. Pauli, Nuovo Cimento 10 (1953) 648. [2] J.L. Martin, Proc. Roy. Soc. A251 (1959) 536. [3] I. Bialynicki-Birula and Z. Iwiniski, Rep. Math. Phys. 4 (1973) 139. [4] J.E. Marsden and A. Weinstein, Physica 4D(1982) 394. [5] R.G. Littlejohn, Singular Poisson tensors, in: Mathematical methods in hydrodynamics, eds. M. Tabor and Y. Treve (AIP, New York, 1982) p. 47. [6] Y. Nambu, Phys. Rev. D7 (1973) 2405. [7] F. Bayen and M. Flato, Phys. Rev. Dll (1975) 3049. [8] I. Cohen and A.J. Kaln~y, Int. J. Theor. Phys. 12 (1975) 61. [9] G.J. Rugged, Int. J. Theor. Phys. 12 (1975) 287. [10] N. Mukunda and E.C.G. Sudatshan, Phys. Rev. D13 (1976) 2846. [11] AJ. Kalnay and R. Tascon, Phys. Rev. D17 (1978) 1552. [12] J.E. Marsden et al., Contemp. Math. 28 (1984) 55. [13] M. Razavy and FJ. Kennedy, Can. J. Phys. 52 (1974) 1532. [14] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Addison-Wesly, Reading, MA, 1959) p. 59.
57
PHYSICS LETTERS
3 February 1986
NAMBU MECHANICS AND GENERALIZED P O I S S O N BRACKETS G.W. K E N T W E L L Department of Theoretical Physics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2601, Australia Received 9 September 1985; accepted for publication 20 November 1985
We show that if a dynamical system has a Nambu bracket representation, it is possible to construct a noncanonical hamiltonian description, by expressing the cosymplectic form in terms of Casimir invariants. This new approach is illustrated for the free-rigid body and the relativistic motion of a charged particle in perpendicular constant electric and magnetic fields.
In recent years there has been a renewed interest in the noncanonical or generalized hamiltonian formulations of mechanics proposed by Pauli [1] and Martin [2]. The basic idea behind noncanonical mechanics is to construct a harniltonian system in which the canonically conjugate variables are replaced by noncanonical variables, which are usually the physical variables of the system. The new algebra is now a dynamical Lie algebra where the physical variables are not c-numbers, but belong to some other, possibly noncommutative, ring. The Poisson bracket associated with this new description still preserves the Lie algebra axioms, but the cosymplectic form is no longer constant and depends on the noncanonical variables. At present there are several methods that are used to construct such a noncanonical description. One method is via a direct change of representation from a canonical hamiltonian system to a noncanonical system by a change of variables [3]. Group theoretical methods have been used by Marsden and Weinstein [4] and proceeds by using the moment mapping of coadjoint group actions of the Lie group for the'configuration space for the system. In general, the generalized Poisson bracket does not produce a hamiltonian phase space in the usual sense, that is a symplectic manifold [5]. It instead produces a family of symplectic manifolds which are parametrized by a set of Casimir invariants, whose number is determined by the corank of the cosymplectic form, that is the Poisson tensor. Since some generalized hamlltonian descriptions 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
are singular, it suggests a connection with a form of mechanics proposed by Nambu [6]. Motivated by Liouville's theorem, Nambu proposed a formulation of mechanics in which the phase space may have odd as well as even dimensions. For this n-dimensional phase space, (n - 1) hamiltonians exist which are constants of motion for the system. As in conventional hamiltonian mechanics a bracket, called the Nambu bracket, can be defmed in the same manner as the Poisson bracket. Bayen and Flato [7], Cohen and Kalnay [8], and Rugged [9] have shown that Nambu mechanics is formally related to singular hamiltonian mechanics. Mukunda and Sudarshan [ 10] and Kalnay and Tascon [ 11 ] have shown that it is possible to imbed Nambu mechanics in a higher dimensional phase space with constraints. In Nambu mechanics, the phase space is spanned by an n-tuple of dynamical variables x i, i = 1,2 ..... n. Given (n - 1) hamfltonians, Casimirs or constants of motion, the equations of motion for the x i are determined by dx i dt
a ( x i , H 1 ..... Hn_ 1) a(x I .....
x n)
= ei]k.., l ~ j n 1 . . . ~ l n n _ l ,
(1)
where ell k l is a generalized Levi-Civita symbol, and ~1 - a/axfi'For the function F i = F l (X i ..... Xn) the Poisson bracket is replaced by { F 1 , . . . , F n } =- eiik...laiFla]F2...alFn .
(2) 55
Volume 114A, number 2
PHYSICS LETTERS
3 February 1986
The canonical relation {qi,Pi} = 6i] where { , } is the canonical Poisson bracket, is replaced by
{F,H}*
(X 1 ..... Xn} = 1,
Ruggeri [9] has investigated, in the finite degree of freedom case, the relationship between Nambu mechanics. From eqs. (1), (6) it follows that
and (qi, qi) = {Pi, Pi) = 0, is replaced by (xl,xl,...,x,}=o.
As an example, Nambu considers the case of a canonical triplet
{.4 ,B,C} = O(A ,B,C)/O(x I ,x2,x3),
(3)
with the evolution equation i:= {F, It 1 ,H).
12°32 = (11 - 13)6°16°3' (5)
Here m is tile angular momentum, o~ is the angular velocity and m i = liw i defines the diagonal moment of inertia tensor I. In the noncanonical, or generalized form, the equations of motion are simply [ 1,2,12] OF 0H dFdt - {F,H)* = k,] ~ Old(X) Ox k O~ .
(6)
It must be shown however that O]k, the cosymplectic form, satisfies the Lie-algebra axioms of antisymmetry and the Jacobi identity {A ,B}*
=
-{B,A}*,
(7)
{.4, {B,C}*}* + {B, {C,A)*)* + {C, {A,B}*)* = 0.
(8) As an example, the free rigid body may be expressed in noncanonical form [12] =
{a),H)*,
where H is given by H = ½m . ~ . The generalized Poisson bracket is 56
(TFX
VH).
(10)
OF OH fi'= {F'HI ..... Hn-1) =]~k O]k OX]. OXk'
(11)
where
O/k = i3,~,in (--1)nelki3 ...i n Oi3H 2 . ..OinHn_ 1 .
12)6o26o3,
13~3 = ( I 2 - I i ) w i c o 2 .
-I.
(12)
(4)
Here F = F(x 1 , x 2 , x 3 ) , and the two constants of motion are H 1 and H. For the free rigid body, there are two conserved quantities: H = ½m. co (the total kinetic energy) and H 1 = ½m2 is a Casimir invariant. Eq. (4) becomes the force free rigid body equations [6] IiCOl = (13
=
(9)
Here one of Nambu's hamiltonians, i.e. H = H I , has been selected to play the role o f H i n eq. (I 1). Eq. (12) also satisfies the Lie algebra axioms. Therefore given the Nambu formulation, it is possible via eq. (12) to obtain the generalized hamiltonian form, via eq. (6). Returning to the example of the free rigid body, eq. (12) and eq. (6) imply eq. (10), which is the generalized Poisson bracket. Thus the Nmbu bracket form provides a simple means to construct the generalized Poisson bracket, which has certain advantages over other approaches. As another example, let us consider the relativistic motion of a charged particle in mutually perpendicular constant electric and magnetic fields [ 13,14]. If H and E are in the x 1 and x 3 directions say, the equations of motion are [14]
~)2 = ~tl-Iv3'
~)3 = }tE(ct) - }tl-Iv2,
ct = LEo 3. (13)
Here ~ is a constant, whose exact form is not necessary for our purposes, while the dot denotes the derivative with respect to the proper time. Razavy and Kennedy [13] have shown how to represent eqs. (13) in Nambu form. The Casimir invariant H 1 is given by H I = u(ct -
X~vo),
(14)
where O is another constant. The hamiltonian H 2 is
H 2 = ~ [kl-lv2/(~2E 2 - 1) - XHo2].
(15)
The equations of motion are represented in Nambu form by 0H 1 0H 2
vi = ~],k eijk Or/ Ovk ,
i,],k = 2,3,4,
(16)
Volume 114A, number 2
PHYSICS LETTERS
where 04 =/act. The generalized Poisson bracket is found to be
{~,~]-* = ~
002 (17)
It is easy to show that using the above bracket, we recover the equations of motion. Eq. (17), by construction, satisfies the Lie-algebra axioms. One may argue that the constants of motion need to be found in the first place if a Nambu prescription is to be found. It is true that this is a limitation, however there exist many dynamical systems which may be integrated. Razavy and Kennedy have shown how to obtain a first integrals of various dynamical systems of the form
~7i = Xi(r/1 ..... r/n) , i = 1,2 .... n, where ~i = dr/i[dt and ×i are arbitrary functions of the dynamical variables. Constants of motion, or first integrals, such that {F, Ci}* = 0 are called Casimir functions (where F is also arbitrary) may also be obtained by group-theoretical means [12]. In any case the above approach provides a convenient and somewhat easy approach to the construction of generalized Poisson brackets. It is also possible to introduce "reduced" Nambu brackets for certain dynamical systems, such as the heavy top equations. This system only has three con-
3 February 1986
stants of motion [12], instead of the five necessary to use the approach of this paper. However by using the method of reduction [12] it is possible to partition the full Nambu bracket into three parts, by utilizing the fact that the configuration space for the heavy-top is the semi-direct product of the euclidean and rotation group. This approach can then be used to obtain the generalized Poisson bracket. The:~e results will be forthcoming, and in a sense the approach of this paper is a special ease of this general theory.
References [1] W. Pauli, Nuovo Cimento 10 (1953) 648. [2] J.L. Martin, Proc. Roy. Soc. A251 (1959) 536. [3] I. Bialynicki-Birula and Z. Iwiniski, Rep. Math. Phys. 4 (1973) 139. [4] J.E. Marsden and A. Weinstein, Physica 4D(1982) 394. [5] R.G. Littlejohn, Singular Poisson tensors, in: Mathematical methods in hydrodynamics, eds. M. Tabor and Y. Treve (AIP, New York, 1982) p. 47. [6] Y. Nambu, Phys. Rev. D7 (1973) 2405. [7] F. Bayen and M. Flato, Phys. Rev. Dll (1975) 3049. [8] I. Cohen and A.J. Kaln~y, Int. J. Theor. Phys. 12 (1975) 61. [9] G.J. Rugged, Int. J. Theor. Phys. 12 (1975) 287. [10] N. Mukunda and E.C.G. Sudatshan, Phys. Rev. D13 (1976) 2846. [11] AJ. Kalnay and R. Tascon, Phys. Rev. D17 (1978) 1552. [12] J.E. Marsden et al., Contemp. Math. 28 (1984) 55. [13] M. Razavy and FJ. Kennedy, Can. J. Phys. 52 (1974) 1532. [14] L.D. Landau and E.M. Lifshitz, The classical theory of fields (Addison-Wesly, Reading, MA, 1959) p. 59.
57