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ISSN 1560-3547, Regular and Chaotic Dynamics, 2012, Vol. 17, No. 6, pp. 580­596. c Pleiades Publishing, Ltd., 2012.

An Extended Hamilton ­ Jacobi Method
Valery V. Kozlov*
V. A. Steklov Mathematical Institute Russian Academy of Sciences ul. Gubkina 8, Moscow, 119991 Russia
Received January 28, 2011; accepted July 14, 2012

Abstract--We develop a new metho d for solving Hamilton's canonical differential equations. The metho d is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton ­ Jacobi metho d. MSC2010 numbers: 70Hxx DOI: 10.1134/S1560354712060093 Keywords: generalized Lamb's equations, vortex manifolds, Clebsch potentials, Lagrange brackets

1. CLEBSCH POTENTIALS In [1] we develop a vortex method for integrating the Hamilton differential equations xk = H , yk yk = - H , xk 1 k n. (1.1)

The Hamiltonian H is a smooth function of coordinates x = (x1 ,... ,xn ), momenta y = (y1 ,... ,yn ) and time t. This method involves the search for a complete solution to the system of generalized Lamb's equations ui + t Here u(x, t) = (u1 (x, t),... ,un (x, t)) is a covector field on the configuration space M n = {x}, vj (x, t) = are the components of the vector field on M n , h(x, t) = H (x, u(x, t),t). Eqs. (1.2) are a closed nonlinear system of first-order partial differential equations with respect to the functions u1 ,... ,un . The solutions to the system (1.2) and only they generate invariant n-dimensional manifolds of the Hamilton equations t = {x, y : y1 = u1 (x, t),... ,yn = un (x, t)}, which are uniquely pro jected onto the configuration space.
*

j

uj ui - xj xi

vj = -

h , xi

1

i

n.

(1.2)

H yj

y =u(x,t)

E-mail: kozlov@pran.ru

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Recall that the complete solution of the system (1.2) is a family of solutions u(x, t, c) which also smoothly depends on n parameters c = (c1 ,... ,cn ), such that det ui = 0. cj (1.3)

The geometric theory of complete solutions of the generalized Lamb equation is developed in [1]. We consider the analytic aspect of this theory with respect to the problem of exact integration of the Hamilton differential equations. We shall seek a solution to the system (1.2) of the following form: = ui dxi = 1 d1 + ... + k dk + dS. (1.4)

Here 1 ,1 ,...,k ,k are the functions of x1 ,... ,x2k , time and parameters c1 ,... ,cn , and the matrix of the Lagrange brackets 0 [x1 ,x2 ] ... [x1 ,x2k ] [x2 ,x1 ] 0 ... [x2 ,x2k ] = .. ... .. ... .. ... ... .. ... .. ... [x2k ,x1 ] [x2k ,x2 ] ...
k

0

is nondegenerate. We recall the definition of the Lagrange bracket: [xi ,xj ] =
s=1

s s s s - . xj xi xi xj

In a certain sense, the Lagrange brackets are dual to the Poisson brackets (see, e.g., [2] for a discussion). In (1.4) there are no differentiations with respect to t and c. Formula (1.4) is equivalent to the following series of relations: S 1 k + 1 + ... + k , u1 = x1 x1 x1 ... ...... (1.5) S 1 k + 1 + ... + k , u2 k = x2k x2k x2k S S ,... ,un = . u2k+1 = x2k+1 xn In analogy with hydrodynamics we shall call the functions 1 ,1 ,...,k ,k and S Clebsch potentials. In a typical case (where the differential 1-form has a constant class) the Clebsch potentials always exist. We emphasize that the function S can depend on all coordinates x1 ,... ,xn . We also set S + h. (1.6) h= t Prop osition 1. Under the above assumptions, the function h is independent of x2k+1 ,... ,xn , and the variables (x1 ,... ,x2k ) = x satisfy the differential equation h u +x = - , t x
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(1.7)


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KOZLOV

where u= i i ,... , x1 i i . x2k

It is clear that the matrix coincides with the matrix of the curl of the covector field u : u - = x u x
T

.

Systems of the form (1.7) are often called Birkhoff systems [3, 4]. If the functions and do not explicitly depend on time, then u / t = 0 and Eq. (1.7) coincides with the ordinary Hamilton equation but represented not in canonical variables. The symplectic structure is defined by the skew-symmetric matrix , and the function h serves as a Hamiltonian. In the variables x1 ,... ,xn the (n - 2k )-dimensional vortex manifolds of the matrix of the curl of the field u are given by the equalities x1 = 1 ,... ,x2k = 2k , j = const.

The system of ordinary differential equations (1.7) is a system on the 2k -dimensional space which is obtained from configuration space by factorization with respect to vortex manifolds. This system is called in [1] a quotient system. Its construction is closely related to the multidimensional generalizations of the classical Bernoulli and Helmholtz theorems from the hydrodynamics of an ideal fluid. We also emphasize that Proposition 1 holds for each fixed value of the parameter c. Proof of Proposition 1. The matrix of the curl of the covector field (1.5) is 0 . 0 0 Further, for all j > 2k uj 2S = . t xj t Therefore, the last n - 2k equations of the Lamb system (1.2) become xj S +h t = 0, j > 2k.

It remains to verify that the first 2k equations of the system (1.2) reduce to the ordinary differential equations (1.7). 2. THE MAIN THEOREM Let b1 ,... ,bn denote the initial coordinates of the Hamiltonian system (1.1): xi (0) = bi (1 i n), x (0) = (b1 ,... ,b2k ) = b .

We recall that the Birkhoff system (1.7) also depends on n parameters c1 ,... ,cn . Let x = X (t, b ,c) be a general solution of the closed system of ordinary differential equations (1.7): X (0,b ,c) = b for all values of c = (c1 ,... ,cn ).
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Theorem 1. Let u(x, t, c) be a complete solution of the system of Lamb's equations (1.2) of the form (1.5) and let det 2S ci xj x
t

= 0,

i, j = 2k +1,... ,n.

(2.2)

If the general solution (2.1) of the system (1.7) is known, then the remaining canonical variables
2k+1

,... ,xn ,y1 ,... ,yn

in the system of Hamilton equations (1.1) are found from the relations S cj =
x =X 0

h + cj aj = S cj

2k i=1

xi c ,

k

s
j s=1

s xi

dt + aj ,
x =X

(2.3)

j = 2k +1,... ,n,

(2.4)

t=0

and y1 = u1 (x, t, c),... ,yn = un (x, t, c). Proof. By Proposition 1, S + H (x1 ,... ,xn ,u1 ,... ,un ,t) = h (x1 ,... ,x2k ,t,c1 ,...,cn ). t We first calculate S cj
·

(2.5)

(2.6)

=

S + t cj

i

2S 2S · xi = + xi cj t cj

i

2 S H . xi cj yi

(2.7)

The vertical bar denotes the substitution (2.5). On the other hand, by (2.6), 2S h + = cj cj t Further, according to formulae (1.5), 2S ui = + ij , cj cj xi where ij = c
k

i

H ui . yi cj

s
j s=1

s ,1 xi

i

2k,

ij = 0,

i > 2k.
k+1

It is clear that the additional terms ij are independent of x2 Hence, 2S h + = cj cj t

,... ,xn .

i

2 S H + cj xi yi

2k i=1

· ij xi .

(2.8)

Comparing (2.7) with (2.8), we arrive at the equality S cj
·

h = - cj

2k i=1

· xi

c

k

s
j s=1

s . xi

(2.9)

Note that this equality holds for all j = 1,... ,n.
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The derivatives x1 ,... , x2k are found parameters c1 ,... ,cn . Substituting in the of ordinary differential equations (1.7) and formula (2.3). Of course, in the expressions x

from (1.7) as functions of x1 ,... ,x2k , right-hand side of (2.9) the solutions of integrating with respect to t, we obtain for S/ cj the substitution (2.1) should a ,... ,xn = bn .

time t and the systems the required lso be made.

For t = 0 the solutions of n - 2k equations (2.3)­(2.4) will obviously be
2k+1

= b2

k+1

Consequently, by the implicit function theorem (in view of condition (2.2)) one can find from (2.3)­ (2.4) the coordinates x2k+1 ,... ,xn as functions of time and 2n parameters b1 ,... ,bn ,c1 ,... ,cn . Substitution of the coordinates in formulae (2.5) gives us parameters (2.10). As a result, we have obtained a general equations, since for t = 0 E 0 (x, y ) = det = det u (b, c) c This proves the theorem. Consider some particular cases. a) Let k = 0. Then all , are equal to zero and h is the function of time and parameters c1 ,... ,cn . By Theorem 1, the general solution of the system of canonical Hamilton equations is defined from the formulae
t

(2.10) momenta as functions of time and solution of the Hamilton differential

ui = 0. cj

S = cj
0

h dt + aj , cj S , xi

aj =

S cj

,
t=0

(2.11) (2.12)

yi = It is clear that
t

i, j = 1,... ,n.

t

h dt = cj c
0

h dt.
j 0

After the substitution
t

S=S-
0

h dt

(2.13)

the relations (2.11)­(2.12) take the form of the classical Jacobi formulae: S = aj , cj yi = S , xi i, j = 1,... ,n.

Note that the second formula of (2.11) does not change: aj = S cj .
t=0

Thus, Theorem 1 is a direct generalization of the Jacobi theorem on a complete integral. In addition, formulae (2.11)­(2.12) can be used without recourse to the gauge transformation (2.13).
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b) Let the functions 1 ,... ,k , 1 ,...,k be independent of time t and parameters c1 ,...,cn . Then the nondegeneracy condition (1.3) of the complete solution (1.5) takes the usual form: det 2S xi c = 0.
j

(2.14)

Eqs. (1.7) become an ordinary Hamiltonian system relative to the symplectic structure with matrix . The general solution of the canonical Hamiltonian system (1.1) is defined by the equalities
t

S = cj
0

h cj

dt +
x =X

S cj

,
t=0

1

j

n,

(2.15)

and (2.5), in which the components of the covector field u(x, t, c) are expressed by (1.5). In view of inequality (2.14) the system of equations (2.15) can be locally solved for all coordinates x1 ,... ,xn . Notice that the first 2k of them have already been found as solutions of the Hamiltonian system (1.7). c) Now let k = 1. The initial Hamiltonian system (1.2) is autonomous and the complete solution (1.5) is also independent of t. In this case, the auxiliary Hamiltonian system (1.7) is easily integrated by quadratures, and after that the solutions of the Hamilton equations (1.2) can be found from the algebraic relations (2.3)­(2.5). We make another remark. Assume that we have found a solution of the system of Lamb's equations which depends not on all parameters c1 ,... ,cn but only on a part of them. More precisely, suppose there is the solution u(x1 ,... ,xn ,t,c
2k+1

,... ,cn ),

such that the condition (2.2) is satisfied. It is easy to see that the relations (2.3)­(2.4) will be satisfied as before. The number of these relations is equal to n - 2k . The first 2k coordinates x1 ,... ,x2k are found as solutions of a closed system of ordinary differential equations. And from Eqs. (2.3)­(2.4) in view of inequality (2.2) the remaining n - 2k coordinates x2k+1 ,... ,xn are found as functions of time depending on n +(n - 2k ) = 2(n - k ) parameters b1 ,... ,bn ,c
2k+1

,... ,cn .

(2.16)

The relations (2.5) (along with (1.5)) give us momenta as functions of time and the same 2(n - k ) parameters (2.16). Thus, under the above assumption we find a 2(n - k )-parameter solution to the initial system of Hamilton differential equations. For k = 0 we obtain a general solution. However, in some cases it can be shown that the existence of this 2(n - k )-parameter family completely solves the problem of integrating the system of canonical Hamilton equations. 3. THE EULER TOP We demonstrate the method of exact integration using the Euler top as an example. We can assume without loss of generality that the constant angular momentum of the rigid body is vertical. Let c ( R) denote the value of this angular momentum. Let us choose the three Euler angles , , as generalized coordinates. According to [1, Chapter III, § 2], the equations of motion of a top admit a family of stationary invariant manifolds, and the form (1.4) is: = cd + c cos d. Hence, S = c , = c cos , = . The angles and serve as the first 2k = 2 coordinates. Their Lagrange bracket is easily calculated: [, ] = c sin , so that the skew-symmetric 2 â 2-matrix is nondegenerate if c = 0.
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(3.2)


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KOZLOV

Due to stationarity the Bernoulli function h coincides with h, which is a restriction of the kinetic energy of the rigid body to an invariant manifold. It is straightforward to verify that h= c2 2 sin2 sin2 sin2 cos2 cos2 + + . I1 I2 I3 (3.3)

Here I1 , I2 , I3 are the principal moments of inertia of the rigid body. Using (3.2) and (3.3), we obtain the system of differential equations (1.7): =c 1 1 - I1 I2 sin sin cos , (3.4)

= c cos

sin2 cos2 1 - - . I3 I1 I2

Since this system is autonomous and admits the nonconstant integral (3.3), it can be integrated by quadratures. The relation (2.9) S c
·

=

h - (c cos ) c c

(3.5)

in view of (3.3) and (3.4) defines the precession angle . After elementary transformations we obtain the equation =c sin2 cos2 + . I1 I2 2S = 1. c Hence, the condition (2.2) of Theorem 1 is obviously satisfied. Although the solution (3.1) contains only one parameter (rather than three), we have actually obtained a general solution of the equations of free rotation of a rigid body with a fixed point, since the choice of the direction of the angular momentum is of no importance. 4. THE LARMOR MOTION As an illustration, we also consider the Marmor problem of the motion of a charged particle in a constant magnetic field. Its motion is governed by the Newton equation x = x â H, ¨ x E3. (4.1) (3.6)

Eqs. (3.4) and (3.6) are, of course, well known in the theory of the Euler top [2]. We also note that

For simplicity of notation, we have set the mass and charge of the particle and the velocity of light equal to 1. In any case, these parameters can be "included" in the magnetic field H = (H1 ,H2 ,H3 ). Eqs. (4.1) admit an obvious family of stationary three-dimensional invariant manifolds x1 = H3 x2 - H2 x3 + c1 , x2 = H1 x3 - H3 x1 + c2 , x3 = H2 x1 - H1 x2 + c3 . (4.2)

In order to apply the method developed above, these manifolds should be represented in canonical variables. Eq. (4.1) is obviously equivalent to the Lagrange equation with Lagrangian L= 1 12 ( 1 + x2 + x2 )+ x (H2 x3 - H3 x1 ) 1 2 3 x 2 2 +(H3 x1 - H1 x3 ) 2 +(H1 x2 - H2 x1 ) 3 . x x
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Knowing the Lagrangian, one can calculate the generalized momenta yi = L/ xi and represent the invariant manifolds (4.2) in the canonical variables: 1 1 (H3 x2 - H2 x3 )+ c1 , y2 = (H1 x3 - H3 x1 )+ c2 , 2 2 (4.3) 1 y3 = (H2 x1 - H1 x2 )+ c3 . 2 The nondegeneracy condition (1.3) is here, of course, satisfied. Can formulae (4.3) be written in the form of (1.5)? In the general case, this is difficult. However, one can avoid this difficulty by assuming, without loss of generality, that the magnetic field is directed, say along the axis x3 . We note that the special choice of variables is also a standard technique in the classical Hamilton­Jacobi method. Thus, let y1 = H1 = H2 = 0, H3 = H (4.4) be the components of the magnetic field. Then formulae (4.3) H H x2 + c1 , y2 = - x1 + c2 , 2 2 already admit the representation (1.5) if one sets y1 = S = c1 x1 + c2 x2 + c3 x3 , = x2 + x2 1 2 H, 2 y3 = c
3

= arctg

x1 . x2

Consequently, we have the particular case b) from Section 2. The Lagrange bracket [x1 ,x2 ] is equal to H . Hence, for H = 0 the matrix is nondegenerate. The Bernoulli function is easily calculated : 1 c2 1 (Hx2 + c1 )2 + (-Hx1 + c2 )2 + 3 . 2 2 2 2 /2. The auxiliary Hamiltonian system (1.7) b ecomes c3 h= x1 = Hx2 + c1 , x2 = -Hx1 + c2 . (4.5)

Obviously, h

It coincides with the system of the two first equations (4.2) after substituting (4.4). The lines of the level set of the first integral (4.5) give the Larmor circles of radius 2h - c2 /H. 3 From (2.15) and in view of (4.5) we obtain a simple law of change of the third coordinate: x3 = c3 t + x3 (0). 5. NONCOMMUTATIVE INTEGRATION The preceding discussion has been concerned with n-parameter families of solutions to Lamb's equations and is based on the theory of n-dimensional invariant manifolds of the Hamilton equations which are uniquely pro jected onto the configuration space. However, other extensions of the Hamilton­Jacobi method in which lower-dimensional invariant manifolds appear are also possible. The basic principles of the theory of such invariant manifolds are given in [5]. The solutions of the corresponding generalized Lamb equations will depend on a larger number of parameters. Thus, suppose that the Hamiltonian system (1.1) with n degrees of freedom admits m n integrals F1 (x, y , t),... ,Fm (x, y , t), (5.1) which are almost everywhere independent. Let 2q be the rank of the skew-symmetric matrix of their Poisson brackets {Fi ,Fj } .
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(5.2)


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KOZLOV

We consider a domain of the phase space where this rank takes the maximum value. The numbers n, m and q are related by the inequality n+q By the classical Poisson theorem, all brackets {Fi ,Fj } (5.4) m. (5.3)

will also be the first integrals of the system (1.1). By virtue of this remark it is natural to consider the closed sets of first integrals (5.1), where their pairwise Poisson brackets (5.4) are functionally expressed in terms of F1 ,...,Fm . According to E. Cartan, any set of first integrals can be supplemented up to a closed one; the necessary calculations require only differentiations [6]. We now consider the integral invariant manifolds 2n-m of the Hamilton differential equations t of dimension 2n - m n, which are defined by the following system of algebraic equations: F1 (x, y , t) = c1 ,...,Fm (x, y , t) = cm . Assume that (F1 ,... ,Fm ) = 0. (y1 ,...,yn ,x1 ,... ,xk ) Here k =m-n 0. (5.6) Then from (5.5) one can find (at least locally) the momenta of the system and the first k generalized coordinates as functions of the remaining generalized coordinates, time and the parameters c = (c1 ,... ,cm ): y1 = u1 (xk+1 ,... ,xn ,t,c), x1 = un+1 (xk+1 ,... ,xn ,t,c), ..., ..., yn = un (xk+1 ,... ,xn ,t,c), xk = un+k (xk+1 ,...,xn ,t,c).
2 n-m t

(5.5)

(5.7)

The variables xk+1 ,... ,xn serve as local coordinates on Let us restrict the fundamental 1-form
n

.

yj dx
j =1

j

to the invariant manifold 2 t

n-m n+1

: + ... + uk du
n+k

= u1 du

+ uk

+1

dx

k+1

+ ... + un dxn .

(5.8)

It is well known that if m = n and the functions (5.1) are pairwise in involution, then = dS. The function S (x, t, c) will be a complete integral of the Hamilton­Jacobi equation (see, e.g., [2]). Our immediate goal is to generalize this classical result to the case m > n. The differential 1-form (5.8) is = Uk
+1

dx

k+1

+ ... + Un dxn .

(5.9)
m

Its coefficients are smooth functions of xk+1 ,... ,xn ; they also depend on t and c1 ,... ,c parameters. These coefficients are calculated from the following formulae: Uk
+1

as

= u1

un+1 un+k + ... + uk + uk xk+1 xk+1

+1

, (5.10)

...... ... un+1 un+k + ... + uk + un . Un = u1 xn xn
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Let 2s be the rank of the exterior 2-form d . In other words, 2s is the rank of the skew-symmetric matrix Ui Uj - , k +1 i, j n. xj xi Our goal is to evaluate the rank of this matrix. Prop osition 2. The fol lowing equality holds s = n - m + q. According to inequality (5.3), s Corollary 1. If n + q = m, (5.12) then k = q and the differential form is also closed: local ly = dS , where S is a smooth function of xk+1 ,... ,xn , t and c1 ,... ,cm . Corollary 2. If m = n, then = dS (x, t, c). In particular, under this condition the functions (5.1) are pairwise in involution. Equality (5.12) is known as the condition of noncommutative integrability of the Hamiltonian system (1.1). Discussions of this range of problems and the appropriate references can be found, e.g., in [7]. Proof of Proposition 2. Since the set of functions (5.1) is closed, by the Lie­Cartan theorem [6] one can find m functions 1 ,..., m , which functionally depend only on F1 ,... ,Fm , such that {1 , 2 } = ... = {2
q -1

(5.11)

0.

, 2q } = 1,

(5.13)

while the other Poisson brackets {i , j } are equal to zero. The number 2q is the rank of the matrix (5.2). In view of inequality (5.3) and Caratheodory's lemma of fil ling one can introduce new canonical coordinates (we denote them again by x1 ,...,xn , y1 ,...,yn ) in such a way that xi = 2
i-1

,

yi = 2i ,

if i q , and yi = i if i > 2q . In view of (5.13) such a transformation of coordinates will be canonical. Up to the differential of some function (which is inessential for analysis of the differential d ) the 1-form (5.5) becomes
n

yi dxi ,
i=1

(5.14)

where the momenta y1 ,...,ym-q (m - q n) and the coordinates x1 ,... ,xq should be replaced with constants. Hence, in terms of the new canonical variables the differential of the form (5.14) becomes
n

dyi dxi .
i=m-q

The rank of this 2-form is obviously equal to 2[n - (m - q )] = 2s. This proves Proposition 2.
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KOZLOV

We now use the Lamb equation (see [5]): + iv d = -dh. t Here v is the vector field on 2 t
n-m

(5.15)

with the components
+1

vk

=

H H ,..., vn = . yk+1 yn ,...,xn ,t,c) = H |.
n-m

(5.16)

The vertical bar denotes the substitution (5.7). In its turn, h(x
k+1

The restriction of the Hamiltonian system (1.1) to the invariant manifold 2 t the parameters c1 ,...,cm ) is xk
+1

(depending on (5.17)

= vk

+1

,..., xn = vn .

If this system is integrated, the other canonical variables are easily found from formulae (5.7). Let the condition of noncommutative integrability (5.12) be satisfied. Then (by Corollary 1) d = 0. Eq. (5.15) gives S + h = 0, x t whence S + h = f (t, c). t The gauge transformation S=S- f (t, c) dt

leads to the generalized Hamilton­Jacobi equation: S + H (un+1 ,... ,un+k ,xk+1 ,... ,xn ,u1 ,... ,un ,t) = 0. (5.18) t For m = n it coincides with the classical Hamilton­Jacobi equation, since in this case k = 0 and uj = S . xj
m

Theorem 2. Assume that among m parameters c1 ,... ,c them by k+1 ,... ,n ), such that 1)
k j =1

there are n - k parameters (we denote

H un+j + xj p p n,

k j =1

H uj - yj p

n i=k+1

H yi p

k

uj un+j =0 xi

j =1

for al l k +1

2) the fol lowing inequality holds det 2S xp q = 0; k +1 p, q n. (5.19)

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Then a complete solution to the canonical differential equations (1.1) wil l be found from n - k algebraic equations S = aj = const, j and n + k relations (5.7). In the condition (5.19) the function S can obviously be replaced with S . The gauge transformation relating the functions S and S has another interpretation: adding the term explicitly depending only on time to the Hamiltonian function does not change the canonical Hamilton equations. The role of 2n arbitrary constants is played by 2n parameters c1 ,... ,cm (m = n + k ) and ak+1 ,...,an . If k = 0 (m = n), then Theorem 2 turns into the classical Jacobi theorem on a complete integral. Strictly speaking, Theorem 2 will be proved if we establish that the derivatives S/j (5.21) k +1 j n (5.20)

are the first integrals of the system of differential equations (5.17). To do so, we calculate the total time derivative of (5.21) taking into account the first condition of Theorem 2 and the relations (5.10). From (5.18) we have S p
·

=

2S + p t

n j =k+1

H 2S = 0. yj p xj

(5.22)

In view of the definition (5.16) of components of the vector field v , we finally obtain from (5.22) t S p
n

+
j =k+1

vj

S = 0. xj p

This proves Theorem 2. Theorem 2 offers a method of explicit noncommutative integration of Hamiltonian systems. An additional restriction is the first condition of the theorem. Can one dispense with it at all? The success of applying Theorem 2 depends on a successful choice of canonical variables. This remark is typical of the Hamilton­Jacobi method in general. In particular, with an appropriate choice of canonical variables the conditions of Theorem 2 are obviously satisfied. It suffices to use the variables introduced above in the proof of Proposition 2 with the help of the Lie­Cartan theorem. Theorem 2 can be reversed. The following theorem holds. Theorem 3. Assume that the Hamilton equations (1.1) admit an m-parameter family of (2n - m)dimensional invariant manifolds given by (5.7) and let the fol lowing conditions be satisfied: 1) (u1 ,... ,un+k ) = 0, n + k = m , (c1 ,... ,cn+k )

2) the differential 1-form (5.8) is a total differential: = dS , where S is a smooth function of xk+1 ,... ,xn , t and c. Among n + k parameters c1 ,... ,c such that 3)
k j =1 n+k

there are n - k parameters (we denote them by k

+1

,... ,n ),

H un+j + xj p p n,

k j =1

H uj - yj p

n i=k+1

H yi p

k j =1

un+j uj =0 xi

for al l k +1

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4) det

2S xp q

= 0,

k +1

p, q

n.

Then a complete solution of the canonical differential equations (1.1) can be found from n - k algebraic equations (5.20) and n + k relations (5.7). If k = 0, then Theorem 3 turns into the classical Jacobi theorem on a complete integral. Keep in mind that to the solutions of the Hamilton­Jacobi equation there correspond invariant Lagrangian manifolds of the Hamilton equations. Theorems 2 and 3 are dual to each other in the following sense. According to the first condition of Theorem 3, from (5.7) one can express (at least locally) the parameters c1 ,...,cn+k in terms of canonical variables and time. As a result, we obtain m = n + k first integrals of the Hamilton differential equations which satisfy the condition of noncommutative integrability (5.12). After this remark Theorem 3 reduces to Theorem 2. We note that the parameters = (k+1 ,... ,n ) can be chosen in a more general way by assuming the initial parameters c1 ,... ,cm to be smooth functions of . This remark lends additional flexibility to the method developed here. 6. THE AUTONOMOUS CASE We show what the generalized Hamilton ­ Jacobi method looks like in the case where the initial Hamiltonian system (1.1) is autonomous and the first integrals (5.1) do not explicitly depend on time. It is natural to assume that there is an energy integral among the functions (5.1); let F1 = H . Now the algebraic system (5.5) does not contain time. Therefore, under the condition (5.15) = dS, where S is a smooth function of xk and, hence, (5.18) becomes S + c 1 = 0, t where (according to the gauge transformation) S = S - g (t, c). By virtue of (6.1), g = -c1 t + G, where G is some smooth function depending only on the parameters c1 ,... ,cm . The relations (5.20) now become c1 S = t + aj , j j Here aj = aj - G , j k +1 j n. (6.2) (6.1)
+1

,... ,xn and c1 ,... ,cm . It is clear that H | = c1 ,

where aj are arbitrary constants. By virtue of arbitrariness in the choice of the constants aj we can set G 0. The nondegeneracy condition (5.19) has the same form, but the function S should be replaced with S . The relations (6.2) are an immediate generalization of the classical Jacobi formulae. Note that the governing equation (6.1) has been derived by us from other considerations.
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Consider the case where the first n - k functions (5.1) are in involution with all others and the integral manifolds n-k are compact. Then, as shown in [8], n-k are diffeomorphic to (n - k )dimensional tori, and in a neighborhood of these tori there exist canonical coordinates I , p, mod 2 , q , such that Is = Is (F1 ,... ,Fn
-k

),

1

s

n - k,

and p, q depend on all Fi . In terms of these coordinates the fundamental 1-form becomes Ij dj + pi dqi ,
-k

and the Hamiltonian function H = F1 depends only on I1 ,... ,In

.

The variables I , p, , and q are "globalized" canonical variables whose existence follows from the Lie­Cartan theorem. What does the conclusion of Theorem 2 in these variables look like? Eqs. (5.1) take the form
0 Ij = Ij ,

pi = p0 , i

0 qi = qi ,

j = 1,... ,n - k, i = 1,... ,k .

Consequently, = whence S=
0 Ij j -k 0 Ij dj = dS,

is a multi-valued function in a neighborhood of the invariant tori n Further, the function S = S - f (t, I 0 ,p0 ,q 0 ) satisfies (5.18): S = -H (I 0 ). t Hence, S = S - H (I 0 )t + g (I 0 ,p0 ,q 0 ).

.

(6.3)

0 0 The parameters k+1 ,... ,n are the initial data of the action variables I1 ,... ,In conditions 1 and 2 of Theorem 2 are obviously satisfied. Hence, according to (5.20),

-k

. The

S 0 = aj , Ij

j = 1,... ,n - k,

whence we obtain the formulae for changing the angle variables with time: j = H g 0 t + aj + I 0 . Ij j

Since aj are arbitrary constants, one can set g = 0.
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In the case at hand Eq. (6.3) becomes S +H t S S ,..., 1 n- = 0.
k

This is the ordinary Hamilton ­ Jacobi equation for the reduced Hamiltonian system with n - k degrees of freedom. This fact is a reflection of the following, more general, observation: the theorems on noncommutative integrable systems reduce to the classical Liouville theorem on completely integrable systems and the Lie ­ Cartan theorem on the closed sets of first integrals. As an illustration of Theorems 2 and 3, we consider the problem of the inertial motion of a material particle on the plane R2 = {x1 ,x2 }. It is clear that the particle moves in a straight line with constant velocity. In this sense the problem is trivial. But on the other hand, the motion of a particle on a plane is a simple example of a noncommutative integrable system, which can be used to illustrate Theorem 2 without recourse to special canonical variables I , p, and q . The Hamiltonian system with Hamiltonian 12 2 H = (y1 + y2 ) 2 admits three first integrals: F1 = y1 , This set of integrals is closed, since {F1 ,F2 } = 0,
2 1 2 2

F2 = y2 ,

F3 = x2 y1 - x1 y2 . {F2 ,F3 } = F1 .

(6.4)

{F1 ,F3 } = -F2 ,

If c + c = 0, then Eqs. (5.1) define a one-dimensional invariant manifold. Otherwise the first integrals (6.4) are dependent. Let c1 = 0. Suppose x1 is a coordinate on 1 . Then c2 x1 + c3 y1 = c1 , y2 = c2 , x2 = c1 and c2 = c1 dx1 + 2 dx1 = dS, c1 whence c2 + c2 2 x1 . S= 1 c1 Further, S= c2 + c2 c2 + c2 1 2 2 t. x1 - 1 c1 2 (6.5)

It can be verified that the first condition of Theorem 2 reduces to the equality c2 c1 - c2 = 0. (6.6) c1 If one takes the polar radius as parameter (c1 = sin , c2 = cos ; since c1 = 0, sin = 0), then the relation (6.6) will automatically be satisfied. Then the function (6.5) takes the form S= 2 x1 - t. sin 2

The second condition of Theorem 2 is, of course, satisfied. Consequently, the relation (5.20) yields a law of change of the first coordinate: x1 - t = a. sin Thus, x1 = c1 t + a , a = const.
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7. HISTORICAL COMMENTS In 1909 V. A. Steklov published three notes on the theory of Hamiltonian systems [9­11]. They have recently been translated into Russian and published in [12]. Unfortunately, contrary to other works of V. A. Steklov, there was little interest in the concisely written papers [9­11] at that time, so that they did not get into the orbit of modern analytical dynamics and the theory of dynamical systems. It turns out that they have much in common with the contents of this paper (especially Sections 5 and 6). Therefore, it is appropriate here to make a short comment on them. The starting point for V. A. Steklov was Jacobi's "Lectures on dynamics", especially lecture 35, which develops the relation of the Hamilton­Jacobi method to the theory of canonical transformations represented in the form of corresponding properties of the Poisson brackets. V. A. Steklov considers m n algebraic equations (5.1) (which do not contain time) and expresses from them the momenta of the system and a part of generalized coordinates in terms of other coordinates (as it is done in § 5; see the relations (5.7)). The following condition is imposed on the functions from the set (5.1): the restriction of the fundamental 1-form yi dxi to the integral manifolds (5.1) will be a closed differential form. From the local point of view (adopted by V. A. Steklov), this is the total differential of some smooth function depending on a part of generalized coordinates: = dS . By Proposition 2, this means that the set m of the functions (5.1) satisfies the condition of noncommutative integrability. Thus, 60 years before the works of N. N. Nekhorochev [8] and A. S. Mishchenko and A. T. Fomenko [13] V. A. Steklov actually introduced the class of noncommutative integrable systems and investigated their local properties. Starting from the initial set of functions (5.1), V. A. Steklov introduced canonical variables similar in their properties to the canonical variables of N. N. Nekhoroshev (variables I , p, , q from § 6) [11]. This result allowed V. A. Steklov to draw the conclusion that the general case of existence of the noncommutative set of first integrals actually reduces to the classical Liouville theorem, which was applied to the reduced Hamiltonian system [11]. The results of V. A. Steklov are "absorbed" by the more general Lie­Cartan theorem, which we have used in the proof of Proposition 2. It is worth noting that the Lie­Cartan theorem was established by Sophus Lie already in 1873. It was set forth by him in [14] (Chapter 9, Theorem 23). Of course, the terminology used by S. Lie will seem unusual to the modern reader. The Lie Theorem was re-proved by E. Cartan in [6] without mentioning the work of S. Lie. E. Cartan used this theorem to solve the problem of reducing the order of Hamiltonian systems with known first integrals. In his notes [9­11] V. A. Steklov mentioned the results of Lie and Meyer but without accurate references. Actually, the notes [9­11] are variations on the theme of the Lie­Cartan theorem which are related to the exact integration of the Hamilton equations. V. A. Steklov constantly sought to render his constructions constructive by using only algebraic operations and simple quadratures. Therefore, in [9­11] he actually formulated the well-known theorem of A. V. Brailov [15] on the integrability of Hamiltonian systems by quadratures with a functionally closed set of first integrals satisfying the condition of noncommutative integrability. Our results contained in Sections 5 and 6 should be regarded as development of the ideas and results of the works of V. A. Steklov [9­11]. We combine the ideas of V. A. Steklov on noncommutative integrable systems with the ideas of the classical Hamilton­Jacobi method and thereby propose another way of explicit integration of the equations of classical mechanics, which, of course, requires further study and development. The author is grateful to A. V. Bolsinov and A. V. Borisov for useful discussions. REFERENCES
1. Kozlov, V. V., General Theory of Vortices, Izhevsk: Izdatel'skij Dom "Udmurtskij Universitet", 1998 [Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003]. 2. Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; with an Introduction to the Problem of Three Bodies, 3rd ed., Cambridge: Cambridge Univ. Press, 1927. 3. Birkhoff, G. D., Dynamical Systems: With an Addendum by J. Moser, rev. ed., American Mathematical Society Collo quium Publications, vol. 9, Providence, R.I.: AMS, 1966. 4. Santilli, R. M., Foundations of Theoretical Mechanics: 2. Birkhoffian Generalization of Hamiltonian Mechanics, Texts Monogr. Phys., New York: Springer, 1983.
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5. Kozlov, V. V., On invariant Manifolds of Hamilton's Equations, Prikl. Mat. Mekh., 2012, vol. 76, no. 4, pp. 526­539 [J. Appl. Math. Mech., in press]. ´ 6. Cartan, E., Le¸ cons sur les invariants int´ aux, Paris: Hermann, 1922. egr 7. Arnold, V. I., Kozlov, V. V., and Ne tadt, A. I., Mathematical Aspects of Classical and Celestial ish Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1­291. 8. Nekhoroshev, N. N., Action-Angle Variables and Their Generalization, Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181­198 [Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180­198]. 9. Stekloff, W., Sur une g´ ´ eneralisation d'un th´ eoreme de Jacobi, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 153­155. 10. Stekloff, W., Application d'un th´ ` eoreme g´ ´ eneralis´ de Jacobi au probl`me de S. Lie­Mayer, C. R. Acad. e e Sci. Paris, 1909, vol. 148, pp. 277­279. 11. Stekloff, W., Application du th´ ` eoreme g´ ´ eneralis´ de Jacobi au probl` e eme de Jacobi­Lie, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 465­468. 12. Steklov, V. A., Works on Mechanics of 1902­1909, Translated from French into Russian. Moscow­Izhevsk: R&C Dynamics, 2011. 13. Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Metho d of Integration of Hamiltonian Systems, Funktsional. Anal. i Prilozhen., 1978, vol. 12, no 2, pp. 46­56 [Funct. Anal. Appl., 1978, vol. 12, no. 2, pp. 113­121]. 14. Lie, S., Theorie der Transformationsgruppen II, Leipzig: Teubner, 1890. 15. Brailov, A. V., Complete Integrability of Some Geo desic Flows and Integrable Systems with Noncommuting Integrals, Dokl. Akad. Nauk SSSR, 1983, vol. 271, no. 2, pp. 273­276 (Russian).

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