Constructing a complete integral of the Hamilton-Jacobi equation on pseudo-Riemannian spaces with simply transitive groups of motions
OOn constructing complete integrals of theHamilton–Jacobi equation on pseudo-Riemannian spaceswith simply transitive groups of motions
A. A. Magazev Omsk State Technical University, Prospect Mira, 11, Omsk, Russia, 644050
Abstract
In this work, a method for constructing a complete integral of the geodesicHamilton–Jacobi equation on pseudo-Riemannian manifolds with simply tran-sitive actions of groups of motions is suggested. The method is based on usinga special transition to canonical coordinates on coadjoint orbits of the group ofmotion. As a non-trivial example, we consider the problem of constructing acomplete integral of the geodesic Hamilton–Jacobi equation in the McLenaghan–Tariq–Tupper spacetime.
Keywords:
Hamilton–Jacobi equation, complete integral, simply transitiveaction, coadjoint orbit
Introduction
Let ( M, g ) be a pseudo-Riemannian manifold of dimension n and let ( x , . . . , x n ) be a local system of coordinates on M . The geodesic Hamilton–Jacobi equation on ( M, g ) is the first-order partial differential equation g ij ∂S∂x i ∂S∂x j = m , (1)where g ij are the contravariant components of the metric, m is a real non-negative parameter. A complete integral of this equation is a solution S ( x ; α ) depending on n real parameters α = ( α , . . . , α n ) such that det (cid:13)(cid:13)(cid:13)(cid:13) ∂ S ( x ; α ) ∂x i ∂α j (cid:13)(cid:13)(cid:13)(cid:13) (cid:54) = 0 . (2)The problem of finding a complete integral of the Hamilton–Jacobi equationis closely related to the problem of solving the geodesic equations (see, for ex-ample, [1]). Indeed, if we know a complete integral of equation (1), then, by Email address: [email protected] (A. A. Magazev)
Preprint submitted to Journal of L A TEX Templates August 28, 2019 a r X i v : . [ m a t h - ph ] A ug urely algebraic manipulations, we can construct integral curves of the Hamil-tonian system with the geodesic Hamiltonian H ( x, p ) = g ij p i p j . This methodof constructing solutions of the geodesic equations, called the Jacobi method , isoften used in analytical mechanics and general relativity [2, 3, 4].The most traditional approach to the construction of a complete integral forthe Hamilton–Jacobi equation is based on the concept of separation of variables .Although the theory of separation of variables has a long history going back toworks of Levi-Civita and St¨ackel, modern physicists and mathematicians stilldemonstrate their interest in this field. We will not give a review of this theoryin this paper; instead, we refer the reader to the works [5, 6, 7, 8, 9, 10] wherethe problem of separation of variables in the geodesic Hamilton–Jacobi equationhas been discussed in detail.The St¨ackel spaces, that is, pseudo-Riemannian manifolds ( M, g ) , on whichthe geodesic Hamilton–Jacobi equation can be solved by separation of variables,have been well studied at present time. In particular, the geodesic equationsin the St¨ackel spaces must necessarily admit dim M pairwise commuting firstintegrals associated with the Killing tensors of the first and second order (see,for instance, [7, 10]). This condition is a rather strong restriction on the pseudo-Riemannian manifold ( M, g ) and is not satisfied in many important cases. Themost famous example of this type is the Euler’s equations describing the ro-tation of a rigid body. As V. I. Arnol’d has shown, these equations can bewritten as the geodesic equations on the rotation group SO (3) [3]. Althoughthis problem admits three mutually commuting first integrals, it can be shownthat the separation of variables for the corresponding geodesic Hamilton–Jacobiequation is impossible. There are also examples of non-St¨ackel spaces satisfyingthe Einstein field equations; the investigation of such cases, based on the clas-sification results by A. Z. Petrov [11], was given in [12, 13]. All of the aboveleads us to the conclusion that the problem of constructing a complete inte-gral for the Hamilton–Jacobi on non-St¨ackel pseudo-Riemannian manifolds is offundamental importance.It should be noted that there are different ways to go beyond the frameworkof the method of separation of variables. For instance, instead of coordinatetransformations in M , one can consider more general canonical transformationsof the whole phase space T ∗ M (see, for example, [14] and references therein).Another way to do it is to use the symplectic reduction machinery. [15, 16]. Themain idea of this method is to apply the noncommutative symmetries associatedwith conservation laws to decrease the number of phase variables of a mechanicalsystem required for describing its dynamics. The difficulty, however, is the factthat the symplectic reduction is a procedure directly applied to Hamiltoniansystems, while the Hamilton–Jacobi equation is a partial differential equation.Nevertheless, there is already a certain success in combining both symplecticreduction theory and Hamilton–Jacoby theory [17, 18, 19].In this paper, we solve the problem of constructing a complete integral for thegeodesic Hamilton–Jacobi equation on a pseudo-Riemannian manifold ( M, g ) ,admitting a simply transitive group of motion G . Following the basic idea ofsymplectic reduction, we reduce this problem to an auxiliary first-order partial2ifferential equation on Lagrangian submanifolds of regular coadjoint orbits ofthe group G . This auxiliary equation involves fewer independent variables thanthe original equation and in some cases can be solved by quadratures. In orderto distinguish these cases, we give a simple algebraic criterion in terms of the Liealgebra of the group of motions G . It should be emphasized that the techniquefor constructing a complete integral of Eq. (1) developed in the paper is fullyconstructive: all steps of the technique are reduced to using only quadraturesand tools of linear algebra.The paper is organized as follows. In Sec. 1, we recall some facts concern-ing pseudo-Riemannian manifolds with simply transitive groups of motions. Inparticular, we write the equation (1) in terms of invariant vector fields on M .Sec. 2 is devoted to coadjoint orbits of Lie groups, especially the problem of con-structing canonical coordinates (the Darboux coordinates) on the orbits. Wepay special attention to a specific class of canonical coordinates that are closelyrelated to polarizations of Lie algebras. In Sec. 3, we prove a theorem containingthe basic result of the present paper. According to this theorem, a completeintegral of Eq. (1) can be constructed by solving an auxiliary Hamilton–Jacobiequation on the invariant Lagrangian submanifolds of regular coadjoint orbitsof G , where G is the isometry group of the pseudo-Riemannian manifold ( M, g ) .For the Lie groups with two-dimensional regular orbits, this result allows us tointegrate Eq. (1) by quadratures alone. In the final section, as an example ofthe application of our method, we consider the problem of constructing a com-plete integral of the geodesic Hamilton–Jacobi equation in the background ofMcLenaghan–Tariq–Tapper spacetime [20, 21]. It is noteworthy that this spaceis non-St¨ackel, that is, the integration of Eq. (1) for the McLenaghan–Tariq–Tapper metric cannot be performed by the method of separation of variables.
1. The Hamilton–Jacobi equation on pseudo-Riemannian manifoldswith simply transitive isometry groups
We start with recalling some facts concerning pseudo-Riemannian manifoldswith simply transitive groups of motions. For additional details and references,we refer the reader to the book of Stephani et al. [22].Let ( M, g ) be a differentiable pseudo-Riemannian manifold admitting an n -dimensional group of motions G : τ ∗ z g = g, z ∈ G. Here τ z : M → M is the transformation associated with the group element z .Everywhere in this paper, we assume that the group G acts on M on the left ,that is τ z z x = τ z τ z x, x ∈ M, z , z ∈ G. Suppose that the action of G on M is simply transitive . This means thatfor any two points x , x ∈ M there exists one and only one group element3 ∈ G such that x = τ z x . Choose a point x ∈ M and define a mapping ψ x : G → M by ψ x ( z ) def = τ z x , z ∈ G. (3)It is clear that this mapping establishes a smooth one-to-one correspondencebetween points of M and elements of G . In particular, the manifold M and theLie group G have the same dimensions: dim M = dim G = n .We recall that the left translation associated with z ∈ G is a mapping L z : G → G such that L z ( z (cid:48) ) = zz (cid:48) for all z (cid:48) ∈ G . From (3) it follows that ψ x ◦ L z = τ z ◦ ψ x , z ∈ G, (4)i.e. the mapping ψ x is equivariant with respect to the actions of G on G and M , respectively. In analogy with the left translation, one can define the righttranslation R z : G → G by R z ( z (cid:48) ) = zz (cid:48) . As can be readily seen, the righttranslations commute with the left ones, R z ◦ L z (cid:48) = L z (cid:48) ◦ R z , z, z (cid:48) ∈ G .Let g be the Lie algebra of the group G associated with the tangent space T e G and let e , . . . , e n be a basis in g . The Lie bracket [ · , · ] on g is completelydetermined by the commutation relations between its basis elements: [ e i , e j ] = C kij e k . The coefficients C kij are known as the structure constants of the group G . Here-after, we assume a summation over the repeated indices (Einstein notation).Consider the left-invariant vector field l i ( z ) = ( L z ) ∗ e i on G corresponding tothe basis vector e i ∈ g and denote by ξ i the image of l i under the mapping ψ x : ξ i ( τ z x ) def = ( ψ x ) ∗ l i ( z ) , z ∈ G. (5)By virtue of (4), the vector filed ξ i is invariant with respect to the action of G on M . Moreover, it is easy to show that [ ξ i , ξ j ] = C kij ξ k , i.e. the set of vector fields ξ i spans the Lie algebra g L ( M ) that is isomorphic tothe Lie algebra g .Similarly, we can also introduce the vector fields η i on M by the formula η i ( τ z x ) def = ( ψ x ) ∗ r i ( z ) , z ∈ G, where r i ( z ) = ( R z ) ∗ e i is the right-invariant vector field on G associated with e i ∈ g . Unlike the vector fields ξ i , the fields η i are not invariant under the groupaction G on M . Furthermore, these vector fields commute with ξ i and form theLie algebra g R ( M ) that is anti-isomorphic to the Lie algebra g : [ η i , ξ j ] = 0 , [ η i , η j ] = − C kij η k . Taking into account that the vector fields − r i are infinitesimal generators of theleft translations, we obtain ( η i ϕ ) ( τ z x ) = ddt ϕ (cid:0) τ exp( te i ) z x (cid:1) (cid:12)(cid:12) t =0 = ddt ϕ (cid:0) τ exp( te i ) τ z x (cid:1) (cid:12)(cid:12) t =0 , ϕ ∈ C ∞ ( M ) . From this, it is clear that the vector fields − η i are the infinitesimal generators of G acting on M . Since the group G actson M by isometries, we conclude that − η i are the Killing vector fields of thepseudo-Riemannian manifold ( M, g ) .Using the relation (4) and the invariance of the metric under the group G ,we obtain g ( ξ i , ξ j ) = G ij , (6)where the constants G ij are defined as G ij def = g (( ψ x ) ∗ e i , ( ψ x ) ∗ e j ) . It follows from (5) that the vector fields ξ i form a basis of the tangent space T x M at each point x of M . Consider the collection of 1-forms ω i on M uniquelydetermined by the vector fields ξ i : (cid:104) ω i , ξ j (cid:105) = δ ij . These 1-forms form a basis ofthe cotangent space T ∗ x M at x ∈ M called the dual to the basis { ξ i } . Using (6),we can express the metric g in terms of the 1-form ω i as follows: g = G ij ω i ω j . (7)If ( x , . . . , x n ) is a local coordinate system on M , then for the covariant andcontravariant components of the metric we obtain g ij ( x ) = G kl ω ki ( x ) ω lj ( x ) , g ij ( x ) = G kl ξ ik ( x ) ξ jl ( x ) . (8)Here ω ki ( x ) and ξ ik ( x ) are the coordinate components of the 1-form ω k and thevector field ξ k , respectively, G ij G jk = δ ik .It is immediately clear from the above results that the Hamilton-Jacobi theequation (1) can be expressed in terms of the invariant vector fields ξ i as G ij ( ξ i S ) ( ξ j S ) = m . (9)Here ξ i S denotes the directional derivative of a function S ∈ C ∞ ( M ) along thevector field ξ i .
2. Canonical coordinates on coadjoint orbits
The method for constructing a complete integral of Eq. (9) developed belowsubstantially uses special canonical coordinates on coadjoint orbits of G . Inthe present section, we recall necessary definitions and describe a convenientalgebraic way to construct such coordinates, following the papers [23, 24, 25].For any z ∈ G , the automorphism R z − ◦ L z : G → G leaves the identityelement e ∈ G fixed. Its differential at e is a linear map of the Lie algebra g (cid:39) T e G into itself. This map is denoted by Ad z def = ( R z − ) ∗ ( L z ) ∗ and is calledthe adjoint representation of the group G . The coadjoint representation of G isa representation on the dual space g ∗ , that is dual to the adjoint representation: Ad ∗ z def = (Ad z − ) ∗ . More explicitly (cid:104) Ad ∗ z λ, Z (cid:105) = (cid:104) λ, Ad z − Z (cid:105) , (10)5here λ ∈ g ∗ , Z ∈ g , and (cid:104)· , ·(cid:105) denote the natural pairing between the spaces g and g ∗ .The dual space g ∗ of the Lie algebra g admits a natural Poisson structure.The corresponding Poisson bracket, called the Lie–Poisson bracket , has the form { ϕ, ψ } ( f ) = C kij f k ∂ϕ ( f ) ∂f i ∂ψ ( f ) ∂f j , ϕ, ψ ∈ C ∞ ( g ∗ ) , (11)where ( f , . . . , f n ) are the coordinates of f ∈ g ∗ with respect to the basis { e i } that is dual to the basis { e i } of g . In general, the Lie–Poisson bracket (11) isdegenerate. This means that the dual space g ∗ admits a stratification by thesymplectic leaves of the Lie–Poisson bracket. It has been shown by A. A. Kirillov[26] and B. Kostants [27] that the symplectic leaves are exactly the coadjointorbits of the group G . Thus, the above-mentioned stratification is, in fact, thedecomposition of g ∗ into coadjoint orbits.Let O λ be the coadjoint orbit passing through λ ∈ g ∗ . From the above,the restriction of the Lie–Poisson bracket to the orbit O λ is non-degenerateand therefore defines the symplectic form ω λ , called the Kirillov–Kostant form ,on it. By construction, the form ω λ is invariant under the coadjoint action of G and hence each coadjoint orbit possesses a canonical G -invariant symplecticstructure.It follows from Darboux’s theorem that there exist local coordinates ( p a , q a ) on O λ in which the Kirillov–Kostant form ω λ takes the form ω λ = dp a ∧ dq a , α = 1 , . . . ,
12 dim O λ . The coordinates ( p a , q a ) are called the canonical coordinates . It is easy to seethat to construct the canonical coordinates on the orbit O λ , we need to definefunctions f i ( q, p ; λ ) such that f i (0 , λ ) = λ i , (12) ∂f i ( q, p ; λ ) ∂p a ∂f j ( q, p ; λ ) ∂q a − ∂f i ( q, p ; λ ) ∂q a ∂f j ( q, p ; λ ) ∂p a = C kij f k ( q, p ; λ ) , (13)and rank (cid:13)(cid:13)(cid:13)(cid:13) ∂f i ( p, q ; λ ) ∂q a , ∂f i ( p, q ; λ ) ∂p a (cid:13)(cid:13)(cid:13)(cid:13) = 12 dim O λ . (14)Let us distinguish a special class of canonical coordinates whose functions f i ( q, p ; λ ) are linear with respect to the "momentum" variables p a : f i ( q, p ; λ ) = ζ ai ( q ) p a + χ i ( q ; λ ) . (15)In this case, the condition (13) can be rewritten in the form ζ ai ( q ) ∂ζ bj ( q ) ∂q a − ζ aj ( q ) ∂ζ bi ( q ) ∂q a = C kij ζ bk ( q ) , (16)6 ai ( q ) ∂χ j ( q ; λ ) ∂q a − ζ aj ( q ) ∂χ i ( q ; λ ) ∂q a = C kij χ k ( q ; λ ) . (17)Furthermore, it follows from (14) that rank (cid:107) ζ ai ( q ) (cid:107) = 12 dim O λ . (18)In the paper [23], it is shown that the system (16) has a solution satisfyingthe condition (18) if and only if there exists a subalgebra h ⊂ g such that dim h = dim g −
12 dim O λ . (19)In this case, the solutions of (16) have the following interpretation: the vectorfields ζ i = ζ ai ( q ) ∂ q a are infinitesimal generators of a local transitive action ofthe group G on some smooth manifold Q with dimension dim O λ / . It will beconvenient to assume that G acts on Q on the right. Then Q is diffeomorphicto the quotient space H \ G , where H is the stationary group of the point q = 0 .Also, note that the manifold Q can be interpreted as a G -invariant Lagrangiansubmanifold of the symplectic manifold O λ .Further, the equations (17) have a solution if and only if the subalgebra h issubordinate to the element λ [23]: (cid:104) λ, [ h , h ] (cid:105) = 0 . (20)The subalgebra h ⊂ g that satisfies the conditions (19) and (20) is called polar-ization of the element λ ∈ g ∗ . Thus, the canonical coordinates on O λ definedby the transition functions (15) exist if and only if the element λ admits apolarization.Currently, the problem of the existence of polarizations in Lie algebras hasbeen well studied. A quite complete list of results on this problem is given inDixmier’s book [28]. In particularly, polarizations always exist for nilpotentand completely solvable Lie algebras. Conversely, for a semisimple Lie algebra,a polarization might not exist. But, in spite of this fact, the canonical transition(15) can still be constructed in this case as well. This requires the considerationof polarizations in the complexifications of Lie algebras (see [23] for details).We note that if a polarization h of λ ∈ g ∗ exists, then the problem of con-structing canonical coordinates on the coadjoint orbit O λ can be constructivelysolved by the tools of linear algebra. Indeed, in the paper [29], it is shown thatinfinitesimal generators of G acting on the homogeneous space Q = H \ G canbe constructed from the structure constants of Lie group G by the computationof matrix inversions and matrix exponents. The functions χ i ( q ; λ ) , which sat-isfy the system (17), can also be found without direct solving the differentialequations. An algebraic method of constructing such functions is described inthe work [24]. 7 . Constructing a complete integral of the Hamilton–Jacobi equation Now we come to the question of how to construct a complete integral forthe geodesic Hamilton–Jacobi equation (9). It turns out that this problem canbe reduced to finding a complete integral for the Hamilton–Jacobi equationon Lagrangian submanifolds of regular coadjoint orbits of G . Thus, solvingthe original differential equation with n = dim G independent variables can bereduced to solving some auxiliary differential equation in which the number ofindependent variables equals r = dim O λ < n .Before formulating the basic result, we introduce some additional construc-tions.Let G be a Lie group and H ⊂ G be its connected closed subgroup. Denoteby g and h the Lie algebras of the groups G and H , respectively. Let us considerthe right homogeneous space Q = H \ G . The group G naturally acts on Q ; wedenote this action as ρ : G × Q → Q , q (cid:55)→ ρ z q , q ∈ Q , z ∈ G . To each basisvector e i of the Lie algebra g , we can associate the infinitesimal generator ζ i ofthe action of G on Q by the formula ( ζ i f ) ( q ) def = ddt f (cid:0) ρ exp( te i ) q (cid:1) (cid:12)(cid:12)(cid:12) t =0 , f ∈ C ∞ ( Q ) . Let us define the map ϕ : M × Q → Q by ϕ ( x, q ) def = ρ ψ − x ( x ) q, x ∈ M, q ∈ Q, (21)where ψ − x : M → G is the inverse of the diffeomorphism (3). Clearly, ϕ ( x , q ) = q . Moreover, from the equality ρ z z = ρ z ρ z it follows that ρ z ϕ ( x, q ) = ϕ (cid:0) ψ x ( ψ − x ( x ) z ) , q (cid:1) , (22)for all q ∈ Q , x ∈ M , and z ∈ G . Let U ⊂ M and V ⊂ Q be the domains suchthat ϕ ( U × V ) ⊂ V and let x , . . . , x n и q , . . . , q r be the systems of local coordi-nates in U and V , respectively. Denote by ϕ a ( x, q ) = ϕ a ( x , . . . , x n , q , . . . , q r ) the collection of functions defining the map ϕ ( x, q ) in the local coordinates, a = 1 , . . . , r . Then, from (22) and the definition of the vector field ξ i , it followsthat ξ ji ( x ) ∂ϕ a ( x, q ) ∂x j = ζ ai ( ϕ ( x, q )) . (23)Here ξ ji ( x ) and ζ ai ( q ) are the coordinate components of the vector fields ξ i and ζ i , respectively.Note that the above relations are correct for any subalgebra h ⊂ g . Next, weapply these results to the case when h is a polarization of an element λ ∈ g ∗ . Aswe saw earlier, the homogeneous space Q = H \ G , in this case, is an invariantLagrangian submanifold of the coadjoint orbit O λ .Before proceeding, we recall some terminology concerning coadjoint orbits.8n element λ ∈ g ∗ is called regular if the coadjoint orbit O λ passing through λ has the maximal dimension in g ∗ . For a regular element λ ∈ g ∗ , the dimensionof O λ can be calculated by the formula dim O λ = dim g − ind g . The non-negative integer ind g , called the index of the Lie algebra g , is definedas ind g def = inf λ ∈ g ∗ corank (cid:107) C kij λ k (cid:107) . (24)Note that for a semisimple Lie algebra g the index ind g coincides with its rank.Let us fix a regular element λ ∈ g ∗ . Since λ is in general position, thecoadjoint orbits close to O λ are diffeomorphic to it. Thus, there exists a smallneighborhood U ⊂ g ∗ of λ that is stratified on the homomorphic G -fibers.Consider the quotient space J = U/G that is the base space of this fibration. Itis clear that dim J = ind g . Let j = ( j , . . . , j ind g ) be the local coordinates onthe manifold J and denote by λ ( j ) some smooth local section of the fibration U → U/G . Then the correspondence j → O λ ( j ) defines some smooth one-to-one parametrization of the coadjoint orbits in the neighborhood U of the regularelement λ .Now we formulate the main result. Теорема 1.
Let λ ( j ) = λ ( j , . . . , j ind g ) be a smooth (local) parametrization ofregular coadjoint orbits in g ∗ , h ⊂ g be a polarization of the element λ ( j ) , and f i ( q, p ; λ ( j )) = ζ ai ( q ) p a + χ i ( q ; λ ( j )) be the functions that define the transitionto canonical coordinates on O λ ( j ) . Let us assume that there exists the closedsubgroup H ⊂ G with the Lie algebra h . On the homogeneous space Q = H \ G we consider the differential equation G ij f i (cid:32) q, ∂ ˜ S∂q ; λ ( j ) (cid:33) f j (cid:32) q, ∂ ˜ S∂q ; λ ( j ) (cid:33) = m . (25) Denote by ˜ S j ( q ; β ) its complete integral depending on the collection of parame-ters β = ( β , . . . , β (dim g − ind g ) / ) . Then the function S ( x ; α ) = ˜ S j ( ϕ ( x, q ); β ) + (cid:90) χ k ( ϕ ( x, q ); λ ( j )) ω k ( x ) (26) is a complete integral of the Hamilton–Jacobi equation (9) . Here α = ( q, j, β ) isthe collection of n parameters, ω k ( x ) are 1-forms dual to the vector fields ξ k ( x ) ,and the mapping ϕ : M × Q → Q is defined by the formula (21) .Proof. First we show that the function (26) satisfies the Hamilton–Jacobi equa-tion (9). From Eq. (23) and the fact that the 1-forms ω k are dual to the vector9elds ξ i , we get ξ ki ( x ) ∂S ( x ; α ) ∂x k = ξ ki ( x ) ∂ϕ a ( x, q ) ∂x k ∂ ˜ S j ( ϕ ( x, q ); β ) ∂ϕ a ( x, q ) + χ k ( ϕ ( x, q ); λ ( j )) (cid:104) ω k , ξ i (cid:105) == ζ ai ( ϕ ( x, q )) ∂ ˜ S j ( ϕ ( x, q ); β ) ∂ϕ a ( x, q ) + χ i ( ϕ ( x, q ); λ ( j )) == f i (cid:32) q (cid:48) , ∂ ˜ S j ( q (cid:48) ; β ) ∂q (cid:48) ; λ ( j ) (cid:33) (cid:12)(cid:12)(cid:12) q (cid:48) = ϕ ( x,q ) . (27)Since ˜ S j ( q ; β ) is a solution of Eq. (25), it is clear that the function (26) satisfiesEq. (9).Now we prove that the function (26) obeys the condition (2). We introducethe notation q (cid:48) a = ϕ ( x, q ) , p (cid:48) a = ∂ ˜ S j ( q (cid:48) ; β ) ∂q (cid:48) a , u = ( q (cid:48) , p (cid:48) , J ) . Using Eq. (27), we obtain ∂ S ( x ; α ) ∂x i ∂α j = ω ki ( x ) ∂f k ( q (cid:48) , p (cid:48) , λ ( j )) ∂α j = ω ki ( x ) ∂f k ( u ) ∂u l ∂u l ∂α j , i.e., det (cid:13)(cid:13)(cid:13)(cid:13) ∂ S ( x ; α ) ∂x i ∂α j (cid:13)(cid:13)(cid:13)(cid:13) = det (cid:107) ω ki ( x ) (cid:107) · det (cid:13)(cid:13)(cid:13)(cid:13) ∂f k ( u ) ∂u l (cid:13)(cid:13)(cid:13)(cid:13) · det (cid:13)(cid:13)(cid:13)(cid:13) ∂u l ∂α j (cid:13)(cid:13)(cid:13)(cid:13) . (28)Clearly, the first determinant on the right-hand side of the last equality is non-zero since the collection of 1-forms { ω k ( x ) } forms a basis in T ∗ x M at any x ∈ M .The second factor on the right-hand side of Eq. (28) is also non-zero, because,by construction, the functions f i ( q, p ; λ ) define a local immersion of orbit O λ inthe dual space g ∗ . In order to show that the third determinant cannot be zero,we rewrite it in the form det (cid:13)(cid:13)(cid:13)(cid:13) ∂u l ∂α j (cid:13)(cid:13)(cid:13)(cid:13) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ ( x,q ) ∂q I ∗ ∗ ∂ ˜ S j ( q ; β ) ∂q ∂β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , From this, we obtain det (cid:13)(cid:13)(cid:13)(cid:13) ∂u l ∂α j (cid:13)(cid:13)(cid:13)(cid:13) = det (cid:13)(cid:13)(cid:13)(cid:13) ∂ϕ a ( x, q ) ∂q b (cid:13)(cid:13)(cid:13)(cid:13) · · det (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ ˜ S j ( q ; β ) ∂q a ∂β b (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . The function ˜ S j ( q ; β ) is a complete integral of Eq. (25); с); hence, the lastequality does not vanish by virtue of the definition of the map ϕ .10he above theorem allows us to reduce the problem of construction of acomplete integral of the Hamilton–Jacobi equation (9) to the problem of findinga complete integral of the subsidiary Hamilton–Jacobi equation on Lagrangiansubmanifolds of regular coadjoint orbits. In particular, if the dimension of reg-ular coadjoint orbits less than or equal to 2, then a complete integral of (9) canbe found in quadratures. Indeed, using the equation G ij f i ( q, p ; λ ( j )) f j ( q, p ; λ ( j )) = m , we can express the variable p as a function q , j , and β = m : p = p ( q ; j, m ) .Thus, for the function ˜ S j ( q ; m ) , we have ˜ S j ( q ; m ) = (cid:90) p ( q ; j, m ) dq. Substituting this function in the formula (26), we obtain a complete integral ofthe Hamilton–Jacobi equation (9).
4. An example: construction of a complete integral for the geodesicHamilton–Jacobi equation in the McLenaghan–Tariq–Tupper space-time
McLenaghan and Tariq found a solution to the Einstein–Maxwell equationswhose electromagnetic tensor does not share the spacetime symmetry [20]. Theline element of the metric can be written as ds = ( dx ) + 2 dx dx − kx dx ( dx + dx )++ (cid:104) k ( x ) − e kx (cid:105) ( dx ) − e − kx d ( x ) , (29)where k is an arbitrary positive real number. The special case of this metriccorresponding to k = 4 was found by Tariq and Tupper [21].As it was shown in [20], the group of motions G of the metric (29) is generatedby the Killing vectors η = ∂ x , η = ∂ x , η = kx ∂ x + ∂ x , η = − ∂ x + ∂ x + k x ∂ x − k x ∂ x , and is simply transitive by virtue of the condition det (cid:107) η ji ( x ) (cid:107) (cid:54) = 0 . The corre-sponding group action ˜ x = τ z ( x ) , expressed in coordinates, has the form ˜ x = x − z + 2 z − ke kz / z x , ˜ x = x − z , ˜ x = x e − kz / − z , ˜ x = x e kz / − z . Here we denote by z = ( z , z , z , z ) the group parameters that are arbitraryreal numbers. 11e fix the point x = (0 , , , ; then the mapping ψ x ( z ) defined by theequality (3) can be written as ψ x ( z ) = 2 z − z , ψ x ( z ) = − z , ψ x ( z ) = − z , ψ x ( z ) = − z . (30)By the formula (5), we get the following expressions for the invariant vectorfields ξ i : ξ = − ∂ x , ξ = − e − kx (cid:0) kx ∂ x + ∂ x (cid:1) , ξ = − e kx ∂ x , ξ = 2 ∂ x − ∂ x . (31)These vector fields generate the Lie algebra g with the commutation relations [ ξ , ξ ] = [ ξ , ξ ] = [ ξ , ξ ] = 0 , [ ξ , ξ ] = kξ , [ ξ , ξ ] = − k ξ , [ ξ , ξ ] = k ξ . It is easy to see that the Lie algebra g is a one-dimensional central extension ofthe three-dimensional algebra (cid:104) ξ , ξ , ξ (cid:105) of the Bianchi type VI .In the tetrad basis (31), the McLenaghan–Tariq–Tupper metric has the form (cid:107) G ij (cid:107) = − − − − . Then, in accordance with (7), we have ds = G ij ω i ω j = ( ω ) − ω ω − ( ω ) − ( ω ) − ( ω ) , (32)where the one-forms ω i are dual to the vector fields ξ i : ω = − dx − dx + kx dx , ω = − e kx dx , ω = − e − kx dx , ω = − dx . (33)It follows from (32) that the geodesic Hamilton–Jacobi equation (1) for theMcLenaghan–Tariq–Tupper metric can be written as ( ξ S ) − ξ S )( ξ S ) − ( ξ S ) − ( ξ S ) − ( ξ S ) = m , or, in the explicit form, − k ( x ) e − kx (cid:18) ∂S∂x (cid:19) + 2 (cid:18) ∂S∂x − kx e − kx ∂S∂x (cid:19) ∂S∂x −− (cid:18) ∂S∂x (cid:19) − e kx (cid:18) ∂S∂x (cid:19) − e − kx (cid:18) ∂S∂x (cid:19) = m . (34)It is important to note that the McLenaghan–Tariq–Tupper metric is not St¨ackel,i.e. the Hamilton–Jacobi equation (34) cannot be solved by the method of sep-aration of variables. It is enough for that to verify that the metric (29) does12ot satisfy the necessary and sufficient conditions of separability of variablesobtained by V. N. Shapovalov [7].For constructing a complete integral of the equation (34), we apply themethod outlined in Sec. 3. We note that this problem can be solved by quadra-tures since it follows from the formula (24) that ind g = 2 ; therefore the dimen-sion of regular coadjoint orbits of G equals dim g − ind g = 2 .The matrix (cid:107) Ad z − (cid:107) of the adjoint representation can be expressed in termsof the Killing vector fields η i and the invariant 1-forms ω i as follows (cid:107) Ad z − (cid:107) ij = ω ik ( x ) η kj ( x ) (cid:12)(cid:12) x = ψ x ( z ) , so that (cid:107) Ad z − (cid:107) = kz − kz k z z / e − kz / kz e − kz / /
20 0 e kz / − kz e kz / /
20 0 0 1 . In accordance with (10), the action of Ad ∗ z on f = ( f , f , f , f ) ∈ g ∗ has theform (Ad ∗ z f ) = f , (Ad ∗ z f ) = kz f + e − kz / f , (Ad ∗ z f ) = − kz f + e kz / f , (35) (Ad ∗ z f ) = k z z f + k z e − kz / f − k z e kz / f + f . (36)It is easy to see that the functions K = f , K = 2 f f + f f are functionally independent invariants of this action; therefore the connectedcomponents of their level sets are coadjoint orbits of G .Let us introduce a local parametrization in the space of regular coadjointorbits: λ ( j ) = ( j , , , j ) , j (cid:54) = 0 . It is easy to verify that the subalgebra h = (cid:104) ξ , ξ , ξ (cid:105) ⊂ g is a polarization of λ ( j ) ∈ g ∗ . The corresponding canonicalcoordinates ( p, q ) on the coadjoint orbit passing through the element λ ( j ) aredefined by the relations f = j , f = p, f = kj q, f = j − k pq. (37)Here q is a local coordinate on the homogeneous space Q = exp( h ) \ G , whichis G -invariant Lagrangian submanifold of the orbit O λ ( j ) .From (37), we obtain that q = f / ( kj ) . Using (35) and (36), we can findthe action ρ : G × Q → Q : ρ z ( q ) = (Ad ∗ z − f ) kj = e − kz / (cid:0) q + z (cid:1) .
13n view of (30), we get the following expression for the function ϕ ( x, q ) (see(21)): ϕ ( x, q ) = e kx / (cid:0) q − x (cid:1) . (38)The equation (25) for the function ˜ S j = ˜ S j ( q ; m ) can be written by usingthe explicit form of the functions f i ( q, p ; λ ) : − (cid:18) k q (cid:19) (cid:32) ∂ ˜ S j ∂q (cid:33) + kq ( j + j ) ∂ ˜ S j ∂q − k j q − j j − j = m . (39)Expressing the derivative of ˜ S j ( q ; m ) with respect to q , we obtain ∂ ˜ S j ( q ; m ) ∂q = 2 k ( j + j ) q + 2 (cid:112) D ( k q ; j, m )4 + k q , (40)where D ( θ ; j, m ) = − j θ − ( m + 3 j ) θ − m + 2 j j + j ) . The equation (40) has a real solution in some open set U ⊂ R if and only if ∆ ≥ , √ ∆ > m + 3 j , where ∆ def = ( m − j + 4 j j )( m − j − j j ) is the discriminant of thequadratic polynomial D ( θ ; j, E ) . In this case, the quadratic equation D ( θ ; j, E ) =0 has the real roots θ + = √ ∆ − ( m + 3 j )2 j > , θ − = − √ ∆ + ( m + 3 j )2 j < , (41)and the domain U is defined as U = { q ∈ R : − (cid:112) θ + k < q < (cid:112) θ + k } . The function ˜ S j ( q ; m ) can be found by integration of the right-hand side of(40) and, after some algebra, can be expressed in terms of incomplete ellipticintegrals: ˜ S j ( q ; m ) = q (cid:90) k ( j + j ) q + 2 (cid:112) D ( k q ; j, m )4 + k q dq == j + j k ln (cid:18) k q (cid:19) + 2 j k (cid:34)(cid:112) θ + − θ − E (cid:32)(cid:115) − k q θ + , θ + θ + − θ − (cid:33) −− θ + (cid:112) θ + − θ − F (cid:32)(cid:115) − k q θ + , θ + θ + − θ − (cid:33) ++ 4 + θ − (cid:112) θ + − θ − Π (cid:32)(cid:115) − k q θ + , θ + θ + , θ + θ + − θ − (cid:33)(cid:35) . (42)14ere the functions F( z, κ ) , E( z, κ ) , and Π( z, a, κ ) are incomplete elliptic inte-grals in the Legendre normal forms of the first, second and third kind, respec-tively [30]. The parameters θ + and θ − are defined by the relations (41).A complete integral of the Hamilton–Jacobi equation (34) is given by theformula (26). Using (33), (37), and (38), we obtain the following expression for S ( x ; α ) : S ( x ; α ) = ˜ S j (cid:16) e kx / ( q − x ); m (cid:17) − j x − (2 j + j ) x + kj ( x − q ) x . Here the function ˜ S j ( q ; m ) is given by (42), and α = ( q, j , j , m ) is a set ofparameters. Acknowledgements
Dr. S. V. Danilova is gratefully acknowledged for careful reading of themanuscript.