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On constructing complete integrals of the Hamilton-Jacobi equation on pseudo-Riemannian spaces with simply transitive groups of motions

In this work, a method for constructing a complete integral of the geodesic Hamilton-Jacobi equation on pseudo-Riemannian manifolds with simply transitive actions of groups of motions is suggested. The method is based on using a special transition to canonical coordinates on coadjoint orbits of the group of motion. As a non-trivial example, we consider the problem of constructing a complete integral of the geodesic Hamilton-Jacobi equation in the McLenaghan-Tariq-Tupper spacetime.


Introduction
Let (M, g) be a pseudo-Riemannian manifold of dimension n and let (x 1 , . . . , 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 where g ij are the contravariant components of the metric, m is a real nonnegative parameter. A complete integral of this equation is a solution S(x; α) depending on n real parameters α = (α 1 , . . . , α n ) such that The problem of finding a complete integral of the Hamilton-Jacobi equation is closely related to the problem of solving the geodesic equations (see, for example, [1]). Indeed, if we know a complete integral of equation (1), then, by purely algebraic manipulations, we can construct integral curves of the Hamiltonian system with the geodesic Hamiltonian H(x, p) = 1 2 g ij p i p j . This method of constructing solutions of the geodesic equations, called the Jacobi method, is often used in analytical mechanics and general relativity [2,3,4].
The most traditional approach to the construction of a complete integral for the 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 to works of Levi-Civita and Stäckel, modern physicists and mathematicians still demonstrate their interest in this field. We will not give a review of this theory in this paper; instead, we refer the reader to the works [5,6,7,8,9,10] where the problem of separation of variables in the geodesic Hamilton-Jacobi equation has been discussed in detail.
The Stäckel spaces, that is, pseudo-Riemannian manifolds (M, g), on which the geodesic Hamilton-Jacobi equation can be solved by separation of variables, have been well studied at present time. In particular, the geodesic equations in the Stäckel spaces must necessarily admit dim M pairwise commuting first integrals 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. The most famous example of this type is the Euler's equations describing the rotation of a rigid body. As V. I. Arnol'd has shown, these equations can be written as the geodesic equations on the rotation group SO(3) [3]. Although this problem admits three mutually commuting first integrals, it can be shown that the separation of variables for the corresponding geodesic Hamilton-Jacobi equation is impossible. There are also examples of non-Stäckel spaces satisfying the Einstein field equations; the investigation of such cases, based on the classification results by A. Z. Petrov [11], was given in [12,13]. All of the above leads us to the conclusion that the problem of constructing a complete integral for the Hamilton-Jacobi on non-Stäckel pseudo-Riemannian manifolds is of fundamental importance.
It should be noted that there are different ways to go beyond the framework of the method of separation of variables. For instance, instead of coordinate transformations in M , one can consider more general canonical transformations of 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]. The main idea of this method is to apply the noncommutative symmetries associated with conservation laws to decrease the number of phase variables of a mechanical system required for describing its dynamics. The difficulty, however, is the fact that the symplectic reduction is a procedure directly applied to Hamiltonian systems, while the Hamilton-Jacobi equation is a partial differential equation. Nevertheless, there is already a certain success in combining both symplectic reduction theory and Hamilton-Jacoby theory [17,18,19].
In this paper, we solve the problem of constructing a complete integral for the geodesic Hamilton-Jacobi equation on a pseudo-Riemannian manifold (M, g), admitting a simply transitive group of motion G. Following the basic idea of symplectic reduction, we reduce this problem to an auxiliary first-order partial differential equation on Lagrangian submanifolds of regular coadjoint orbits of the group G. This auxiliary equation involves fewer independent variables than the original equation and in some cases can be solved by quadratures. In order to distinguish these cases, we give a simple algebraic criterion in terms of the Lie algebra of the group of motions G. It should be emphasized that the technique for constructing a complete integral of Eq. (1) developed in the paper is fully constructive: all steps of the technique are reduced to using only quadratures and tools of linear algebra.
The paper is organized as follows. In Sec. 1, we recall some facts concerning pseudo-Riemannian manifolds with simply transitive groups of motions. In particular, 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 constructing canonical coordinates (the Darboux coordinates) on the orbits. We pay special attention to a specific class of canonical coordinates that are closely related to polarizations of Lie algebras. In Sec. 3, we prove a theorem containing the basic result of the present paper. According to this theorem, a complete integral of Eq. (1) can be constructed by solving an auxiliary Hamilton-Jacobi equation on the invariant Lagrangian submanifolds of regular coadjoint orbits of 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 to integrate Eq. (1) by quadratures alone. In the final section, as an example of the application of our method, we consider the problem of constructing a complete integral of the geodesic Hamilton-Jacobi equation in the background of McLenaghan-Tariq-Tapper spacetime [20,21]. It is noteworthy that this space is non-Stäckel, that is, the integration of Eq. (1) for the McLenaghan-Tariq-Tapper metric cannot be performed by the method of separation of variables.

The Hamilton-Jacobi equation on pseudo-Riemannian manifolds with simply transitive isometry groups
We start with recalling some facts concerning pseudo-Riemannian manifolds with 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: 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 Suppose that the action of G on M is simply transitive. This means that for any two points x 1 , x 2 ∈ M there exists one and only one group element z ∈ G such that x 2 = τ z x 1 . Choose a point x 0 ∈ M and define a mapping It is clear that this mapping establishes a smooth one-to-one correspondence between points of M and elements of G. In particular, the manifold M and the Lie 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 : i.e. the mapping ψ x0 is equivariant with respect to the actions of G on G and M , respectively. In analogy with the left translation, one can define the right translation R z : G → G by R z (z ) = zz . As can be readily seen, the right translations commute with the left ones, Let g be the Lie algebra of the group G associated with the tangent space T e G and let e 1 , . . . , e n be a basis in g. The Lie bracket [·, ·] on g is completely determined by the commutation relations between its basis elements: The coefficients C k ij are known as the structure constants of the group G. Hereafter, 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 to the basis vector e i ∈ g and denote by ξ i the image of l i under the mapping ψ x0 : 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 e. the set of vector fields ξ i spans the Lie algebra g L (M ) that is isomorphic to the Lie algebra g.
Similarly, we can also introduce the vector fields η i on M by the formula 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 group action G on M . Furthermore, these vector fields commute with ξ i and form the Lie algebra g R (M ) that is anti-isomorphic to the Lie algebra g: Taking into account that the vector fields −r i are infinitesimal generators of the left translations, we obtain for any function 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 acts on M by isometries, we conclude that −η i are the Killing vector fields of the pseudo-Riemannian manifold (M, g).
Using the relation (4) and the invariance of the metric under the group G, we obtain where the constants G ij are defined as It follows from (5) (6), we can express the metric g in terms of the 1-form ω i as follows: If (x 1 , . . . , x n ) is a local coordinate system on M , then for the covariant and contravariant components of the metric we obtain Here ω k i (x) and ξ i k (x) are the coordinate components of the 1-form ω k and the vector field ξ k , respectively, G ij G jk = δ i k . It is immediately clear from the above results that the Hamilton-Jacobi the equation (1) can be expressed in terms of the invariant vector fields ξ i as Here ξ i S denotes the directional derivative of a function S ∈ C ∞ (M ) along the vector field ξ i .

Canonical coordinates on coadjoint orbits
The method for constructing a complete integral of Eq. (9) developed below substantially uses special canonical coordinates on coadjoint orbits of G. In the present section, we recall necessary definitions and describe a convenient algebraic way to construct such coordinates, following the papers [23,24,25].
For any z ∈ G, the automorphism R z −1 • L z : G → G leaves the identity element e ∈ G fixed. Its differential at e is a linear map of the Lie algebra g T e G into itself. This map is denoted by Ad z def = (R z −1 ) * (L z ) * and is called the adjoint representation of the group G. The coadjoint representation of G is a representation on the dual space g * , that is dual to the adjoint representation: where λ ∈ g * , Z ∈ g, and ·, · 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 where (f 1 , . . . , 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) is degenerate. This means that the dual space g * admits a stratification by the symplectic 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 coadjoint orbits of the group G. Thus, the above-mentioned stratification is, in fact, the decomposition 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-degenerate and 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 symplectic structure.
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, . . . , The coordinates (p a , q a ) are called the canonical coordinates. It is easy to see that to construct the canonical coordinates on the orbit O λ , we need to define functions f i (q, p; λ) such that and 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 : In this case, the condition (13) can be rewritten in the form Furthermore, it follows from (14) that In the paper [23], it is shown that the system (16) has a solution satisfying the condition (18) if and only if there exists a subalgebra h ⊂ g such that In this case, the solutions of (16) have the following interpretation: the vector fields ζ i = ζ a i (q)∂ q a are infinitesimal generators of a local transitive action of the group G on some smooth manifold Q with dimension dim O λ /2. It will be convenient to assume that G acts on Q on the right. Then Q is diffeomorphic to 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 Lagrangian submanifold of the symplectic manifold O λ .
Further, the equations (17) The subalgebra h ⊂ g that satisfies the conditions (19) and (20) is called polarization of the element λ ∈ g * . Thus, the canonical coordinates on O λ defined by the transition functions (15) exist if and only if the element λ admits a polarization.
Currently, the problem of the existence of polarizations in Lie algebras has been well studied. A quite complete list of results on this problem is given in Dixmier's book [28]. In particularly, polarizations always exist for nilpotent and 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 consideration of 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 constructing canonical coordinates on the coadjoint orbit O λ can be constructively solved by the tools of linear algebra. Indeed, in the paper [29], it is shown that infinitesimal generators of G acting on the homogeneous space Q = H \ G can be constructed from the structure constants of Lie group G by the computation of matrix inversions and matrix exponents. The functions χ i (q; λ), which satisfy the system (17), can also be found without direct solving the differential equations. An algebraic method of constructing such functions is described in the work [24].

Constructing a complete integral of the Hamilton-Jacobi equation
Now we come to the question of how to construct a complete integral for the geodesic Hamilton-Jacobi equation (9). It turns out that this problem can be reduced to finding a complete integral for the Hamilton-Jacobi equation on Lagrangian submanifolds of regular coadjoint orbits of G. Thus, solving the original differential equation with n = dim G independent variables can be reduced to solving some auxiliary differential equation in which the number of independent variables equals r = 1 2 dim O λ < n. Before formulating the basic result, we introduce some additional constructions.
Let G be a Lie group and H ⊂ G be its connected closed subgroup. Denote by g and h the Lie algebras of the groups G and H, respectively. Let us consider the right homogeneous space Q = H \ G. The group G naturally acts on Q; we denote this action as ρ : G × Q → Q, q → ρ z q, q ∈ Q, z ∈ G. To each basis vector e i of the Lie algebra g, we can associate the infinitesimal generator ζ i of the action of G on Q by the formula Let us define the map ϕ : where ψ −1 x0 : M → G is the inverse of the diffeomorphism (3). Clearly, ϕ(x 0 , q) = q. Moreover, from the equality ρ z1z2 = ρ z2 ρ z1 it follows that for all q ∈ Q, x ∈ M , and z ∈ G. Let U ⊂ M and V ⊂ Q be the domains such that ϕ(U ×V ) ⊂ V and let x 1 , . . . , x n и q 1 , . . . , q r be the systems of local coordinates in U and V , respectively. Denote by ϕ a (x, q) = ϕ a (x 1 , . . . , x n , q 1 , . . . , 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 follows that Here ξ j i (x) and ζ a i (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, we apply these results to the case when h is a polarization of an element λ ∈ g * . As we saw earlier, the homogeneous space Q = H \ G, in this case, is an invariant Lagrangian submanifold of the coadjoint orbit O λ .
Before proceeding, we recall some terminology concerning coadjoint orbits.
An 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 dimension of O λ can be calculated by the formula The non-negative integer ind g, called the index of the Lie algebra g, is defined as Note that for a semisimple Lie algebra g the index ind g coincides with its rank. Let us fix a regular element λ 0 ∈ g * . Since λ 0 is in general position, the coadjoint orbits close to O λ0 are diffeomorphic to it. Thus, there exists a small neighborhood U ⊂ g * of λ 0 that is stratified on the homomorphic G-fibers. Consider the quotient space J = U/G that is the base space of this fibration. It is clear that dim J = ind g. Let j = (j 1 , . . . , j ind g ) be the local coordinates on the 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-toone parametrization of the coadjoint orbits in the neighborhood U of the regular element λ 0 . Now we formulate the main result.
Теорема 1. Let λ(j) = λ(j 1 , . . . , j ind g ) be a smooth (local) parametrization of regular coadjoint orbits in g * , h ⊂ g be a polarization of the element λ(j), and f i (q, p; λ(j)) = ζ a i (q)p a + χ i (q; λ(j)) be the functions that define the transition to canonical coordinates on O λ(j) . Let us assume that there exists the closed subgroup H ⊂ G with the Lie algebra h. On the homogeneous space Q = H \ G we consider the differential equation Denote byS j (q; β) its complete integral depending on the collection of parameters β = (β 1 , . . . , β (dim g−ind g)/2 ). Then the function is a complete integral of the Hamilton-Jacobi equation (9). Here α = (q, j, β) is the 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 equation (9). From Eq. (23) and the fact that the 1-forms ω k are dual to the vector fields ξ i , we get . (27) SinceS j (q; β) is a solution of Eq. (25), it is clear that the function (26) satisfies Eq. (9). Now we prove that the function (26) obeys the condition (2). We introduce the notation q a = ϕ(x, q), p a = ∂S j (q ; β) ∂q a , u = (q , p , J).
Using Eq. (27), we obtain Clearly, the first determinant on the right-hand side of the last equality is nonzero 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 in the dual space g * . In order to show that the third determinant cannot be zero, we rewrite it in the form The functionS j (q; β) is a complete integral of Eq. (25); с); hence, the last equality does not vanish by virtue of the definition of the map ϕ.
The above theorem allows us to reduce the problem of construction of a complete integral of the Hamilton-Jacobi equation (9) to the problem of finding a complete integral of the subsidiary Hamilton-Jacobi equation on Lagrangian submanifolds of regular coadjoint orbits. In particular, if the dimension of regular coadjoint orbits less than or equal to 2, then a complete integral of (9) can be found in quadratures. Indeed, using the equation we can express the variable p as a function q, j, and β = m: p = p(q; j, m). Thus, for the functionS j (q; m), we havẽ S j (q; m) = p(q; j, m) dq. (26), we obtain a complete integral of the Hamilton-Jacobi equation (9).

An example: construction of a complete integral for the geodesic Hamilton-Jacobi equation in the McLenaghan-Tariq-Tupper spacetime
McLenaghan and Tariq found a solution to the Einstein-Maxwell equations whose electromagnetic tensor does not share the spacetime symmetry [20]. The line element of the metric can be written as where k is an arbitrary positive real number. The special case of this metric corresponding 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 generated by the Killing vectors and is simply transitive by virtue of the condition det η j i (x) = 0. The corresponding group actionx = τ z (x), expressed in coordinates, has the form Here we denote by z = (z 1 , z 2 , z 3 , z 4 ) the group parameters that are arbitrary real numbers.
We fix the point x 0 = (0, 0, 0, 0); then the mapping ψ x0 (z) defined by the equality (3) can be written as By the formula (5), we get the following expressions for the invariant vector fields ξ i : (31) These vector fields generate the Lie algebra g with the commutation relations It is easy to see that the Lie algebra g is a one-dimensional central extension of the three-dimensional algebra ξ 2 , ξ 3 , ξ 4 of the Bianchi type VI 0 . In the tetrad basis (31), the McLenaghan-Tariq-Tupper metric has the form Then, in accordance with (7), we have where the one-forms ω i are dual to the vector fields ξ i : In view of (30), we get the following expression for the function ϕ(x, q) (see (21)): ϕ(x, q) = e kx 2 /2 q − x 4 .