1.Introduction

The standard method for solving the Hamilton-Jacobi equation is that of separation of variables, and, therefore, it is interesting to know the conditions for the existence of complete separable solutions for this equation. A partial answer to this question is provided by the Stäckel theorem (see, e.g., Refs.^1-4), which gives necessary and sufficient conditions for the existence of complete separable solutions of the Hamilton-Jacobi equation for certain orthogonal systems (here orthogonal means that the kinetic energy has the form (1/2)∑i=1nq̇i2/ci, where the c_i are functions of the generalized coordinates only).

A more restricted class of systems than those included in the Stäckel theorem are the Liouville systems (see, e.g., Refs.^1,2,4). In a recent paper ⁵, it is shown that the form of the Lagrangian of the Liouville systems can be obtained starting from a sum of one-dimensional Lagrangians expressed in terms of a fictitious (or local) time, which replaces the real time. The aim of this paper is to show that, in a similar way, the conditions of the Stäckel theorem are related to several fictitious times.

In Sec.2, the Stäckel theorem is enunciated, following Refs. ^1,2, and we find a general expression for 𝑛 constants of motion possessed by any Lagrangian satisfying the conditions of the Stäckel theorem. These constants of motion are essentially the separation constants appearing in the solution of the Hamilton-Jacobi equation. In Sec. 3, we show that the motion of a system satisfying the conditions of the Stäckel theorem can be viewed as the composition of n independent one-dimensional systems by introducing a possibly different fictitious (or local) time for each coordinate q_i.

2.The Stäckel theorem

The Stäckel theorem is applicable to a system described by a Lagrangian of the form

where the functions c_i may depend on the n generalized coordinates q1,q2,…,qn only, and w_i is a function of the single variable q_i (throughout this paper, the summation convention is not employed; all the sums are explicitly indicated). According to this theorem, the Hamilton-Jacobi equation corresponding to (1) admits separable solutions in this coordinate system if and only if there exists a non-singular n x n matrix (uij), where uij is a function of q_i only (that is, the entries in the first row of (uij) are functions of q₁ only, and so on) such that

This last condition implies that if (ũij) is the inverse of the matrix (uij), then

For example, for a charged particle in the field of a point dipole, in terms of the spherical coordinates (r,θ,ϕ), the natural Lagrangian is

where k is a constant. This Lagrangian has the form (1) with

c1=1m,  c2=1mr2,  c3=1mr2sin2θ,

if we take (q1,q2,q3)=(r,θ,ϕ), and we can choose (though this is not the only possibility)

w1(r)=0,  w2(θ)=mkcosθ,  w3(ϕ)=0.

Then, a matrix (uij) that satisfies condition (2) is

(Note that the first row of this matrix is a function of r only, the second row is a function of θ only, and the third row is a function of ϕ only. Contrariwise to the claim in Ref. ⁶, it is a straightforward matter to verify if the conditions of the Stäckel theorem are satisfied; see also the example at the end of this section).

Thus, the Stäckel theorem guarantees that the Hamilton-Jacobi equation for this problem, in these coordinates, admits complete separable solutions (see, e.g., Ref.⁷). However, instead of exhibiting this separability, we shall consider the more elementary problem of solving the Lagrange equations: Substituting the Lagrangian (4) into the Lagrange equations we obtain

The last of these equations means that the quantity inside the parentheses is a constant of motion:

where l3 is a constant (that represents the z-component of the angular momentum and the momentum conjugate to ϕ). Making use of this relation in order to eliminate ϕ̇ from (7), we obtain

Multiplying this equation by 2mr2θ̇, we find that

0=2mr2θ˙ddt(mr2θ˙)-2l32cos⁡θθ˙sin3⁡θ-2mksin⁡θθ˙=ddt[mr2θ˙]2+l32sin2⁡θ+2mk cos⁡θ,

which yields a second constant of motion:

(when k = 0, 𝑀 reduces to the square of the angular momentum).

With the aid of Eqs. (9) and (11) now we can eliminate ϕ̇ and θ̇ from Eq. (6); the result is

(not only ϕ̇ and θ̇ were eliminated, ϕ and θ also disappeared) hence, multiplying by ṙ,

0=r˙ddt(mr˙)-Mr˙mr3=ddt(m2r˙2+M2mr2).

This means that the expression inside the parentheses is also a constant of motion, which turns out to be the total energy [see Eqs. (4), (9) and (11)]:

where E is a third constant of motion (whose existence follows directly from the fact that the Lagrangian does not depend explicitly of the time).

Thus, despite the mixture of the coordinates in Eqs. (6)-(8), miraculously, one obtains three first-order differential equations or, equivalently, three first integrals [Eqs. (9), (11), and (13)]. As we shall show now, this is a consequence of the fulfillment of the conditions of the Stäckel theorem, and a similar reduction can be obtained for any Lagrangian of the form (1), if the conditions of the Stäckel theorem are satisfied.

Proposition. For a Lagrangian of the form (1), satisfying the conditions of the Stäckel theorem, the 𝑛 quantities

are constants of motion. In particular, a₁ is the Jacobi integral ∑i=1nq̇i2/2ci+ciwi.

Proof. Making use of the definition ([p1]), one finds that the time derivative of a_i is

where we have made use of the fact that w_j and ujk are functions of the single variable q_j and that, for any non-singular matrix A, dA-1=-A-1(dA)A-1. On the other hand, substitution of the Lagrangian (1) into the Lagrange equations gives, for j=1,2,…,n,

Taking the partial derivative with respect to q_j on both sides of Eq. (2), one obtains

which implies that

Substituting Eqs. (16) and (18) into (15) one finds that, in effect, dαi/dt=0.

Finally, with the aid of Eqs. (14) and (3) we see that

α1=∑j=1ncj12q˙jcj2+wj,

which is the Jacobi integral corresponding to the Lagrangian (1).

Thus, we have the expression for n functionally independent constants of motion, which reduce the equations of motion to quadratures. In fact, Eqs. (14) can be inverted to give

(Remember that w_j and ujk are functions of the single variable q_j.)

Going back to the case of the Lagrangian (4), one readily finds that the inverse of the matrix (5) is given by

(ũij)=1/m1/mr21/mr2sin2θ00-1011/sin2θ

[cf. Eq. (3)] and that the constants of motion (14) are related to the constants of motion previously obtained, (9), (11), and (13), by a₁ = E, α2=-(1/2)l32, and α3=(1/2)M.

Another illustrative example is given by the Lagrangian

where (u,v,ϕ) are parabolic coordinates, which are related to the Cartesian ones by x=uvcosϕ, y=uvsinϕ,z=(1/2)(u2-v2), and k, γ are constants. (This Lagrangian corresponds to a charged particle of mass m, in the field of a point charge placed at the origin, and a uniform field in the z-direction.) Clearly, in this example,

c1=1m(u2+v2)=c2,  c3=1mu2v2,

and the one-variable functions w_j can be taken as

w1(u)=-mk-mγ2u4,  w2(v)=-mk+mγ2v4,w3(ϕ)=0.

Equation (2) is satisfied by the matrix

(uij)=mu21/u21mv21/v2-10-10,

and, therefore, the conditions of the Stäckel theorem are met.

The matrix (ũij) is

1m(u2+v2)111/u2+1/v200-m(u2+v2)mv2-mu2m(v2/u2-u2/v2),

and, from Eq. (14), we have immediately the two constants of motion (apart from the total energy a₁),

α2=-pϕ22,α3=m22[u2+v2v2u̇2-u2v̇2+u2v2v2-u2ϕ̇2]-mγ2u2v2+mku2-v2u2+v2,

where pϕ is the momentum conjugate to ϕ (which is conserved as a consequence of the fact that 𝜙 is an ignorable coordinate). The constant of motion α₃ is not related to obvious symmetries and, when γ is set equal to zero, reduces to minus the z-component of the Laplace-Runge-Lenz vector. (Of course, as is well known, any constant of motion corresponds to a one-parameter group of symmetries of the Hamiltonian in the phase space; however, α₃ is not related to a variational symmetry of the Lagrangian; in fact, the Laplace-Runge-Lenz vector is the standard example of a constant of motion associated with a hidden or accidental symmetry.)

In this case, Eqs. (19) amount to

12m(u2+v2) u̇2-mk-mγ2u4=α1mu2+α2u2+α3,12m(u2+v2) v̇2-mk+mγ2v4=α1mv2+α2v2-α3,12(mu2v2 ϕ̇)2=-α2

[cf. Eqs. (9), (11) and (13)]. These equations are equivalent to those obtained using the Hamilton-Jacobi equation.

3.Local times

Even though Eqs. (9), (11) and (12) contain the time derivative of a single coordinate, Eqs. (9) and (11) still contain mixtures of the coordinates. This fact can be conveniently hidden by introducing local (or fictitious) times, τi, defined by

(see, e.g., Refs. ^8,5, and the references cited therein).

Considering again the example provided by the Lagrangian (4), we have the three local times

dτ1=1mdt,dτ2=1mr2dt,dτ3=1mr2sin2⁡θdt.

We see that only dτ1 is an exact differential, and τ1 is, up to a constant factor, the real time. For instance, in terms of τ2, Eqs. (10) and (11) take the form

and

respectively, which determine θ as a function of τ2 alone. Furthermore, one can readily verify that Eq. (22) is the Lagrange equation for the one-dimensional Lagrangian

treating l3 as a constant, and that the left-hand side of Eq.(23) is, essentially, the Jacobi integral corresponding to L₂, multiplied by 2.

Again, this is not an exceptional behavior; making use of Eqs. (21), (14) and (17) we see that the Lagrange equations (16) are equivalent to

(j=1,2,…,n), which, for each value of j, is the Lagrange equation for the one-dimensional Lagrangian

provided that the αk are treated as constants. (Recall that w_j and ujk are functions of the single variable q_j 𝑞 𝑗 .) Making use of Eqs. (26), (21), (2), and (1), one finds that the one-dimensional Lagrangians (26) are related to the original Lagrangian (1) by

[cf. Ref. ⁵, Eq. (21)]. The combination (L+E)dt appears in the principle of least action (instead of Ldt, considered in Hamilton’s principle), which is applicable when E is conserved on the actual and the varied paths (see, e.g., Refs.^9,10).

Since the Lagrangian L_j does not depend on τj, the corresponding Jacobi integral is a constant of motion. Making use of the definitions (26), (21), and (14), one finds that these constants of motion are identically equal to zero. (By contrast, in the case of a Liouville system, where a single local time is enough, the Jacobi integrals associated with the one-dimensional Lagrangians lead to n - 1 nontrivial constants of motion ⁵).

4.Concluding remarks

With the only exception of α1, which is related to the time-independence of the Lagrangian (1), the other constants of motion, α2,…,αn, need not be associated with ignorable coordinates.

As we have shown, the Stäckel theorem, which is related to the separability of the Hamilton-Jacobi equation, turns out to be directly relevant in the Lagrangian formalism. This fact must not be surprising since, in both formalisms, one is dealing with the same problem. In order to have an analog of the Stäckel theorem in the Lagrangian formalism, without reference to the Hamilton-Jacobi equation, it would be necessary to have an analog of the concept of separability.