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
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 qi.
2.The Stäckel theorem
The Stäckel theorem is applicable to a system described by a Lagrangian of the form
where the functions ci may depend on the n generalized coordinates
This last condition implies that if
For example, for a charged particle in the field of a point dipole, in terms of the
spherical coordinates
where k is a constant. This Lagrangian has the form (1) with
if we take
Then, a matrix
(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
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
Multiplying this equation by
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
(not only
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
Proof. Making use of the definition ([p1]), one finds that the time derivative of ai is
where we have made use of the fact that wj and
Taking the partial derivative with respect to qj on both sides of Eq. (2), one obtains
which implies that
Substituting Eqs. (16) and (18) into (15) one finds that, in effect,
Finally, with the aid of Eqs. (14) and (3) we see that
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 wj and
Going back to the case of the Lagrangian (4), one readily finds that the inverse of the matrix (5) is given by
[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,
Another illustrative example is given by the Lagrangian
where
and the one-variable functions wj can be taken as
Equation (2) is satisfied by the matrix
and, therefore, the conditions of the Stäckel theorem are met.
The matrix
and, from Eq. (14), we have immediately the two constants of motion (apart from the total energy a1),
where
In this case, Eqs. (19) amount to
[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,
(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
We see that only
and
respectively, which determine θ as a function of
treating
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
(
provided that the
[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 Lj does not depend on
4.Concluding remarks
With the only exception of
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.