On reciprocal equivalence of St"ackel systems

In this paper we ivestigate St"ackel transforms between different classes of parameter-dependent St"ackel separable systems of the same dimension. We show that the set of all St"ackel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single St"ackel transform. We also give an explicit formula relating solutions of two St"ackel-related systems. These results show in particular that any two geodesic St"ackel systems are St"ackel equivalent in the sense that it is possible to transform one into another by a single St"ackel transform. We also simplify proofs of some known statements about multiparameter St"ackel transform.

💡 Research Summary

The paper investigates Stäckel transforms between families of parameter‑dependent Stäckel‑separable Hamiltonian systems that share the same number of degrees of freedom. A Stäckel system is a Hamiltonian whose kinetic part is quadratic in the momenta and whose potential depends linearly on a set of parameters; such systems admit separation of variables in the Hamilton‑Jacobi equation. The authors first formalize the notion of a Stäckel transform as a linear change of the parameter vector together with a corresponding linear recombination of the potentials, governed by a non‑singular matrix (T). Under this transformation the Stäckel matrix (S) of one system is related to the Stäckel matrix (\tilde S) of the transformed system by (\tilde S = T^{-1} S T). This matrix relation guarantees that the separation relations, conserved quantities, and the underlying Poisson‑Lagrange structures are preserved up to a linear re‑labeling.

The central result, Theorem 1, proves that all Stäckel systems of a given dimension fall into equivalence classes defined by the invertibility of their Stäckel matrices. Within each class any two members can be connected by a single Stäckel transform. The proof relies on the fact that the Stäckel matrix encodes the full set of separation relations; if the matrix is non‑degenerate, its inverse provides the coefficients needed to express any other admissible set of parameters and potentials as a linear combination of the original ones. Consequently, the space of Stäckel systems is partitioned into disjoint orbits of the group (GL(m,\mathbb{R})) acting via Stäckel transforms, where (m) is the number of independent parameters.

A particularly striking corollary concerns geodesic Stäckel systems, i.e., systems without a potential term. Although such systems lack explicit parameters, their Stäckel matrices still possess a full rank and thus generate a non‑trivial transformation group. The authors demonstrate that any two geodesic Stäckel systems are Stäckel‑equivalent: by selecting an appropriate matrix (T) one can map the kinetic coefficients of one system onto those of another, effectively showing that all geodesic Stäckel metrics belong to a single equivalence class. This result refines earlier statements that any two geodesic Stäckel metrics are related by a sequence of transformations, by showing that a single transformation suffices.

The paper also provides an explicit formula linking the solutions (action functions) of two Stäckel‑related systems. If (S(q,\lambda)) solves the Hamilton‑Jacobi equation for the original system, then the solution (\tilde S(q,\tilde\lambda)) for the transformed system is given by
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