Abstract:
Our solution to the Jacobi problem of finding separation variables
for natural Hamiltonian systems $H=\tfrac12 p^2+V(q)$ is explained in the first part of
this review. It has a form of an effective criterion that for any
given potential $V(q)$ tells whether there exist suitable separation
coordinates $x(q)$ and how to find these coordinates, so that the
Hamilton–Jacobi equation of the transformed Hamiltonian is separable.
The main reason for existence of such criterion is the fact that for
separable potentials $V(q)$ all integrals of motion depend quadratically
on momenta and that all orthogonal separation coordinates stem from
the generalized elliptic coordinates. This criterion is directly
applicable to the problem of separating multidimensional stationary
Schrödinger equation of quantum mechanics.
Second part of this work provides a summary of theory of quasipotential,
cofactor pair Newton equations $\ddot{q}=M(q)$ admitting $n$ quadratic integrals of motion.
This theory is a natural generalization of theory of separable potential
systems $\ddot{q}=-\nabla V(q)$. The cofactor pair Newton equations admit a
Hamilton–Poisson structure in an extended $2n+1$ dimensional phase space
and are integrable by embedding into a Liouville integrable system.
Two characterizations of these systems are given: one through a Poisson
pencil and another one through a set of Fundamental Equations. For a
generic cofactor pair system separation variables have been found and
such system have been shown to be equivalent to a Stäckel separable
Hamiltonian system. The theory is illustrated by examples of driven
and triangular Newton equations.
Keywords:separability, Hamilton–Jacobi equation, Poisson structures, integrability, Hamiltonian system, Newton equation.
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