XL Workshop on Geometric Methods in PhysicsBiałowieża, 2-8.07.2023XII School on Geometry and PhysicsBiałystok, 26-30.06.2023
Stefan Rauch
Driven cofactor systems of Newton equations and separability of time dependent potentials
By Newton equation we mean equation of the form
, , that is acceleration = force
not depending on time or velocities. It is driven when it has the form
, with in which a solution
of the first -subsystem "drives" the force of the second -subsystem, called "driven" equation.
Theory of such equations becomes interesting when there are known one or two energy-like integrals of motion
(which are quadratic in velocities) of cofactor type. If Newton equation admits two such quadratic cofactor integrals then this Newton equation is completely integrable and is separable in a non-standard sense.
If the whole Newton system has only one quadratic cofactor integral then it has further quadratic integrals,
due to special structure of equations. In particular the driving subsystem inherits a quadratic cofactor
integral of motion depending only on -variables.
When a solution of the driving subsystem is known and the driven subsystem force has a potential, then
for one obtains a system generated by time dependent potential .
It appears that such system has separable time-dependent Hamilton-Jacobi equation and is solvable by natural extension of the Stäckel separability procedure. It gives rise to time-dependent separation variables.
This procedure is illustrated by a simple example.