Constants of motion associated with canonoid transformations for non-autonomous systems

Abstract

We consider non-autonomous systems of ordinary differential equations that can be expressed in Hamiltonian form in terms of two different coordinate systems, not related by a canonical transformation. We show that the relationship between these coordinate systems leads to a, possibly time-dependent, tensor field, $S^{\alpha}_{\beta}$, whose eigenvalues are constants of motion. We prove that if the Nijenhuis torsion tensor of $S^{\alpha}_{\beta}$ is equal to zero then the eigenvalues of $S^{\alpha}_{\beta}$ are in involution, and that these eigenvalues may be in involution even if the Nijenhuis tensor is not zero.