Intrinsic geometric characterization of dynamical systems

Identify intrinsic geometric objects on the underlying manifold or bundle that uniquely characterize a dynamical system, in the sense of providing a canonical geometric invariant whose specification determines the system’s dynamics among all dynamically equivalent variational and geometric formulations.

Background

The paper develops a method that, starting from a variational principle, constructs geometric structures (e.g., cosymplectic, cocontact) whose equations of motion reproduce the same set of solutions. In several steps of the method, auxiliary choices (such as vector fields) lead to different but dynamically equivalent geometric structures, and the authors do not establish uniqueness of the resulting geometry.

This motivates an explicit unresolved question: whether there exist intrinsic geometric objects—independent of arbitrary choices—that uniquely and canonically characterize a dynamical system’s dynamics across all equivalent formulations. Establishing such objects would provide a unifying classification and remove ambiguities in assigning geometries to variational problems.