A differential equations approach to Hamiltonian systems (Q1853826)

The authors explore the geometric model for time-dependent Hamiltonian systems which starts from a cosymplectic manifold \(M\). Constraints correspond to a given submanifold \(N\subset M\) and have to be handled in such a model via a Gotay-Nester type algorithm. Assuming \(M\) is fibred over a one-dimensional base and the Reeb vector field is transversal to the fibration, Hamiltonian differential equations are identified with a fibred submanifold of the first-jet bundle \(J_1M\). The main result of the paper is the proof of equivalence of the aforementioned constraint algorithm with the completion algorithm for obtaining formally integrable differential equations in the sense of the Cartan-Kuranishi theorem. As a kind of application, some aspects of a covariant formulation of classical mechanics are discussed. The paper ends with a brief outline of the difficulties one has to address for a generalization to field theories.