Hamiltonian formalism and time transformations in the extended phase space \(T^*(Q \times \mathbb{R})\)
The general theory of Hamiltonian systems has as its natural setting the cotangent bundle, \(T^*Q\), over the coordinate manifold \(Q\). This includes time-dependent symplectic (canonical) transformations. However there are instances where one wishes to exchange the time and phase space variables and the present paper approaches this problem by using the symplectic structure of the extended phase space \(T^*(Q\times\mathbb{R})\), where \(\mathbb{R}\) represents the time coordinate. In order to restrict attention to systems which arise from bona fide non-autonomous Hamiltonian systems on \(T^*Q\) the authors introduce the idea of a \(\mu\)-projectable vector field on \(T^*(Q\times\mathbb{R})\) and from these select those that are locally Hamiltonian for the extended symplectic form. This restricts the form of the Hamiltonian function on \(T^*(Q\times\mathbb{R})\) and allows one to identify a foliation of \(T^*(Q\times\mathbb{R})\) by contact manifolds isomorphic to \(T^*Q\times\mathbb{R}\). Canonical transformations on \(T^*(Q\times\mathbb{R})\) must preserve \(T^*Q\times\mathbb{R}\) as the zero section. The authors give the form that such a transformation must take. A canonoid transformation, on the other hand, is one that preserves the Hamiltonian nature of a particular system rather than all systems and these are appropriately defined in the extended case. Some results concerning constants of motion are given before the authors focus on ``non-standard'' canonoid transformations, namely those where the new time depends upon the \(Q\)'s and the \(P\)'s. They give two examples of their use in simplifying non- autonomous Hamiltonian systems.
scientific article; zbMATH DE number 89987
scientific article