Lifts of Time Dependent Complex Hamiltonian Mechanical Systems

TL;DR

This paper extends the geometric framework of time-dependent complex Hamiltonian systems by developing higher order lifts of structures on contact manifolds, providing new insights into their geometric properties.

Contribution

It introduces higher order lifts of geometric structures and Hamiltonian equations on contact manifolds, advancing the geometric understanding of complex Hamiltonian systems.

Findings

Higher order vertical and complete lifts of Hamiltonian equations are formulated.

Extended contact manifolds are defined with lifted structures.

The geometric meaning of Hamiltonian systems is clarified.

Abstract

In this study, firstly, the k-th order extension of complex product manifold is consid- ered. Then the higher order vertical, complete lifts of geometric structures on product manifold to its extended spaces are given. Also higher order lifts of tensor field of type (1,1) are presented. And then extended contact manifolds are defined. Finally higher order vertical and complete lifts of time dependent complex Hamiltonian equations on contact manifold to its extensions are introduced. In conclusion, geometric meaning of Hamiltonian mechanical systems is discussed.