A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems

Abstract

The time-scale non-shifted Hamiltonian dynamics are investigated, including both general holonomic systems and nonholonomic systems. The time-scale non-shifted Hamilton principle is presented and extended to nonconservative system, and the dynamic equations in Hamiltonian framework are deduced. The Noether symmetry, including its definition and criteria, for time-scale non-shifted Hamiltonian dynamics is put forward, and Noether's theorems for both holonomic and nonholonomic systems are presented and proved. The non-shifted Noether conservation laws are given. The validity of the theorems is verified by two examples.

Canonical Transformations and Poisson Theory for Dynamics with Non-Standard Lagrangians

“…Musielak [5] constructed Lagrangians for variable-coefficient dissipative dynamics systems, including standard and non-standard Lagrangians. Symmetry and conserved quantity have always been a hot spot in analytical mechanics [6][7][8] . Recently, Zhang and his collaborators have studied Noether theorems for systems with non-standard Lagrangians [9,10] , non-standard Hamiltonians [11] , non-standard Birkhoffians [12] , and Lie symmetry [13,14] , Mei symmetry [15] , first integral and method of reduction [16,17] .…”