Approximate Mei Symmetries and Invariants of the Hamiltonian

Abstract

It is known that corresponding to each Noether symmetry there is a conserved quantity. Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries. However, the two sets of symmetries may give different conserved quantities. In this paper, a procedure of finding approximate Mei symmetries and invariants of the perturbed/approximate Hamiltonian is presented that can be used in different fields of study where approximate Hamiltonians are under consideration. The results are presented in the form of theorems along with their proofs. A simple example of mechanics is considered to elaborate the method of finding these symmetries and the related Mei invariants. At the end, a comparison of approximate Mei symmetries and approximate Noether symmetries is also given. The comparison shows that there is only one common symmetry in both sets of symmetries. Hence, rest of the symmetries in the two sets correspond to two different sets of conserved quantities.

First-Order Approximate Mei Symmetries and Invariants of the Lagrangian

“…At the end, approximate Mei symmetries and invariants of Lagrangian of DHO are obtained as an example. The exact and approximate Mei symmetries and invariants of DHO corresponding to the Hamiltonian are already calculated in [14]…”

“…Furthermore, a relation between approximate Noether symmetries and collineations of the underlying manifolds is also discussed here. The authors of [14] discussed the formulation of approximate Mei symmetries and invariants relative to a Hamiltonian. The determining equations, criterion and theorems to construct Mei symmetries of the Hamiltonian is presented here.…”