Approximate Mei Symmetries and Invariants of the Hamiltonian

Umara Kausar,Tooba Feroze

doi:10.3390/math9222910

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.

Highlights

Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries
A simple example of mechanics is considered to elaborate the method of finding these symmetries and the related Mei invariants
We focus on the formulation of approximate Mei symmetries and invariants

Summary

Here Ea is called the Euler operator defined as d

Proof of Theorem 1. To prove the above relations, Equations (1) and (2), apply the first order prolongation of Z, i.e., Z[1] = Z0 + eZ1 on H = H0 + eH1 to have Z[1] H = (Z0 + eZ1 )( H0 + eH1 ).

Neglecting the higher order terms in e yields

Rearranging the above expression as a a a a

Conclusions and Discussions

Approximate Mei Symmetries