The Approximate Mei Symmetries and their Applications

Abstract

The Mei symmetries, a class of symmetries, correspond to conserved quantities just like Noether symmetries. However, the two sets of symmetries result in different conserved quantities. The formulation of first-order approximate Mei symmetries of the associated perturbed Lagrangian is presented in this thesis. Theorems and determining equations are given to evaluate approximate Mei symmetries, as well as approximate Mei invariants relative to each symmetry of the associated Lagrangian. The stated approach is illustrated using the linear equation of motion of a damped harmonic oscillator (DHO). Furthermore, a method for determining approximate Mei symmetries and invariants of the perturbed Hamiltonian is described, which can be employed in various fields of study where approximate Hamiltonian are considered. The Legendre transformation is used to convert Lagrangian into Hamiltonian. The results are provided as theorems with proof. To elaborate on the method of determining these symmetries and the related Mei invariants, a basic example of DHO is presented. Moreover, a comparison of approximate Mei symmetries with approximate Noether symmetries is provided. The comparison indicates that both sets of symmetries have only one common symmetry. Furthermore, the number of approximate Mei symmetries exceeds the number of approximate Noether symmetries. As a result, the remaining symmetries in the two sets correspond to two distinct sets of conserved quantities. The Mei symmetries associated with the Lagrangian and Hamiltonian of DHO are compared. First-order approximate Mei symmetries of the geodesics Lagrangian are determined as an application of approximate Mei symmetries for particular classes of ppwave spacetimes. These classes of pp-wave spacetimes include plane wave spacetimes in which (i).A(u) = α 2 (ii). A(u) = αu−2 (iii). A(u) = α 2u −4 and for pp-wave space times (iv). h = αxn (where h is called scale factor and α is a constant). After that, approximate Mei invariants are calculated corresponding to each case.