Ermakov-Lewis dynamic invariants with some applications

TL;DR

This paper explores Ermakov-Lewis invariants, their theoretical foundations, and diverse applications in physics, including cosmology and optics, highlighting their connection to Noether's theorem and potential generalizations.

Contribution

It provides a comprehensive analysis of Ermakov-Lewis invariants, linking them to symmetries and extending their applicability across different physical contexts.

Findings

Ermakov-Lewis invariants are connected to Noether's theorem.

Applications demonstrated in cosmology and optics.

Potential for generalizations of Ermakov's method.

Abstract

Contents: Introduction(3).The method of Ermakov(4).The method of Milne(7). Pinney's result(8).Lewis' results(8). The interpretation of Eliezer and Gray(14). The connection of the Ermakov invariant with N\"other's theorem(17). Possible generalizations of Ermakov's method(20). Geometrical angles and phases in Ermakov's problem(22). Application to the minisuperspace cosmology(26). Application to physical optics(42).Conclusions(47). Appendix A: Calculation of the integral of I(48). Appendix B: Calculation of <\hat{H}> in eigenstates of \hat{I}(49).References(50).