| dc.description.abstract | **Please note that the full text is embargoed until 05/13/2024** ABSTRACT: The three known families of Lagrangians are standard, non-standard, and null Lagrangians. While Lagrangians are widely used in Physics for their ability to characterize physical systems, most of this work uses only standard Lagrangians. As such, standard Lagrangians have been studied in Physics for around three hundred years; however, non-standard Lagrangians for systems in Physics have been considered only for a few decades, and null have been ignored almost entirely. Although only some Lagrangians are null Lagrangians, all Lagrangians can be categorized as either standard or non-standard Lagrangians, meaning that there exist standard null Lagrangians and non-standard null Lagrangians. My dissertation is devoted to the study of null Lagrangians in Physics, with some additional applications of non-standard Lagrangians.
Non-standard Lagrangians, which are Lagrangians with forms different from standard Lagrangians, have been studied in Physics, but significantly less so than standard Lagrangians. This family of Lagrangians is known for having indiscernible. kinetic and potential energy terms. However, it is shown herein that the non-standard Lagrangian for the Law of Inertia preserves its Galilean invariance, which is notably different from the standard Lagrangian formulation.
Null Lagrangians are a special family of Lagrangians known for yielding identically zero from the Euler-Lagrange equation, and, in this way, not contributing to the equation of motion. Although this special but lesser-known family of Lagrangians has been studied in Mathematics since the 1960s, very little work has been done concerning them outside of this field. Though it might seem that as a result they would be of little use for physical systems, this could not be farther from the truth. The work presented in this dissertation comprises what I hope will become the first step in a larger body of work exploring the role of null and non-standard Lagrangians for physical systems.
I show how the addition of a null Lagrangian is sufficient to introduce force to a system, converting an undriven system to a driven one. In this way, forces naturally arise out of the gauges terms corresponding to these null Lagrangians. A formalism for constructing null Lagrangians for systems in dynamics, along with a generalized approach showing how null Lagrangians and their gauge functions can be linked to known forces, is developed and presented. Further, this formalism for introducing forces is extended to dissipated systems through an application to the Bateman oscillator system. It is then shown how nonlinearities can also be introduced through null Lagrangians, including the Duffing oscillator.
Compelling results from the application of null Lagrangians to physical systems of increasing complexity are presented and discussed. Particular attention is given to applications to oscillators, including the Bateman oscillator system, to illustrate the physical implications of the work. Null Lagrangians are found for equations with special function solutions in mathematical physics, including Bessel functions, and Legendre and Hermite polynomials. The connection between a system’s Galilean invariance and how it behaves under the introduction of specific gauge functions corresponding to null Lagrangians is discussed and explored for multiple equations of key interest in Physics.
As this work is not constrained to classical dynamics, I then show how to formulate physically consistent null Lagrangians for systems in Quantum Mechanics. The Galilean invariance of the Schrödinger Lagrangian is investigated. Null Lagrangians of a similar form to the Schrödinger Lagrangian are presented and whether they can be used to replace the phase factor required for the aforementioned Galilean invariance is discussed.
As we seek to better understand our universe, null Lagrangians have the potential to be a powerful new lens with which to view and investigate physical phenomena; what underlying symmetries might be uncovered using these new tools? The work presented in this dissertation shows that the investigation of null Lagrangians for systems in physics has already yielded exciting results, and that this promising area of research should be further explored. | |
