Parameter invariant Lagrangian formulation of Kawaguchi geometry

TL;DR

This thesis develops a parameter-invariant Lagrangian framework on Kawaguchi manifolds, extending Finsler geometry to higher derivatives and parameter spaces, with potential applications in fundamental physics theories.

Contribution

It introduces a novel parameter-invariant Lagrangian formulation on Kawaguchi manifolds, generalizing Finsler geometry to higher order derivatives and multi-dimensional parameters.

Findings

Global Lagrangian constructed for first and second order cases.

Lagrangian remains reparameterisation invariant despite higher order extensions.

Locally, conventional Lagrangians can be reformulated as parameter-independent.

Abstract

This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter extension is also called areal geometry in some references. On Finsler (Kawaguchi) manifold, we can define a reparameterisation invariant 1 (k)-dimensional area by the Hilbert (Kawaguchi) form, which we take as an action. The equation of motion obtained from such action also has the property of reparameterisation invariance. In this framework, the solution manifold of the Euler-Lagrange equation is realised as a submanifold of Finsler/Kawaguchi manifold, and no fibered structure over the parameter space is needed. We also show that for the case of first order k-dimensional parameter space and second order 1-dimensional parameter space, a global Lagrangian could be constructed. For second order k-dimensional parameter space, Lagrangian is not global but it still has the reparameterisation invariant property. Such theory is expected to provide the ideal stage for formulating fundamental theories of physics, especially for cases such as when one needs to consider the mixing of spacetime and field variables. It is shown that locally, any conventional Lagrangian could be reformulated by the parameter independent Lagrangian. Furthermore, the parameter independent property will gives us the freedom of choosing a parameter, which in some cases turns out to be useful in finding solutions and symmetries.