Geometrical Formulation of Relativistic Mechanics

Abstract

The relativistic Lagrangian in presence of potentials was formulated directly from the metric, with the classical Lagrangian shown embedded within it. Using it we formulated covariant equations of motion, a deformed Euler-Lagrange equation, and relativistic Hamiltonian mechanics. We also formulate a modified local Lorentz transformation, such that the metric at a point is invariant only under the transformation defined at that point, and derive the formulae for time-dilation, length contraction, and gravitational redshift. Then we compare our formulation under non-relativistic approximations to the conventional ad-hoc formulation, and we briefly analyze the relativistic Lienard oscillator and the spacetime it implies.

24 pages

References in corpus (10)

• Contact Hamiltonian Mechanics
• The Jacobi-metric for timelike geodesics in static spacetimes
• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Bertrand spacetimes as Kepler/oscillator potentials
• Jacobi-Maupertuis-Eisenhart metric and geodesic flows
• Ultra-relativistic spinning particle and a rotating body in external fields
• World-line geometry probed by fast spinning particle
• On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications
• Jacobi-Maupertius metric and Kepler equation

Cited by in corpus (6)

• Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect
• Destructive relativity
• Transient chaos under coordinate transformations in relativistic systems
• Ergodic decay laws in Newtonian and relativistic chaotic scattering
• A Transfer Operator Approach to Relativistic Quantum Wavefunction
• More on Jacobi metric: Randers-Finsler metrics, frame dragging and geometrisation techniques