From physical principles to relativistic classical Hamiltonian and Lagrangian particle mechanics

TL;DR

This paper derives relativistic classical particle mechanics from fundamental assumptions, linking mathematical and physical concepts to provide a foundation compatible with future quantum and field theories.

Contribution

It introduces a novel derivation of relativistic Hamiltonian and Lagrangian mechanics based on assumptions of reducibility, determinism, and kinematic equivalence.

Findings

Classical mechanics can be derived from fundamental assumptions.

The approach links concepts across mathematics and physics.

Provides a foundation compatible with quantum mechanics and field theories.

Abstract

We show that classical particle mechanics (Hamiltonian and Lagrangian consistent with relativistic electromagnetism) can be derived from three fundamental assumptions: infinite reducibility, deterministic and reversible evolution, and kinematic equivalence. The core idea is that deterministic and reversible systems preserve the cardinality of a set of states, which puts considerable constraints on the equations of motion. This perspective links different concepts from different branches of math and physics (e.g. cardinality of a set, cotangent bundle for phase space, Hamiltonian flow, locally Minkowskian space-time manifold), providing new insights. The derivation strives to use definitions and mathematical concepts compatible with future extensions to field theories and quantum mechanics.