# From physical assumptions to classical and quantum Hamiltonian and Lagrangian particle mechanics

## TL;DR

This paper derives classical and quantum particle mechanics from fundamental physical assumptions, showing how different assumptions lead to Hamiltonian and Lagrangian formulations.

## Contribution

It introduces a unified framework deriving classical and quantum mechanics from simple physical principles, clarifying their foundational differences.

## Key findings

- Classical mechanics emerges from infinitesimal reducibility assumption.
- Quantum mechanics arises from irreducibility assumption.
- Lagrangian mechanics results from the kinematic equivalence assumption.

## Abstract

The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible---specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts---will give us a classical Hamiltonian system. Assuming the system is irreducible---specifying the state and the dynamics of the whole system tells us nothing about the state and the dynamics of its substructure---will give us a quantum Hamiltonian system. Assuming kinematic equivalence, that studying trajectories is equivalent to studying state evolution, will give us Lagrangian mechanics and limit the form of the Hamiltonian/Lagrangian to the one with scalar and vector potential forces.