  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.
