Classical Mechanics

Key idea: we study a finite number of freedom degrees. If you want to introduce locality and causality you need to use Classical Field Theory, what I think is tightly related to Continuum Mechanics.

Several approaches:

A personal approach: axiomatic Newtonian mechanics

One approach: Lagrangian Mechanics

Another approach: Hamiltonian mechanics

One more approach: Koopman-von Neumann classical mechanics.

See this comparison with Quantum mechanics.