Researched
Lagrangian & Hamiltonian Mechanics
Lagrange's Mécanique analytique (1788) derives all of mechanics from one quantity; Hamilton (1833) recasts it in positions and momenta, the form that quantum and statistical physics later use.
Open in the interactive tree →Lagrange merged the work of Euler, d'Alembert and others into one general method for any mechanical problem, and he noted that his book contains no diagrams. Hamilton's 1833 reformulation replaced velocities by momenta, which put position and momentum on the same footing. The Hamiltonian is the starting point of the Schrödinger equation, the Poisson bracket becomes the quantum commutator, and the same action principle carries over to relativity and to field theories.
Prerequisites
- Calculus (Newton & Leibniz)1684The calculus of variations supplies the method
- Newtonian Mechanics1687Lagrange recasts Newton's laws as one variational principle
Unlocks
- Statistical Mechanics1877Boltzmann and Gibbs describe gases as points in a phase space of positions and momenta
- Noether's Theorem1918Noether's theorem is stated for systems described by a Lagrangian
- Quantum Mechanics1925Canonical quantization and the Schrödinger equation start from the Hamiltonian form