Lagrangian mechanics

A formulation of classical mechanics in which motion follows from a system’s Lagrangian, usually kinetic energy minus potential energy, through the principle of stationary action.

Principle of stationary action: A physical system follows a path that makes the action stationary under small variations of that path. It is the variational rule from which Lagrangian equations of motion are derived.
Calculus of variations: The mathematical study of extrema of functionals, which map functions to numbers. Lagrangian mechanics applies variational methods to the action integral.
Double pendulum: A mechanical system of two pendulums connected end to end, with nonlinear coupled motion. Its angular coordinates and coupled energies make a compact Lagrangian model natural.
Newtonian mechanics: Classical mechanics formulated through forces and Newton’s laws of motion. It derives trajectories from force balances rather than stationary action.
Conservation law: A statement that a physical quantity remains constant during evolution under specified conditions. Continuous symmetries of the action generate conserved quantities in the resulting motion.
Euler–Lagrange equation: A differential equation giving extrema of a functional that depends on a function and its derivatives. Applying it to the action yields the equations for generalized coordinates.
Generalized force: A force component defined with respect to a generalized coordinate, doing work through its corresponding displacement. It extends the equations to include nonconservative forces and applied inputs.
Rigid body dynamics: The study of motion and forces of bodies whose internal distances remain fixed. Lagrangian coordinates handle rotational degrees of freedom and constraints systematically.
Hamiltonian mechanics: A formulation of classical mechanics using coordinates, conjugate momenta, and a Hamiltonian function. It replaces velocity-based dynamics with phase-space evolution after a Legendre transform.
Canonical momentum: The momentum conjugate to a generalized coordinate, defined as the Lagrangian’s derivative with respect to its velocity. Its definition provides the bridge from Lagrangian equations to Hamiltonian mechanics.

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