Analytical mechanics

Analytical mechanics formulates classical motion through generalized coordinates and principles such as stationary action. It replaces force-by-force vector equations with scalar or variational methods that adapt to constraints and coordinate systems.

Lagrangian mechanics: A formulation of classical mechanics that derives equations of motion from a system’s Lagrangian and generalized coordinates. It obtains motion equations by making the action stationary.
Generalized coordinates: Independent variables that specify a mechanical system’s configuration, often chosen to match its constraints. They let analytical mechanics describe motion without relying on Cartesian coordinates.
Joseph-Louis Lagrange: An Italian-French mathematician and astronomer whose work developed variational methods and Lagrangian mechanics. His 1788 treatise systematized mechanics through generalized coordinates and kinetic energy.
Rigid body dynamics: The study of the translational and rotational motion of bodies treated as undeformable. Euler angles and generalized coordinates simplify rotation equations for rigid bodies.
Newtonian mechanics: Classical mechanics based on Newton’s laws relating forces, mass, and acceleration. It derives trajectories from local force equations rather than variational structure.
Hamiltonian mechanics: A formulation of mechanics that describes motion in phase space using a Hamiltonian function. It recasts dynamics in terms of coordinates, momenta, and energy.
Configuration space: The space whose points represent all possible configurations of a mechanical system. A system’s path through this space is the object varied in action-based formulations.
Mécanique analytique: Joseph-Louis Lagrange’s 1788 treatise presenting mechanics through analytical methods and generalized coordinates. This work gave the field its name and a systematic mathematical presentation.
Celestial mechanics: The study of the motions and gravitational interactions of astronomical bodies. Analytical methods organize orbital motion and perturbations in multi-body systems.
D'Alembert's principle: A principle expressing constrained dynamics as a balance of virtual work and inertial forces. It bridges force-based mechanics and the variational formulation of constrained motion.

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