Lagrangian mechanics

In Newtonian mechanics, motion is often computed by resolving forces along spatial axes. Lagrangian mechanics offers an alternative: describe the system by coordinates suited to its degrees of freedom, then form a Lagrangian L. For a common class of conservative systems, L = T − V, where T is kinetic energy and V is potential energy. The Lagrangian is not generally the total energy; it is a function used to derive the motion equations.

A generalized coordinate can be a distance, an angle, or another variable that specifies a system's configuration. Choosing coordinates that already respect geometric constraints can reduce the number of unknowns. For each coordinate qᵢ, the Euler–Lagrange equation is d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0 for a system with no nonconservative generalized force; such forces can be included on the right-hand side when present. These equations describe how the coordinates change over time. The formalism is not limited to the simple T − V case, and the appropriate Lagrangian depends on the physical system.

The equations can be derived from the principle of stationary action. The action is the time integral of the Lagrangian along a possible history of the system. Among nearby histories that share fixed endpoints, the physical history makes the action stationary to small variations. Stationary does not mean that the action must always be a minimum; it can be a maximum or a saddle point. This variational viewpoint gives a compact route from a system's Lagrangian to its equations of motion.

Lagrangian mechanics is particularly convenient for linked bodies, pendulums, rolling systems, and other problems where constraints make direct force bookkeeping cumbersome. A well-chosen set of generalized coordinates incorporates constraints into the description, rather than requiring a separate force equation for each constraint. MIT OpenCourseWare illustrates the approach in a cart-and-pendulum recitation. Lagrangian methods also provide a foundation for more advanced formulations of physics, although classical Lagrangian mechanics has a defined domain and does not replace every other method.