Lagrangiaan

The Lagrangian, sometimes spelled Lagrangiaan in some languages, is a function used to describe the dynamics of a physical system in Lagrangian mechanics. It is typically denoted by L and depends on the generalized coordinates q_i(t), their time derivatives q̇_i(t), and possibly time t. For many mechanical systems, a common form is L = T − V, where T is the kinetic energy and V is the potential energy.

The action S is the time integral of the Lagrangian: S[q] = ∫ L(q_i, q̇_i, t) dt. The actual motion of the system is determined by the principle of stationary action, which states that S is stationary with respect to small variations of the path q_i(t). This leads to the Euler–Lagrange equations: d/dt (∂L/∂q̇_i) − ∂L/∂q_i = 0 for each generalized coordinate i.

In the standard form with kinetic and potential energies, L = T − V. For a single particle of mass m in a potential V(q), L = 1/2 m q̇^2 − V(q). Using generalized coordinates allows constraints to be incorporated naturally; constraints can also be imposed via Lagrange multipliers.

The Lagrangian is closely related to the Hamiltonian H, defined as H = ∑ p_i q̇_i − L with p_i = ∂L/∂q̇_i. The equations of motion in Hamiltonian form are Hamilton’s equations.

Applications extend from classical mechanics to field theory and quantum mechanics. In field theory, the Lagrangian is replaced by a Lagrangian density, and the action principle yields the field equations. In quantum mechanics, the path integral formulation uses exp(iS/ħ). Noether’s theorem connects symmetries of L to conserved quantities. The Lagrangian formalism is named after Joseph-Louis Lagrange, developed in the 18th century.