Lagrangian - Explanation

Explanation

The Lagrangian in many classical systems is a function of generalized coordinates q_i and their velocities dq_i/dt. These coordinates (and velocities) are, in their turn, parametric functions of time. In the classical view, time is an independent variable and q_i (and dq_i/dt) are dependent variables as is often seen in phase space explanations of systems. This formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a manifold. And the dependent variables q are replaced by φ the value of a field at that point in spacetime so that the equations of motion are obtained by means of an action principle, written as:

where the action, is a functional of the dependent variables φ_i(s) with their derivatives and s itself

and where s = { sα} denotes the set of n independent variables of the system, indexed by α = 1, 2, 3,..., n. Notice L is used in the case of one independent variable (t) and is used in the case of multiple independent variables (usually four: x, y, z, t).

The equations of motion obtained from this functional derivative are the Euler–Lagrange equations of this action. For example, in the classical mechanics of particles, the only independent variable is time, t. So the Euler-Lagrange equations are

Dynamical systems whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as Lagrangian dynamical systems. Examples of Lagrangian dynamical systems range from the classical version of the Standard Model, to Newton's equations, to purely mathematical problems such as geodesic equations and Plateau's problem.