Lagrangian

The Lagrangian L of a system is simply the kinetic energy minus the potential energy:

L = T - V

This might seem like an uninteresting quantity, but it is the basis of much of classical mechanics. Let L be a function of q and q′, a generalized position variable and its time derivative. The Euler-Lagrange equation is then

∂L/ ∂q = d/dt (∂ L/∂ q′ )

If L is a function of more than one position variable q_i and its time derivative, there is one Euler-Lagrange equation for each q_i. It is left as an exercise to the reader to show that this is equivalent to Newtons Laws of Motion. In practice, it is often easier to solve a problem using the Euler-Lagrange equations, by choosing appropriate generalized coordinates to exploit the symmetries of the system.

The action, denoted by S, is the time derivative of the Lagrangian:

S = ∫ dt L

Let q₀ and q₁ be the position at respective initial and final times t₀ and t₁). Using the [calculus of variations]?, it can be shown that the Euler-Lagrange equations are equivalent to the statement that the trajectory of q between t₀ and t₀ is such that S is minimized. This is written as

δ S = 0