The Lagrangian L of a system is simply the kinetic energy minus the potential energy:
- L = T - V
Though this seems an uninteresting quantity, 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
- ∂L/ ∂q = d/dt (∂ L/∂ q′ )
If L is a function of more than one position variable qi and its time derivative, there is one Euler-Lagrange equation for each qi. It can be shown 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.
Principle of Least Action
The action, denoted by S, is the time integral 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 equation is equivalent to the statement that the system undergoes the trajectory between t0 and t1 that minimizes the action. This is known as the Principle of Least Action, and written as
- δ S = 0
Thus, instead of thinking about particles accelerating in reponse to applied forces, one might think of them picking out the path with the least possible action.
Feynman 's path integral formulation extends the Principle of Least Action to quantum mechanics. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. This point of view is influential in modern particle physics, and underlies much of quantum field theory.
A quantity related to the Lagrangian is the Hamiltonian, denoted by H. It is equivalent to the total energy function, T + V; formally, H is obtained by performing a Laplace transform on the Lagrangian. The Hamiltonian is a particularly ubiquitous quantity in the usual (not path integral) formulation of quantum mechanics.