Least Action Principle -- from Eric Weisstein's World of Physics

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Least Action Principle

Let be the -variation Eric Weisstein's World of Math (which is distinct from -variation Eric Weisstein's World of Math) of the Lagrangian L, so

where is the varied path

and and are the initial and final times, respectively. Integrating by parts then yields

where is a generalized coordinate. From Lagrange's equations, the term in brackets vanishes, so

since

where is a generalized momentum. The two variations are connected by

so

where H is the Hamiltonian.

Now, require that

1. L (and therefore H) are not explicit functions of time, so H is conserved.

2. H is conserved along the varied as well as actual path.

3. vanish at the end-points.

These conditions lead to

But the definition of the action integral is

so

If the defining equations for generalized coordinates do not involve time explicitly, then the kinetic energy T is a quadratic function of the s,

If the potential is not velocity dependent, then

so

If there are no external forces, then T is conserved, and

which is a generalization of Fermat's principle.

Fermat's Principle