Let be the -variation (which is distinct from -variation ) of the Lagrangian L, so

(1)

where is the varied path

(2)

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

	(3)

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

(4)

since

(5)

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

(6)

so

			(7)

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.

(8)

But the definition of the action integral is

(9)

(10)

so

(11)

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

(12)

If the potential is not velocity dependent, then

(13)

so

(14)

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

(15)

which is a generalization of Fermat's principle.

Fermat's Principle