According to Lagrange’s equations we have that

Now suppose that *L* is independent of *t*, that is, *L _{t}=0*. Then we can use a variant of Beltrami’s identitet (see below for derivation) and rewrite the equations as

which by integration becomes

where *C* is a constant. This is called a *conservation law *and the quantity

is called the *Hamiltonian* of the system. It normally represents the total energy of the system. The analysis above thus means that if *L* is time independent then the totla energy of the system is preserved.

**Derivation:**

since