We can use the above transformation theory to develop an approach to symmetries that sheds light on the significance of time evolution in classical mechanics. Consider an infinitesimal transformation function of type F2 of the form

where e is an infinitesimal parameter and G is some chosen function of q, P, and t. Then we find from (14.18) that

Given that e is infinitesimal, we deduce that, to lowest order in e,

which can we rewritten in terms of the Poisson brackets as follows:

When a transformation is expressed in this way, we say that the function eG is a generator of the infinitesimal canonical transformation (14.22).

Two important results follow from this analysis.

Conserved quantities

Suppose G has no explicit time dependence, that is, G = G(q, p). Further, suppose that G is a symmetry of the Hamiltonian, that is,

Now the Hamilton-Poisson equation of motion for G is

But both terms on the right-hand side of (14.24) are zero. We then deduce that if G is explicitly independent of time and generates a symmetry of the Hamiltonian, then G is conserved, that is, is constant along true trajectories in phase space.

Translation in time

Suppose we choose the infinitesimal generator eG to be eH, where H is the Hamiltonian itself. Then we have

If now we take e ^ 0, we deduce

but we also know that the Hamilton-Poisson equations of motion are

This leads us to interpret the Hamiltonian as the generator of translations in time, a fundamental insight into the relationship between time and energy.