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.
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.