Although the Hamiltonian formulation appears to accord equal status to coordinates and momenta, it is clear that these cannot be entirely independent. In fact, it is clear that the coordinates and momenta must be connected by the fact that the total energy is conserved. In the Hamiltonian formulation, this means that
where E is a constant. In order to show that this is true, we only need to show that dH/dt = 0. The proof is straightforward using the chain rule:
Then, using Hamilton's equations to substitute in for
and 
, this gives
In addition, the time derivative of any function of coordinates and momenta, A(p,q) is
Let us define a bracket:
which is known as the Poisson Bracket. In general, the Poisson bracket between any two quantities A and B is
This is the classical analog of the quantum mechanical commutator, [A,B] = AB-BA. Thus, we see that the time evolution of any quantities A is determined by the Poisson bracket of A with the Hamiltonian:
Thus, if A is conserved, dA/dt = 0. However, this means that the Poisson bracket between A and H must vanish as well. That is, if A is conserved, then
Example: Consider a system of N particles in one dimension with only internal interactions (that is, interactions among the particles, no external interactions). An example would be a set of particles interacting via springs with nearest neighbor interactions only. The condition of no external interactions means that the sum of all the forces is 0 by Newton's third law:
The Hamiltonian is given by
Show that the total momentum
is conserved. In order to show this, all we need to do is compute 
.
Hence, dP/dt = 0, which we were able to show without computing or solving any equations of motion.
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Mark Tuckerman