# Quantum Mechanics

## 2. A First Approach to Classical Mechanics

### 2.5 Poisson Brackets and Hamiltonian Mechanics

Let and be two smooth functions on , where an element of is thought of as a pair , with

• representing position of a particle representing the momentum of a particle

Then the Poisson bracket of and , denoted is the function on given by

For all smooth functions , and on we have the following:

1. for all
2. Jacobi identity:

The position and momentum functions satisfy the following Poisson bracket relations:

If a particle in has the usual sort of energy function (kinetic energy plus potential energy), we have

Wit the Hamiltonian, and as usual, having , we can write Netwon's laws as:

These equations we refer to has Hamilton's equations.

If is a solution of the Hamilton's equation, then for any function on , we have

Call a smooth function on a conserved quantity if is independent of for each solution of Hamilton's equations.

Then is a conserved quantity if and only if

In particular, the Hamiltonian is a conserved quantity.

Solving Hamilton's equatons on gives rise to a flow on , that is, a family of diffeomorphisms of , where is equal to the solution at time of Hamilton's equations with initial conditions .

Since it is possible (depending on the choice of potential function ) that a particle can escape to infinity in finite time, the maps are not necessarily defined on all of , but only on some subset therof.

If is defined on all of we say it's complete.

The flow associated with Hamilton's equations, for an arbitrary Hamitonian function , preserves the (2n)-dimensional volume measure

What this means, more precisely, is that if a measurable set is contained in the domain of for some , then the volume of is equal to the volume of .