Finding Hamiltonian's equations of motion

We can find the Hamiltonian equations of motion for an an-harmonic oscillator.  We are given the potential energy to be:

where k and b are both constants.

First, we need to find the Lagrangian.  The equation for this is:

 where U is the potential energy and T is the kinetic energy.  We know the potential, but we need to find the kinetic energy.  For this and most cases, we can represent this by:

Next, we need to plug this into the Hamiltonian equation, which is:

where p is the momentum, q dot is the velocity and L is the Lagrangian.  We already know the Lagrangian, so we plug it into the equation.  We have to write all velocities in terms of momentum. If we use the equation:

,

then we can solve for x dot.  we get this value to be:

Now, we plug that into the Hamiltonian equation.  After simplifying, we get our Hamiltonian to equal:

Finally, we need to find the Hamiltonian equations of motion.  To find these, we use the following equations:

  

Plugging in our Hamiltonian, we will get the following equations as our equations of motion: