A standing electrostatic wave has equation

To find the potential energy, note that

and



The kinetic energy is

Now, the equations defining the generalized coordinates x and p do not depend on time. Furthermore, the force is derivable from a conservative potential, so it follows that H is a constant of motion given by

Now non-dimensionalize this with a dimensionless parameter ,

Define




Then the equation becomes

Hamilton's equations then become



The fixed points occur when


so they are

However, there are also fixed points at