Hamilton Function

The elegance of the classical description of the motion of a point-mass motion can be observed by using the Hamilton function H(p,q) with momentum coordinate p and position coordinate q. In the most simple case the Hamilton function is the sum of  kinetic and potential energies: H(p,q) = Ekin(p)+V(q). The motion equations then can be described as follows:
 

dq/dt  =  ∂H/∂p

 
dp/dt  =  − ∂H/∂q

This can be explained by using the "Harmonic Oscillator" example:
The potential is given by V(x) = ½ kx² with the strength constant k and distance x. The total energy E is E = Ekin + V(x). Since Ekin = ½mv² and p = mv Þ Ekin = p²/2mÞ  E = p²/2m + ½ kx². For the linear deviation x there will be just q = x (for pendulum q would be deviation angle) and p is the momentum coordinate:
 

H = p²/2m + ½kx²

Now we can write down the following equation for the momentum's change in time (2. Hamilton equation):

 dp/dt  = -H/∂x  =  − kx

and for the coordinate change in time (1. Hamilton equation):

dx/dt  = ∂H/∂p  =  p/m

Or:
d²x/dt²  =  (dp/dt)/m
and finally for dp/dt:
d²x/dt²  = (dp/dt)/m  = −(k/m)x

The solution:  x = A · sin wt    with  w = (k/m)½
it follows:  p = A m w · cos wt

This is a well-known vibration equation.


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