The especial elegance in the classical description of a mass point motion can
be reached by using the Hamilton function H(p,q) for
generalized impulse coordinate p and position coordinate q. When we have the
simplest case then the Hamilton function is the sum of kinetic and potential
energies: H(p,q) = Ekin(p)+V(q). The motion equations are then as
follows:
dq/dt = ∂H/∂p |
dp/dt = − ∂H/∂q |
We would like to explain it using the "Harmonic Oscillator" example:
The potential is given by V(x) = ½ kx² with the strength
constant k and deviation 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 impulse coordinate:
H = p²/2m + ½kx² |
Now we can write down the following equation for the impulse 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
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.