# Hamiltonian function of undamped spring (ODE)

1. Aug 31, 2009

### zass

1. The problem statement, all variables and given/known data

Derive the Hamiltonian function H(A,B) such that A'=HB and B'=-HA. Plot the contours of this function in the range -0.005 =< H =< 0.01. Identify the approximate position an type of each of the three critical points which occur.

We can look for solutions of the form
x = A(t)cos(0.92t) + B(t)sin(0.92t)

2. Relevant equations

Undamped spring modelled by:
x'' + x - x3 = 0.02cos(0.92t)

Forcing period = T= 2pi/0.92

Initial conditions:
x(0) = A0
x'(0) = B0

A' = -B*(-0.08 + (3/8)*(A2 + B2))
B' = (0.02/2) + A*(-0.08 + (3/8)*(A2 + B2))

3. The attempt at a solution

From integrating A' and B' i got an equation for H;
H = (-1/2)(-0.08B2 + (3/8)B2A2 + (3/16)B4)

and

H = (-1/2)(0.02 -0.08A2 + (3/8)B2A2 + (3/16)A4)

I'm not sure where to go from here. Is this the Hamiltonian function they're talking about? If so, how do i draw the contours between H=-0.005 and H=0.01?

If someone could answer this that'd be great! I think i'd have an idea how to classify the crit. points if i saw them, its just getting to that stage that's killing me!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution