Hamiltonian of Van der Pol Oscillator

[pokethis]

Hi, wondering if anyone can help with this;

we have the Hamiltonian of a linear harmonic oscillator

H=(p²+w²q²)/2

now we apply the frictional force F=ap(bq-1) where a and b are constants. how do we alter the Hamiltonian to take that into account?

if it helps, the total time derivative of H is Fp.

also does anyone know how to calculate the action variable I for the linear case.

I is the path integral of pdq and I=H/w the book I am using says this is "clear" but I am not sure how they get it, particularly when H is dependent on p² how come I is linearly dependent on H?

thanks if anyone can shed some light on either of these points.

pokethis

 

[eljose79]

Hi pokethis,

To take the frictional force into account, we can add it to the Hamiltonian as a potential term. The updated Hamiltonian would be:

H=(p2+w2q2)/2 + V(q)

where V(q) is the potential due to the frictional force, given by V(q)=ap(bq-1).

To calculate the action variable I for the linear case, we can use the action integral formula:

I = ∫pdq

In the case of a linear harmonic oscillator, the momentum p is proportional to the velocity v, which is given by v=ωq. Therefore, we can rewrite the action integral as:

I = ∫ωq dq

Integrating this over one period T, we get:

I = ∫ωq dq = ∫ωqωT dt = ωT∫q dt = ωTq

Since H=ωI, we can rewrite this as:

I = H/ω

which is the formula given in your book.

I hope this helps clarify things for you. Let me know if you have any further questions.