# Help finding equations of motion from Hamiltonian with integral of motion

I've given this Hamiltonian with one degree of freedom: $$H(x,p)=\frac{p^2}{2}+\frac{\omega_0^2x^2}{2}+\lambda\left(\frac{p^2}{2}+\frac{\omega_{0}^2x^2}{2}\right)^2$$ I need to find the general solution for the motion of this particle $(x(t),p(t))$. It's required to solve the equations of motion, not only find them. I showed that $f = p^2+\omega_{0}^2x^2$ is time independent by computing: $$\{f,H\}=0$$ But I'm having a hard getting the final solution. I tried doing: $$2p\dot{p}+2\omega_{0}^2x\dot{x}=\dot{f}=0$$ $$\dot{p}=-\frac{\omega_{0}^2x\dot{x}}{p}$$ And then: $$\newcommand{\pder}[]{\frac{\partial#1}{\partial#2}} \dot{p}=-\pder[H]{x}$$ Same for $\dot{x}$. Yet, I don't obtain nicer differential equations and I don't think it's the right approach. Could someone give me a hint?

• Hi. Do you need to find the equations of motion or solve the equations of motion? Thanks. Nov 22, 2016 at 20:34
• I need to solve them Nov 22, 2016 at 20:34

I want simplification, so I'll take $\omega_0 = 1$. You can rescale $x$ and always get this kind of simplification, if $\omega_0$ is a constant. The Hamiltonian then becomes

$$H(x,p) = \frac{p^2}{2} + \frac{q^2}{2} + \lambda \left( \frac{p^2}{2} + \frac{q^2}{2} \right)^2 \, .$$

Hamilton's equations are given by

\begin{align} \begin{cases} \dot{q} &= p + 2 \lambda p \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \\ \dot{p} &= -q - 2 \lambda q \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \end{cases} \quad \Rightarrow \quad \begin{cases} \dot{q} &= p \left[1 + 2 \lambda \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \right] \\ \dot{p} &= -q \left[1 + 2 \lambda \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \right] \end{cases} \, . \end{align}

Assuming $q, \dot{p} \neq 0$, divide one equation by the other to get

$$\frac{ \left( \frac{dq}{dt} \right)}{ \left(\frac{dp}{dt} \right)} = \frac{p \left[1 + 2 \lambda \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \right]}{-q \left[1 + 2 \lambda \left( \frac{p^2}{2} + \frac{q^2}{2} \right) \right]} = -\frac{p}{q} \, .$$

The solution will, therefore, be the same as for the SHO, with the exception that the new frequency will increase as the orbit gets farther away from the origin.

As I believe this is homework, I'll give no further details. The problem is basically solved.

what if same question is asked to find equation of motion. i know equations of Hamilton. i am having hard time in finding meaning full solutions. i think my apporoch is not good. can you suggest similar other example? thanks

• This does not really answer the question. If you have a different question, you can ask it by clicking Ask Question. To get notified when this question gets new answers, you can follow this question. Once you have enough reputation, you can also add a bounty to draw more attention to this question. - From Review Sep 25, 2021 at 9:44