(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine the Hamiltonian corresponding to the an-harmonic oscillator having the Lagrangian [itex]L(x,\dot x )=\frac{\dot x ^2}{2}-\frac{\omega ^2 x^2}{2}-\alpha x^3 + \beta x \dot x ^2[/itex].

2. Relevant equations

[itex]H(q,p,t)=\sum p_i \dot q _i -L[/itex].

[itex]p _i=\frac{\partial L}{\partial \dot q _i}[/itex].

3. The attempt at a solution

Using the equations in 2), I get [itex]p_x= \dot x +2 \beta x \dot x[/itex].

So that [itex]p_x ^2= \dot x ^2 (1+4 \beta x +4 \beta ^2 x^2)[/itex].

Now if I can write the following function in function of only p, q and t then it would be the Hamiltonian, but I couldn't do it.

Here's the function: [itex]\dot x^2 \left ( \frac{1}{2} +\beta x \right ) +\omega ^2 \frac{x^2}{2}+\alpha x^3[/itex].

I see absolutely no way to get rid of the x's terms (or q in this case).

Any help is appreciated. Thanks!

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# Homework Help: Finding the Hamiltonian if I'm given the Lagrangian

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