I didn´t understand how to use Hamiltonian for some mechanical problems, in particular in a two-body $(m_1, m_2)$ attached by a string $(k,l).

First, calculating The lagrangian: $$L=T-U=\dfrac{1}{2}m_1 \dot x_1^2+\dfrac{1}{2}m_2 \dot x_2^2-\dfrac{1}{2}k(x_2-x_1-l)^2$$ Then: $$p_1=\dfrac{\partial L}{\partial \dot x_1}=m_1 \dot x_1\implies\dot x_1=\dfrac{p_1}{m_1}$$ $$p_2=\dfrac{\partial L}{\partial \dot x_2}=m_2 \dot x_2\implies\dot x_2=\dfrac{p_2}{m_2}$$

$$\implies L=\dfrac{p_1}{2m_1}+\dfrac{p_2}{2m_2}-\dfrac{1}{2}k(\Delta x)^2$$ $$\Delta x=x_2-x_1-l$$

But i didn´t understand the relation with hamilton.


closed as unclear what you're asking by ACuriousMind, Daniel Griscom, John Rennie, Kyle Kanos, Martin Jan 8 '16 at 12:10

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  • $\begingroup$ What do you mean that you don't "understand the relation with hamilton?" Well where's your Hamiltonian? You've started your Legendre transform, but you've not completed it. $\endgroup$ – Kyle Kanos Jan 8 '16 at 11:27

There are two mistakes:

  • You stil have to have $x_1$, $x_2$ otherwise the conjugate momenta $p_1$, $p_2$ does not make any sense (they are conjugate momenta - they must be connected to some $x$).

  • You haven't done the full transformation. You just have a lagrangian in weird variables, but that is just half the work. See the wikipedia article.

I will not provide the full detailed answer since this is homework and excercises question.


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