I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot p=-mgR\sin\theta$$ $$\dot\theta=\frac{p}{mR^2}$$ I know one of the points of Hamiltonian formalism is to get first order diff. equations instead of second order that Lagrangian formalism gives you, but how can I proceed from here without just derivating again wrt. $\dot\theta$ and substituting $\dot p$ to get the same equation that I get with the Lagrangian formulation? Or is that the way to do it? And how could I get the path of the system on the phase space with those equations?
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asked Jan 22, 2013 at 16:31
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Generally both formulations (Largangian and Hamiltonian) are equivalent, but in your case, if $\theta$ is small, you have a simplified equation for $p$ and you can use a solution ansatz like $e^{i\omega t}$ for both $p$ and $\theta$.
To draw a path in the phase space, you have to solve the equations and/or manage to express $p(\theta)$ or $\theta(p)$.
answered Jan 22, 2013 at 16:39
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$\begingroup$ Thanks for the info, so I will get to the same second order differential equation? And about the phase space? $\endgroup$ Commented Jan 22, 2013 at 16:41
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$\begingroup$ Not obligatory, you can solve $\dot{p}\propto \theta$ and $\dot{\theta}\propto p$ by the ansatz. $\endgroup$ Commented Jan 22, 2013 at 16:48
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2$\begingroup$ To add to what Vladimir said, if you consider a system with $n$ generalized coordinates (in this case $n=1$ since your system is described by the coordinate $\theta$), then you will obtain $2n$ first order differential equations, $n$ for the coordinates and $n$ for their corresponding canonical momenta. To get the path in phase space, you can do what Vladimir suggests; this will allow you to obtain $\theta(t)$ and $p(t)$, and then you simply plot these functions as a parametric curve on the $\theta$-$p$ plane. Cheers! $\endgroup$ Commented Jan 22, 2013 at 16:52