How Are Hamiltonian and Lagrangian Functions Related?

[Shafikae]

The hamilton function of a particle in two dimensions is given by

H = (p$$\stackrel{2}{x}$$)/2m + (p$$\stackrel{2}{y}$$)/2m + ap_xp_y + U(x,y)
Obtain the Hamiltonian equations of motion. Find the corresponding Lagrange function and Lagrange equations.

Would it be p_x = dH/dp_y (of course it would be partial)
and p_y = - dH/dp_x ?
and how do we take into account the potential?

 

[gabbagabbahey]

Would it be p_x = dH/dp_y (of course it would be partial)
and p_y = - dH/dp_x ?
and how do we take into account the potential?

No, Hamilton's equations of motion are $$\dot{p_i}=-\frac{\partial H}{\partial q_i}$$ qnd $$\dot{q_i}=\frac{\partial H}{\partial p_i}$$, where $q_i$ are the generalized coordinates and $p_i$ are there corresponding momenta.

In this case, your generalized coordinates are $x$ and $y$ (i.e. $q_1=x$ and $q_2=y$) )and there corresponding momenta are $p_x$ and $p_y$ (i.e. $p_1=p_x$ and $p_2=p_y$)...