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)...