Hamilton and Lagrange functions

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

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

Would it be px = dH/dpy (of course it would be partial)
and py = - dH/dpx ?
and how do we take into account the potential?
 

gabbagabbahey

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Would it be px = dH/dpy (of course it would be partial)
and py = - dH/dpx ?
and how do we take into account the potential?
No, Hamilton's equations of motion are [tex]\dot{p_i}=-\frac{\partial H}{\partial q_i}[/tex] qnd [tex]\dot{q_i}=\frac{\partial H}{\partial p_i}[/tex], where [itex]q_i[/itex] are the generalized coordinates and [itex]p_i[/itex] are there corresponding momenta.

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

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