Let P be a point of mass m free to move on the surface of a smooth paraboloid $x^2 + y^2 = z$. An ideal spring of constant factor characteristic k connects the point P with the Focus of the paraboloid:
The question is to find the Lagrangian function of the system.
Of course we know that L = T - V where T is the kinetic energy and V is the potential, so I tried to compute first T and then V.
P is free to move only on the paraboloid so its coordinates are $P=(x,y,x^2 + y^2)$;
The kinetic energy is $T=\frac{1}{2}m$$||\frac{dP}{dt}||^2$, while the potential is $V=mg(x^2+y^2)$ + $\frac{1}{2}k$$||P-F||^2$ where F is the Focus of the paraboloid $F=(0,0,\frac{1}{4})$.
Is it correct? I feel it is not but I can't find my mistake so a help would be very useful;
thanks
You should change your degrees of freedom first, so you may deal with an unconstraint motion. In the cylindrical coordinates your constrain reads $z = \alpha r^2$ (i included the inverse length $\alpha$ for dimensionality reasons, since otherwise your $z$ has the deimension length$^2$), so the range of your motion is parametrized via $$[x,y,z](r,\phi) = [r\cos(\phi),r\sin(\phi),\alpha r^2]$$ Where $r$ and $\phi$ are time dependent. In terms of these new degree of freedom the potential reads $$V(r,\phi) = -mg\alpha r^2 + \frac{k}{2}(r^2+(\alpha r^2 - \frac{1}{4} )^2)$$ and the kinetic energy $$T = \frac{m}{2}(\dot{r}^2+r^2\dot{\phi}^2+4\alpha^2 r^2\dot{r}^2)$$ Now you may apply the Lagrange-formalism wrt. your new coordinates $r$ and $\phi$.