# Write Equations of motion of a point particle that moves on the surface of a paraboloid [closed]

I'm not a physics major student and I call myself a total noob in physics, however, I have this issue to solve. The problem is that I don't even know where to start from. One little thing to add: this problem was translated from another language so forgive me for some misunderstandings. I'm not asking for a full solution(even though it would be highly appreciated), I need more like a guide where to dig.

A point particle of mass m moves under the force of gravity on the surface of a paraboloid.

$$z = x^2 + y^2$$

Write the equations of motion.

• you can simply write down three equations ma=F for all three components, where you substitute the force you want and constraint you also want (the constraint is already solved for z) – Umaxo Oct 17 '19 at 15:04

You can use the lagrangian approach to solve it. You know that it is fixed to the surface so you can use $$x, y$$ as the independent coordinates. So the lagrangian is: $$L = T-V =\frac{1}{2}m\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-mgz$$
Where $$T$$ is kinetic energy and $$V$$ is potential energy. You now use the fact that $$z$$ can be expressed in terms of $$x$$ and $$y$$,
$$L = \frac{1}{2}m\left[\dot{x}^{2}+\dot{y}^{2}+\left(2x\dot{x}+2y\dot{y}\right)^{2}\right]-mg(x^{2}+y^{2})$$
$$\frac{\partial L}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_i} = 0$$
Where $$q_i$$ can be either $$x$$ or $$y$$. I trust that you are able to do this last step.