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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.

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  • $\begingroup$ 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) $\endgroup$ – Umaxo Oct 17 '19 at 15:04
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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 $y$,

$$L = \frac{1}{2}m\left[\dot{x}^{2}+\dot{y}^{2}+\left(2x\dot{x}+2y\dot{y}\right)^{2}\right]-mg(x^{2}+y^{2})$$

And finally, you use the Euler-Lagrange equations to obtain the equations of motion.

$$\frac{\partial L}{\partial q_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_i} = 0$$

Where $q_i$ can be either $x$ or $y$. I trust that you are able to do this last step.

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