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## Homework Statement

A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system.

## Homework Equations

$$\frac{\partial\mathcal{L}}{\partial q_i}-\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial \dot{q_i}}$$

## The Attempt at a Solution

Potential energy is given by, $$V=mgz=mgb\rho^2.$$ Kinetic energy in cylindrical coordinates is given by, $$T=\dot{\rho}^2+\rho^2\dot{\theta}^2+\dot{z}^2=\dot{\rho}^2+\rho^2\dot{\theta}^2+4b^2\rho^2\dot{\rho}^2.$$ Then the Lagrangian is, $$\mathcal{L}=\dot{\rho}^2+\rho^2\dot{\theta}^2+4b^2\rho^2\dot{\rho}^2-mgb\rho^2.$$ Then, we can write the Euler-Lagrange equations for both ##\theta## and ##\rho##. The ##\theta## equation is very simple. We have, $$\frac{\partial\mathcal{L}}{\partial\theta}=0,$$ and $$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=\frac{d}{dt}\bigg(2\rho^2\dot{\theta}\bigg)=4\rho\dot{\rho}\dot{\theta}+2\rho^2\ddot{\theta}.$$ So the Euler-Lagrange equation reads, $$4\rho\dot{\rho}\dot{\theta}+2\rho^2\ddot{\theta}=0.$$ I'm getting messy with my derivatives in the ##\rho## equations. Any help would be appreciated.