- #1

- 4

- 0

## Homework Statement

A particle slides on the outer surface of an inverted hemisphere. Using Lagrangian multipliers, determine the reaction force on the particle. Where does the particle leave the hemispherical surface?

L - Lagrangian

q

_{i}- Generalized ith coordinate

f(r) - Holonomic constraint

Q

_{i}- Generalized force of constraint on the ith particle

λ

_{j}- Lagrangian multiplier

## Homework Equations

[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = \sum_{j=0}^k \lambda_j \frac{\partial f_j}{\partial q_i},\ i = 1, ..., n \ j = 1,... k\\ \\

x = r * \sin\theta * \cos\phi\\

y = r * \sin\theta * \sin\phi\\

z = r * \cos\theta\\

f(r) = r - R = 0\\

Q_i = \sum_{j=0}^k \lambda_j \frac{\partial f_j}{\partial q_i}[/tex]

## The Attempt at a Solution

So I was able to derive the potential energy [tex]U = mgr \cos\theta[/tex] the kinetic energy [tex]T = \frac{1}{2}m(\dot r^2 + r^2 \dot\theta^2 + r^2 \dot\phi^2 \sin^2\theta)[/tex]I then inserted them into the Lagrangian and got the two equations

[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = \lambda_r * \frac{\partial f}{\partial r} \\

\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = \lambda_\theta * \frac{\partial f}{\partial \theta}[/tex]

Where [tex]\frac{\partial f}{\partial r} = 1[/tex] and [tex]\frac{\partial f}{\partial \theta} = 0[/tex]

But after inputting L into these two equations I get:

[tex]Q_r = m \ddot r - mg\sin\theta - mr \dot \theta^2 - mr \dot \phi^2 \sin^2\theta[/tex]

I think there is something wrong here, but it may just be me. Also, I already searched through other threads to find this answer.