# Particle confined to move on the surface of sphere

Tags:
1. Aug 19, 2016

### kimpossible

1. The problem statement, all variables and given/known data
what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that
2. Relevant equations
Euler-lagranges equation of motion
d/dt(∂L/∂{dq/dt})-∂L/∂q=0

3. The attempt at a solution
Z=(R+R0e^at)cosθ
X=(R+R0e^at)sinθcosΦ
Y=(R+R0e^at)sinθsinΦ
Lagrangian L=T-U
T=1/2(mv^2)=1/2m{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}
and
U=mgz
I just want to know whether i'm going on right track or not? and are the position coordinates right? Is the answer goes with the spherical pendulum?

#### Attached Files:

• ###### hamiltonian.jpg
File size:
7.1 KB
Views:
26
2. Aug 19, 2016

### BvU

Hello Kim,

I see nothing wrong with your approach. I take it you choose $\theta$ and $\phi$ as generalized coordinates.
For $R_0 = 0$ you have the spherical pendulum case, so it's good to check with those expressions.

3. Aug 19, 2016

### Ray Vickson

Are your "generalized coordinates" $q$ just $\theta$ and $\phi$? If so, you need to express the Lagrangian in terms of them, so you need to figure out $v^2$ and $U$ in terms of $\theta, \phi, \dot{\theta}, \dot{\phi}$.

4. Aug 19, 2016

### kimpossible

yeah i expressed v and U in terms of generalized coordinates, but i'm not sure if Φ varies (i mean equation of motion for Φ)

5. Aug 20, 2016

### kimpossible

and let me correct the problem-particle confined to move on the surface of sphere whose radius is expanding such a that R(t)=R+R0e^at

6. Aug 20, 2016

### Ray Vickson

In general there is no a priori reason to assume anything about $\phi$. There are two Lagrange differential equations, one for $\theta$ and one for $\phi$. Write them down and see what they tell you.

7. Aug 20, 2016

### kimpossible

thanks for your view but are my x,y,z coordinates are correct if radius R is expanding such a that R(t)=R0e^at where t is time and a,R0 are constants? Was my approach correct as i described in- The attempt at a solution

8. Aug 21, 2016

### Ray Vickson

It looks like the very beginning of a possibly correct approach, but is far from finished.