Particle confined to move on the surface of sphere

[kimpossible]

Homework Statement

what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that

Homework Equations

Euler-lagranges equation of motion
d/dt(∂L/∂{dq/dt})-∂L/∂q=0

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?

 

[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.

 

[Ray Vickson]

Homework Statement

what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that

Homework Equations

Euler-lagranges equation of motion
d/dt(∂L/∂{dq/dt})-∂L/∂q=0

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?

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 $\theta, \phi, \dot{\theta}, \dot{\phi}$.

 

[kimpossible]

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 $\theta, \phi, \dot{\theta}, \dot{\phi}$.

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)

 

[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

 

[Ray Vickson]

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)

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.

 

[kimpossible]

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.

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

 

[Ray Vickson]

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

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