Lagrangian of a Particle in Spherical Coordinates (Is this correct?)

[Xyius]

Homework Statement

a.) Set up the Lagrange Equations of motion in spherical coordinates, ρ,θ, $\phi$ for a particle of mass m subject to a force whose spherical components are $F_{\rho},F_{\theta},F_{\phi}$.

This is just the first part of the problem but the other parts do not seem so bad.

Homework Equations

Lagrangian equations of motion.
Since the problem doesn't state it is in any uniform conservative field from which it would have a potential function, I assume I use the more general form of the Lagrangian.
$$\frac{d}{dt} \frac{∂T}{∂\dot{q}}-\frac{∂T}{∂q}=Q_k$$

The Attempt at a Solution

So my three general forces would be..
$$Q_{\rho}=F_{\rho}$$
$$Q_{\theta}=\rho F_{\theta}$$
$$Q_{\phi}=F_{\phi}$$

The kinetic energy in Spherical coordinates is..
$$T=\frac{1}{2}m\left( \dot{\rho}^2 +\rho^2 \dot{\theta}^2+\rho^2 \dot{\phi}^2 sin^2\theta \right)$$
Thus the three equations of motion are..
$$\frac{d}{dt} \frac{∂T}{∂ \dot{\rho}}-\frac{∂T}{∂\rho}=Q_k=F_{\rho}$$
$$m \ddot{\rho}-m \rho \dot{\rho} \dot{\theta}^2-2 \rho \dot{\rho} \dot{\phi}^2sin^2 \theta=F{\rho}$$
and..
$$\frac{d}{dt} \frac{∂T}{∂ \dot{\theta}}-\frac{∂T}{∂\theta}=Q_k= \rho F_{\theta}$$
$$\frac{d}{dt}\left( m \rho^2 \dot{\theta}^2 \right)-m \dot{\theta} \rho^2 \dot{\phi}^2sin \theta cos \theta$$
$$2m \rho \dot{\theta} \dot{\rho}+m \rho^2 \ddot{\theta}-m \rho^2 \dot{\phi}^2sin\theta cos\theta$$
and finally..
$$\frac{d}{dt} \frac{∂T}{∂ \dot{\phi}}-\frac{∂T}{∂\phi}=Q_k=F_{\phi}$$
$$\frac{d}{dt}\left( m \rho^2 \dot{\phi}^2 sin^2 \theta \right)$$
$$2m \rho \dot{\rho} \dot{\phi}sin^2 \theta+m \rho^2 \ddot{\phi}sin^2 \theta +2m \rho^2 \dot{\phi}\dot{\theta}sin \theta cos \theta$$

Would these be correct? The rest of the problem says to substitute $\dot{\theta}$ for $\omega$ and then find the corresponding centrifugal and Coriolis forces. I do not recall how I would recognize these terms. I know these are both fictitious forces that are only a product of mass times acceleration. Their form is more obvious is polar form though.. If anyone could help me I would very much appreciate it. :]
Thanks!

 

[Liquidxlax]

use the spherical gradient when taking derivatives in spherical coordinates

http://en.wikipedia.org/wiki/Gradient

 

[Liquidxlax]

man i hate not being able to edit some posts...

i said that because it does not look like that you differentiated in spherical coordinates

 

[Xyius]

What do you mean? I started out in Spherical coordinates from the Kinetic energy didn't I? From there on out it was just the chain rule a bunch of times.

 

[paranormal]

I would say that your setup of the Lagrangian equations of motion in spherical coordinates looks correct. You have correctly identified the kinetic energy in spherical coordinates and have set up the Lagrangian equations for each coordinate. However, it would be helpful to include the potential energy in your Lagrangian, as it is important in determining the forces in the system.

For the second part of the problem, you would need to substitute ω for θ in your equations and then determine the corresponding centrifugal and Coriolis forces. The centrifugal force is a fictitious force that arises due to the rotation of the coordinate system, and it is given by F_cent = mω^2ρsin^2θ. The Coriolis force, on the other hand, is a fictitious force that arises due to the motion of the particle in the rotating coordinate system, and it is given by F_cor = -2mωρsinθcosθ(dθ/dt). You can then substitute these forces into your equations and solve for the equations of motion.

I hope this helps! Let me know if you have any further questions.