Is Phi a Generalized Coordinate in Lagrangian Equations?

[M. next]

If we considered some coordinate as being a generalized one, like when we are considering spherical coordinates-let us suppose that I chose theta and phi as generalized coordinates. After deriving the Lagrangian equation it turned out that the equation doesn't depend on phi. Which means that derivative of the Lagrangian by phi is zero. Does this mean it is not a generalized coordinate? If not, what does it mean? And lastly, what's the difference if the case was that the equation doesn't depend on phi dot(the time derivative of phi)
thanks in advance

 

[jfizzix]

If we considered some coordinate as being a generalized one, like when we are considering spherical coordinates-let us suppose that I chose theta and phi as generalized coordinates. After deriving the Lagrangian equation it turned out that the equation doesn't depend on phi. Which means that derivative of the Lagrangian by phi is zero. Does this mean it is not a generalized coordinate? If not, what does it mean? And lastly, what's the difference if the case was that the equation doesn't depend on phi dot(the time derivative of phi)
thanks in advance

Let's say we have some Lagrangian

$L(\theta, \phi, \dot{\theta}, \dot{\phi})$

Let's look at the equation of motion for $\phi$. Recall that the generalized conjugate momentum to the coordinate $\phi$ is

$p_{\phi}=(\frac{\partial L}{\partial \dot{\phi}})$

and so our equation of motion becomes

$\frac{d}{d t}(\frac{\partial L}{\partial \dot{\phi}}) -(\frac{\partial L}{\partial \phi}) = \frac{d}{d t} p_{\phi} - (\frac{\partial L}{\partial \phi})= 0$

If L does not depend on $\phi$, then $(\frac{\partial L}{\partial \phi})= 0$, and so $p_{\phi}$ is constant in time; it is a conserved quantity.


As for what happens when L does not depend on $\dot{\phi}$ we can look to the same equation of motion. This would be saying the same thing as $p_{\phi} = 0$. As far as dynamics go, I think what it means is that there would be no relevant dynamics in the $\phi$ direction, and you would instead look at how things are changing in the $\theta$ direction.


hope this helps,

-James

 

[Couchyam]

Phi is still a generalized coordinate, and when you're considering your final equations of motion you will need to consider the time-dependence of the phi coordinate in your answer.

The fact that the derivative of the Lagrangian is zero means, as was mentioned above, that you have a conserved quantity. For example, consider the Lagrangian for a small planet orbiting a very large star: assuming that Newtonian gravity is valid, you will get a potential energy that will have no angular dependence at all. If you were free to choose whatever reference frame you wanted, then you would be foolish not to choose one in which one of the generalized coordinates is zero. However, you can imagine a potential that is slightly more complicated in which there is no "phi" dependence, but some "theta" dependence (like the gravitational potential energy of a galactic disk). In this case, the fact that dL/d(phi) = 0 means that angular momentum is conserved about the axis of symmetry. This does not reduce the effective dimensionality of the solution set though.

 

[M. next]

I apologize for the REAL delay in replying, but I thank you both.

 

[PJC01]

I can say that the choice of generalized coordinates in Lagrangian equations is not always straightforward and can depend on the specific system being studied. In this case, it appears that phi is a generalized coordinate, as it was chosen as one and is part of the set of variables that describe the system.

The fact that the derivative of the Lagrangian with respect to phi is zero does not necessarily mean that phi is not a generalized coordinate. It simply means that phi does not have a direct influence on the dynamics of the system. In other words, the equations of motion for the system do not depend on phi and therefore, it can be considered a "passive" coordinate.

On the other hand, if the equation did not depend on the time derivative of phi (phi dot), it would mean that phi is a constant in time and does not change during the motion of the system. In this case, phi would still be a generalized coordinate, but its value would be fixed throughout the motion.

Overall, the difference between the two cases is that in the first one, phi is a passive coordinate that does not directly affect the dynamics of the system, while in the second case, phi is a constant that does not change during the motion. Both cases are valid and can be used in Lagrangian equations, depending on the specific system being studied.