- #1

- 114

- 0

## Homework Statement

A particle is confined to move on the surface of a circular cone with its axis on the vertical z axis, vertex at the origin (pointing down), and half-angle a.

(a) Write the Lagrangian L in terms of the spherical polar coordinates r and ø.

(b) Find the two equations of motion. Interpret the ø equation in terms of the angular momentum

*l*[tex]_{z}[/tex], and use it to eliminate ø-dot from the r equation in favor of the constant

*l*[tex]_{z}[/tex]. Does your r equation make sense in the case that

*l*[tex]_{z}[/tex] = 0? Find the value r[tex]_{o}[/tex] of r at which the particle can remain in a horizontal circular path.

(c) Suppose that the particle is given a small radial kick, so that r(t) = r[tex]_{o}[/tex] + ε(t), where ε(t) is small. Use the r equation to decide whether the circular path is stable. If so, with what frequency does r oscillate about r[tex]_{o}[/tex]?

## Homework Equations

v[tex]^{2}[/tex] = [tex]\dot{r}[/tex][tex]^{2}[/tex] + r[tex]^{2}[/tex]sin[tex]^{2}[/tex]([tex]\phi[/tex])[tex]\dot{\theta}[/tex][tex]^{2}[/tex] + r[tex]^{2}[/tex][tex]\dot{\phi}[/tex][tex]^{2}[/tex]

*l*= r X mv

## The Attempt at a Solution

Okay, so the langrangian L = T - U.

U is easy enough, saying the only potential energy is gravitational energy, so U = mgrcos[tex]\phi[/tex].

But T = (1/2)mv[tex]^{2}[/tex], and v[tex]^{2}[/tex] = [tex]\dot{r}[/tex][tex]^{2}[/tex] + r[tex]^{2}[/tex]sin[tex]^{2}[/tex]([tex]\phi[/tex])[tex]\dot{\theta}[/tex][tex]^{2}[/tex] + r[tex]^{2}[/tex][tex]\dot{\phi}[/tex][tex]^{2}[/tex]

Now, I'm told that the cone on which this particle moves has a half-angle of [tex]\alpha[/tex]. Then, I know, [tex]\phi[/tex] = [tex]\alpha[/tex] = const., so [tex]\dot{\phi}[/tex] = 0. Right?

With [tex]\dot{\phi}[/tex] being zero, v[tex]^{2}[/tex] reduces to [tex]\dot{r}[/tex][tex]^{2}[/tex] + r[tex]^{2}[/tex]sin[tex]^{2}[/tex]([tex]\phi[/tex])[tex]\dot{\theta}[/tex][tex]^{2}[/tex].

But this still has a theta coordinate in it. How can I express the Legrangian in just r and [tex]\phi[/tex]?

And then, after that, how do I relate

*l*and [tex]\dot{\phi}[/tex] so as to eliminate the latter from the r equation of motion?

But first: How can I express the Legrangian in just r and [tex]\phi[/tex]?