- #1

dracobook

- 23

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## Homework Statement

A point particle moves in space under the inﬂuence of a force derivable from

a generalized potential of the form

[tex] U(r, v) = V (r) + \sigma \cdot L [/tex]

where [tex] r [/tex] is the radius vector from a ﬁxed point, [tex] L [/tex] is the angular momentum about that point, and [tex] \sigma [/tex] is a ﬁxed vector in space.

Find the components of the force on the particle in both Cartesian and

spherical poloar coordinates, on the basis of Lagrangian’s equations with

a generalized potential.

## Homework Equations

[tex] Q_j = -\frac{\partial U}{\partial q_j} + \frac{d}{dt} (\frac{\partial U}{\partial \dot{q_j}})[/tex]

## The Attempt at a Solution

Using Cartesian, I get :

[tex] F = 2m(\mathbf{\sigma} \times \mathbf{v}) - V' \frac{\mathbf{r}}{r} [/tex] I let [tex] \sigma [/tex] point in the z direction, then it simplifies my cartesian to

[tex] U = V(r) + m\sigma (x\dot{y} - y\dot{x}) [/tex]

If I plug in the x in terms of [tex] r, \theta, \phi [/tex] I get

[tex] Q_r = - \frac{dV}{dr} - 2 m \sigma r \text{sin}^2 \theta \dot{\phi} [/tex].

My Question is, is it possible to go the other way around? That is, what if I plug in the x, y z-> r, theta, phi equivalence of the generalized potential. How can I then restrict it so that sigma is faced in the z-axis? Sorry I know this is perhaps somewhat irrelevant to the question...but it's something that is really bothering me.

In addition if someone could give me an example of situation in which a point particle would undergo such a force that would be great (the angular momentum dot sigma really throws off my intuition).

Any help would be greatly appreciated.

-draco