# Goldstein Mechanics chpt 1, Exercise 15

1. Mar 12, 2012

### dracobook

1. The problem statement, all variables and given/known data

A point particle moves in space under the inﬂuence of a force derivable from
a generalized potential of the form
$$U(r, v) = V (r) + \sigma \cdot L$$
where $$r$$ is the radius vector from a ﬁxed point, $$L$$ is the angular momentum about that point, and $$\sigma$$ 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.
2. Relevant equations
$$Q_j = -\frac{\partial U}{\partial q_j} + \frac{d}{dt} (\frac{\partial U}{\partial \dot{q_j}})$$

3. The attempt at a solution
Using Cartesian, I get :
$$F = 2m(\mathbf{\sigma} \times \mathbf{v}) - V' \frac{\mathbf{r}}{r}$$ I let $$\sigma$$ point in the z direction, then it simplifies my cartesian to
$$U = V(r) + m\sigma (x\dot{y} - y\dot{x})$$
If I plug in the x in terms of $$r, \theta, \phi$$ I get
$$Q_r = - \frac{dV}{dr} - 2 m \sigma r \text{sin}^2 \theta \dot{\phi}$$.
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