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Goldstein Mechanics chpt 1, Exercise 15

  1. Mar 12, 2012 #1
    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
    [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.
    2. Relevant equations
    [tex] Q_j = -\frac{\partial U}{\partial q_j} + \frac{d}{dt} (\frac{\partial U}{\partial \dot{q_j}})[/tex]


    3. 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
     
  2. jcsd
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