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Equation of trajetory+conserved quantities of a motion over a sphere

  1. Apr 23, 2010 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    Determine the equation of the trajectory and the conserved quantities in the motion of a particle constrained to move freely over the surface of a sphere.


    2. Relevant equations
    Not sure.



    3. The attempt at a solution
    I think it is convenient to use spherical coordinates [tex](r, \phi , \theta)[/tex].
    I notice that a rotation over any diameter of the sphere shouldn't change the dynamics of the particle, hence the angular momentum is conserved?
    Anyway, I want to write the Lagrangian. My problem resides in writing the position vector of the particle in spherical coordinates. I know that [tex]r=r \hat r[/tex]. But I'm not sure how to write [tex]\phi[/tex] and [tex]\theta[/tex] in terms of [tex]\hat r[/tex], [tex]\hat \theta[/tex] and [tex]\hat \phi[/tex].
    I realize that the Lagrangian must depend explicitly on [tex]\phi[/tex] and [tex]\theta[/tex] and not on [tex]r[/tex] since they are variables depending on time. I also believe the energy is conserved, but I have to show it I believe using the Lagrangian of the particle.
    Any correction of my thoughts and help about how to write the position vector is welcome.


    Edit: OK I just saw in wikipedia that [tex]\vec r=r\hat r[/tex] and "thus" [tex]\dot \vec r = \dot r \hat r + r \dot \theta \hat \theta + r \dot \theta \sin (\theta) \hat \phi[/tex]. I'm actually trying to understand this implication.


    Edit 2: Ok, assuming the last formula for [tex]\dot \vec r[/tex], since r is constant I have that [tex]\dot \vec r =r \dot \theta \hat \theta + r \dot \theta \sin (\phi) \hat \phi[/tex]. I can get the Lagrangian. I know that [tex]E= \sum _i \frac{\partial L}{\partial \dot q_i} \dot q_i -L[/tex].
    Now if someone can explain me how to get the expression given in wikipedia, you'll save me hours. Thanks in advance.
     
    Last edited: Apr 23, 2010
  2. jcsd
  3. Apr 23, 2010 #2
    Taking the time derivative of the unit vectors in spherical coords can be tricky. I find it easiest to convert the unit vector in spherical coords to cartesian coords. Take the time derivative there, and convert back to spherical coords.
     
  4. Apr 25, 2010 #3

    fluidistic

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    It might be nice and easy method, yet still very complicated to me. Unfortunately the conversion of spherical unit vectors into cartesian's one cannot be found in this table: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates.
    And I've tried the derivation, I get lost. I've found no website with such a derivation.
     
  5. Apr 26, 2010 #4

    lanedance

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  6. Apr 26, 2010 #5

    gabbagabbahey

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    A handy reference for converting between Cartesian, Cylindrical and Spherical coordinates can be found in the back cover of Griffiths Introduction to Electrodynamics, if you have that text.
     
  7. Apr 26, 2010 #6

    lanedance

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    otherwise for the general formulation, my math method text (riley hobson bence) has a good section under - general curvilinear coordinates, which i found pretty informatiev to understand how all the derivatives are calaulated - though i'm sure you could probably google it
     
  8. Apr 26, 2010 #7

    fluidistic

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    Thanks a lot. I've checked out both Griffith and Riley's book. It is much of help. Especially Riley's since it is explained how can one derive the unit vectors i, j and k expressed in cylindrical and spherical coordinates. So instead of memorizing the whole formulas given in wikipedia and Griffith, if I keep in mind that x= rho cos (phi), y=rho sin (phi) and z=z for cylindrical coordinates, I can derive the corresponding unit vectors with simple partial derivatives (Jacobian) and a normalization. It also allow me to derive them without making a single sketch of the different coordinate systems, which is pretty nice since I get confused a lot on 2 dimensional paper sheets.
    So your suggestions really helped me.
     
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