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Lagrangian equations of particle in rotational paraboloid

  1. Nov 4, 2017 #1
    Hello. I solve this problem:

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

    The particles of mass m moves without friction on the inner wall of the axially symmetric vessel with the equation of the rotational paraboloid:

    paraboild.png

    where b>0.

    a) The particle moves along the circular trajectory at a height of z = z(0).

    express:
    - Lagrangian
    - the equation of motion for the polar coordinate r
    - energy of the particle (with m, z(0), b and g only)
    - angular momentum of the particle (with m, z(0), b and g only)

    b) We slightly deflect the particles downwards. Find the frequency of small oscillations around the original intact trajectory.

    2. Relevant equation
    Lagrangian equations, equations for energy in conservative field and angular momentum.

    3. The attempt at a solution
    a) I didn't have any problem with Lagrangian and equation of motion for the polar coordinate r. Here is the result, I know how to do it:
    - Lagrangian: lagrangian.png (1)
    - equation of motion for the polar coordinate r: equation2.png (2)

    I have problem to express the energy and angular momentrum od particle. I show you my attempt:
    equations3.png
    So, the problem is that I dont know, how to express the time derivation of Θ.
    Could I just say, that the time derivation of Θ is the (gb)^(1/2) because the equation (2) is in standart form where ω^2=gb?

    b) Well, I don't have any idea. I just have the result:
    frequention.png
    And it doesn't make sence if I told that ω^2=gb.

    So, please, could you tell me what is wrong and what to do?

    Thank you.
     
  2. jcsd
  3. Nov 4, 2017 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    You should have two equations of motion, one for the ##\phi## coordinate and another for ##r##.

    You should also have ##\ddot{\phi} = 0## in you EOM.

    Edit: your equation (2) seems wrong to me you should be getting:

    $$0 = (1+b^2r^2)\ddot{r} + 2b^2 r\dot{r}^2+gbr - r\dot{\phi}^2=0$$

    But I don't see how you can find ##\dot{\phi}##.
     
    Last edited: Nov 4, 2017
  4. Nov 6, 2017 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Check the sign of the second term.

    Apply the ##r## equation of motion to the special case of circular motion. (What are ##\dot r## and ##\ddot r## for this case?)

    Note: Be sure not to confuse the angular velocity of the circular motion of part (a) with the angular frequency of small oscillations in part (b).
     
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