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Homework Help: Classical Mechanics problem

  1. Feb 2, 2008 #1
    1. The problem statement, all variables and given/known data

    Problem: Derive eigenvalue equation of motion of a system undergoing free small oscillation in three dimensional space.

    2. Relevant equations

    3. The attempt at a solution

    I want to know if I have correctly written the values of T and V---

    [tex]\ T=[/tex][tex]\frac{1}{2}[/tex][tex]\ m [/tex] [[tex]\dot{x}^2 +\dot{y}^2+ \dot{z}^2[/tex]]

    And [tex]\ V=[/tex][tex]\frac{1}{2}[/tex][tex]\ k [/tex][[tex]\ x^2+\ y^2+\ z^2 [/tex]]

    I m not sure if I have written the write thing...what should be the sign of k.Should I write -ve?And is it OK to assume that the force constants are equal?What would be if they were not equal?Would they simply add like (1/2)(k1+k2) x^2?

    If I can form T and V,I am confident that I will be able to do the rest of calculations.
    Last edited: Feb 2, 2008
  2. jcsd
  3. Feb 2, 2008 #2
    It appears to me that the potential energy would be

    [tex]\ V= [/tex][tex]\frac{1}{2}[/tex][[tex]\ k_1+k_2 [/tex]][tex]\ x^2 [/tex]+...

    From the elimentary knowledge, we know when a spring of natural length is compressed or elongated, workdone on the particle is -(1/2)kx^2. So,...(1/2)kx^2 amount of energy is stored within the spring as potential energy.So, V would be +ve.


    So, we are to omit the factor of half in V in the first post. Please tell me if I am correct.
    Last edited: Feb 2, 2008
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