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Lagrange multipliers in a simple pendulum

  1. Feb 27, 2012 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Not really a homework question, just want to check out if what I'm doing is right. I challenged myself to find the equation of motion and the forces in the simple pendulum system but with using the Lagrange multipliers and the constraint equation.


    2. Relevant equations
    In next part.


    3. The attempt at a solution
    Let [itex]l[/itex] be the length of the pendulum. I use polar coordinates and set my system of reference in the place where the mass is when at rest. The constraint equation is, I believe, [itex]l=\text{constant}=C[/itex].
    I write a modified Lagrangian which takes into account the constraint equation and the Lagrange multiplier. [itex]\tilde L =L-\lambda _ l (l-C)[/itex].
    The kinetic energy of the system is [itex]T=\frac{m\dot \theta ^2 l^2}{2}[/itex] while the potential energy is [itex]V=l-l\cos \theta[/itex].
    So that [itex]\tilde L=\frac{m\dot \theta ^2 l^2}{2}-(l-l\cos \theta)-\lambda _ l (l-C)[/itex]. I have thus 3 "generalized coordinates", [itex]\lambda _l[/itex] which should be worth the force of constraint, theta and [itex]l[/itex].
    Euler-Lagrange equations yield [itex]l=C[/itex] when considering [itex]\lambda_l[/itex] as generalized coordinate, [itex]\theta ^2 +\frac{g}{l}\sin \theta =0[/itex] where I had to consider that [itex]l=C[/itex] so that [itex]\dot l=0[/itex] (that was when considering theta as generalized coordinate) and finally for [itex]l[/itex] they yield [itex]\lambda _l=m \dot \theta ^2 l -mg (1+\cos \theta )[/itex] which has units of newton. So I am guessing this is the tension force or something like that, I am not really sure. This is a generalized force but I'm not sure with respect to what generalized coordinate. Any explanation is appreciated.
    I wonder if all I did is ok. Could someone clarify things up? Thank you!
    However I do not know how to get the generalized force, [itex]\lambda _l[/itex].
     
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