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Varying inclination on plane: Undetermined multipliers

  1. Oct 27, 2015 #1
    1. The problem statement, all variables and given/known data
    A particle of mass ##m## rests on a smooth plane. The plane is raised to an inclination ##\theta## at constant rate ##\alpha##. Find the constraint force.

    2. Relevant equations


    3. The attempt at a solution
    ##L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-mgy## Lagrangian

    ##f=\frac{y}{x}-\tan\alpha t = 0 ## constraint equation

    ##\partial_y f = \frac{1}{x}##

    ##\partial_x f = \frac{-y}{x^2}##

    ##\partial_q L - d_t \partial_\dot{q} L + \lambda \partial_q f = 0 ## Method of undetermined multipliers formula.

    ##\to \boxed{m\ddot{x}+\lambda \frac{y}{x^2} = 0} \quad \boxed{mg+m\ddot{y}=\frac{\lambda}{x}}##

    Using tedious manipulation I've gotten to the point where I can say

    ##\ddot{x}x+\ddot{y}y+gy=0##

    And haven't found any other useful formula.

    I know I could switch to a polar coordinate basis and find ##r(t)## there and solve ##x## and ##y## and indirectly find constraint forces, but I'm not interested in that. Unless I'm clearly using the Lagrange undetermined multipliers.
     
    Last edited: Oct 27, 2015
  2. jcsd
  3. Oct 29, 2015 #2

    Geofleur

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    Science Advisor
    Gold Member

    Here's an idea: Write the Lagrangian in terms of polar coordinates, but keep both ## r ## and ## \phi ## as the generalized coordinates - do not incorporate the constraint into the kinetic or potential energy terms. Rather, incorporate it through the Lagrange multiplier technique.
     
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