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Stable points of a particle in a 2d potential field.

  1. Aug 28, 2014 #1
    1. The problem statement, all variables and given/known data

    Let a particle of mass m moving in 2d space in a potential V (x, y) = -1/2 kx2 + 1/2 λ0 x2y2 + 1/4λ1x where k,λ01 > 0.At what point (x0, y0) is the particle in stable equilibrium? 2 marks

    2. Relevant equations
    ∂V/∂x=0 ∂V/∂y=0; ∂2V/∂x2 > 0;∂2V/ ∂y2 > 0



    3. The attempt at a solution
    After using relevant equations I get x= λ1/4 (k-y2 λ0)
    and λ0 x2 y =0.
    If I take x or y=0 then the point become unstable.
    Don't know what to do next.The problem however simplifies very much for a 2 mark question if the first term of the potential right side is positive.Does anybody think that it could be a typo error? Or is this problem solvable in the given form?
    If the first term were to be positive then stable point would be (λ1 /4k , 0) .This would satisfy all the conditions.
     
  2. jcsd
  3. Aug 28, 2014 #2

    BvU

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    That's funny: I get ## x = {\lambda_1/4 \over k-\lambda_0 y^2}## (but that's probably what you mean, right ?).
    There is no taking x = 0 because it doesn't satisfy that equation. So all that's left is ##y = 0, x = {\lambda_1 \over 4k}## .
    Indeed, at that point the second derivative wrt x is negative, so the answer is: nowhere.

    I'm with you is suspecting this is a typo. Ask teacher !
     
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