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

1. Aug 28, 2014

### sudipmaity

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. Aug 28, 2014

### BvU

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 !