(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle of mass 'm' slides on a smooth surface, the shape of which is given by [itex] y = Ax^{2} [/itex] where A is a positive constant of suitable dimensions and y is measured along the vertical direction. The particle is moved slightly away from the position of equilibrium and then released. Write down the equation of motion of the particle.

2. Relevant equations

Lagrangian

[tex] T = \frac{1}{2}m(\dot{y})^{2} [/tex]

[tex] V = mgy [/tex]

3. The attempt at a solution

Using the chain rule for my derivatives and plugging in the functions for the lagrangian I got:

[tex] L = \frac{1}{2}m(2Ax\dot{x})^{2} - mgAx^{2} [/tex]

Applying this to the Euler-Lagrange equations:

[tex] \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = 8mA^{2}x(\dot{x})^{2} + 4mA^2x^{2}\ddot{x} [/tex]

[tex] \frac{\partial L}{\partial x} = -2mgAx [/tex]

End up with, after some cancelling,:

[tex] 2A^{2}x^{2}\ddot{x} + 4A^{2}x\dot{x}^{2} + gAx = 0 [/tex]

However the solution I have says the final answer should be:

[tex] (1 + 4A^{2}x^{2})\ddot{x} + 4A^{2}x\dot{x}^{2} + 2Agx = 0 [/tex]

I can show more detail in my work if needed, I just didn't want to type it all out.

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# Simple Lagrangian question, not getting right answer

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