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fluidistic

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## Homework Statement

A bead of mass m slides without friction along a wire which has the shape of a parabola y=Ax² with axis vertical in the Earth's gravitational field g.

a)Find the Lagrangian, taking as generalized coordinate the horizontal displacement x.

b)Write down the Lagrange's equation of motion.

## Homework Equations

Solved the problem.

## The Attempt at a Solution

a)[itex]L=\frac{m\dot x ^2}{2}(1+4A^2 x^2)-mgAx^2[/itex].

b)[itex]\ddot x (1+4A^2x^2 )-8A^2 \dot x ^2 x +2gAx=0[/itex].

For the fun of it, I wanted to get some information about the motion of the particle, by looking at either small oscilations or by solving the equation of motion.

However:

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1)For small oscillations I'd need to approximate the potential energy function by a quadratic function but it's already done so I'm "lucky" on that part. This is supposed to make the equation of motion more friendly.

2)I'm currently self studying mathematical methods in physics and I must say that the equation of motion equation looks really terrible! It's non linear and of degree two.

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So I was wondering if I could either solve this differential equation via some method or at least get some info about the behaviour of the motion. Using intuition the motion must be periodic, but I find nothing more than this. Any help is appreciated.

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