I have a classical mechanics question I couldn't conclude. The reason seems to be mathematical. It's this:(adsbygoogle = window.adsbygoogle || []).push({});

There's a paraboloid shaped plane of mass M, which is standing on a frictionless surface and can slide freely. It's surface is [tex]y=ax^2[/tex]. A point mass m is place on the plane. Solve the Euler-Lagrange equations of the system for little mass.

My (unsuccessful) solution is as follows:

I chose [tex]x_m[/tex] as the x (horizontal) coordiante of point mass, and for sliding, plane it's [tex]x_M[/tex]. I defined another coordinate for point mass: [tex]x[/tex] is the horizontal distance of point mass from the center (or bottom) of the parabol. So, coordinates of m are

[tex]x_m = x_M + x[/tex]

[tex]y_m = ax^2[/tex]

so velocities are

[tex]\dot{x_m} = \dot{x_M} + \dot{x}[/tex]

[tex]y_m = a2x\dot{x}[/tex]

So kinetic energy of the system is:

[tex]T = \frac{m\dot{x_m}^2}{2} + \frac{M\dot{x_M}^2}{2}[/tex]

[tex]T = \frac{m ( \dot{x_M}^2 + \dot{x}^2 + 2\dot{x}\dot{x_M} + 4a^2x^2\dot{x}^2 )}{2} + \frac{M\dot{x_M}^2}{2}[/tex]

and potential is

[tex]V = mgy = mgax^2[/tex]

and

[tex]{\cal L} = T-V[/tex]

Now, Euler-Lagrange equations are:

(1) [tex]\frac{d}{dt} \frac{\partial {\cal L}}{\partial \dot{x_M}} - \frac{\partial {\cal L}}{\partial x_M}[/tex]

(2) [tex]\frac{d}{dt} \frac{\partial {\cal L}}{\partial \dot{x}} - \frac{\partial {\cal L}}{\partial x}[/tex]

The solution of the first one yields (of course, if I done the math correctly):

[tex]\ddot{x_M}(m+M) + \ddot{x}m = 0[/tex]

and second is

[tex]\ddot{x} + \ddot{x_M} + 4a^2x\dot{x}^2 4a^2x^2\ddot{x} + 2gax = 0[/tex]

I tried combining two solutions and got this non-linear differential equation:

[tex]\ddot{x}(1- \frac{m}{m+M} + 4a^2x^2) + 4a^2x\dot{x}^2 + 2gax = 0[/tex]

Maybe there was a mathematical mistake in my solution, maybe this differential equation could be solved with some tricks, or maybe I'm downright wrong by choosing such coordinates. I don't know.

Is there someone who can solve this question?

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# Homework Help: Euler-Lagrange equation for paraboloid plane

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