- #1

citheo

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## Homework Statement

Suppose you have an object of mass m that is constrained to move on an ellipsoid with a constraint function [itex]f(x,y,z) = x^2+4y^2+4z^2 -1=0[/itex]. Aside from the force of constraint, the only force acting on the mass is an elastic force [itex]\vec{F}=-kx\hat{x}[/itex]. Find the Lagrangian, the Hamiltonian and the integrals of motion.

## Homework Equations

Euler-Lagrange (EL) equation, Lagrange multipliers, Legendre transform, any equations related to classical mechanics..

## The Attempt at a Solution

The kinetic energy is [itex]K = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}m\dot{y}^2 + \frac{1}{2}m\dot{z}^2[/itex]. The potential energy is just the elastic potential energy [itex]U=\frac{1}{2}kx^2[/itex]. The Lagrangian L can be found in the usual way as [itex]L=K-U[/itex]. By applying a Legendre transform I get a Hamiltonian [itex]H = p_x\dot{x}+p_y\dot{y}+p_z\dot{z} - L[/itex] where I can find the generalised momenta by solving [itex]p_i = \frac{\partial L}{\partial q_i}[/itex] for p. Plugging p back into H I can summarize what I have found so far:

[tex]

L = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}m\dot{y}^2 + \frac{1}{2}m\dot{z}^2 - \frac{1}{2}kx^2 \\

H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{p_z^2}{2m} + \frac{1}{2}kx^2

[/tex]

Ok, so in order to find the integrals of motion, I basically have to find the equations of motion and integrate them if I'm not mistaken. I plug in the Lagrangian into the EL equation, taking into account the constraint with the help of a Lagrange multiplier

[tex]

\frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = \lambda\frac{\partial f}{\partial q_i}

[/tex]

which gives the equations of motion

[tex]

m\ddot{x}+kx+2\lambda x = 0 \\

m\ddot{y} + 8\lambda y = 0 \\

m\ddot{z} + 8\lambda z = 0.

[/tex]

I can find the integrals of motion if I multiply each EOM with [itex]Q_i(q_i) = \dot{q}_i[/itex], rearranging, integrating etc..

The problem for me lies in finding [itex]\lambda[/itex]; it's just a mess of fractions and radicals. Is there perhaps a way to describe the system in a nicer set of coordinates or maybe some other method altogether?