Lagrange multipliers rotating masses connected by spring

[kanne]

Homework Statement

A particle of mass, m1, is constrained to move in a circle with radius a at z=0 and another particle of mass, m2, moves in a circle of radius b at z=c. For this we wish to write up the Lagrangian introucing the constraints by lagrange multipliers and solve the following equations of motion.

Homework Equations

Equations of constraint.

$$z1=0\ \ x1^2+y1^2=a^2$$
$$z2=c \ \ x2^2+y2^2=b^2$$

The Attempt at a Solution

I am working on the Lagrangian given by

$$L=T-V+\lambda_1\left(-z_1\right)+\lambda_2\left(c-z_2\right)+\lambda_3\left(a^2-x_1^2-y_1^2\right)+\lambda_4\left(b^2-x_2^2-y_2^2\right)$$

with
$$T=\frac{1}{2}m_1\left(\dot{x}_1^2+\dot{y}_1^2+\dot{z}_2^2\right)+\frac{1}{2}m_2\left(\dot{x}_2^2+\dot{y}_2^2+\dot{z}_2^2\right)$$
and
$$V=\frac{1}{2}k\left(\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2\right)$$

From this we get the equations of motion:

$$m_1\ddot{x_1}=-k\left(x_2-x_1\right)-2\lambda_3x_1$$

$$m_1\ddot{y_1}=-k\left(y_2-y_1\right)-2\lambda_3y_1$$

$$m_1\ddot{z_1}=-k\left(z_2-z_1\right)-\lambda_1$$

$$m_2\ddot{x_2}=k\left(x_2-x_1\right)-2\lambda_4x_2$$

$$m_2\ddot{y_2}=k\left(y_2-y_1\right)-2\lambda_4y_2$$

$$m_2\ddot{z_2}=k\left(z_2-z_1\right)-\lambda_2 [\tex]<br /> <br /> <br /> Anyone know how to solve these equations of motion$$

 

[Irid]

You could reduce the number of coordinates by introducing polar coordinates for each mass. There would be less equations to solve.