Lagrange multipliers rotating masses connected by spring

  • #1
kanne
3
0

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.

[tex]z1=0\ \ x1^2+y1^2=a^2[/tex]
[tex]z2=c \ \ x2^2+y2^2=b^2[/tex]

The Attempt at a Solution



I am working on the Lagrangian given by

[tex]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)[/tex]

with
[tex]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)[/tex]
and
[tex]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)[/tex]

From this we get the equations of motion:

[tex]m_1\ddot{x_1}=-k\left(x_2-x_1\right)-2\lambda_3x_1[/tex]

[tex]m_1\ddot{y_1}=-k\left(y_2-y_1\right)-2\lambda_3y_1[/tex]

[tex]m_1\ddot{z_1}=-k\left(z_2-z_1\right)-\lambda_1[/tex]

[tex]m_2\ddot{x_2}=k\left(x_2-x_1\right)-2\lambda_4x_2[/tex]

[tex]m_2\ddot{y_2}=k\left(y_2-y_1\right)-2\lambda_4y_2[/tex]

[tex]m_2\ddot{z_2}=k\left(z_2-z_1\right)-\lambda_2 [\tex]


Anyone know how to solve these equations of motion
 
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  • #2
You could reduce the number of coordinates by introducing polar coordinates for each mass. There would be less equations to solve.
 
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