Ex 1.21 The dumbbell, 3

Structure and Interpretation of Classical Mechanics

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c. Make a change of coordinates to a coordinate system with center of mass coordinates x_{cm}, y_{cm}, angle \theta, distance between the particles c, and tension force F. Write the Lagrangian in these coordinates, and write the Lagrange equations.

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[guess]

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{m_0 y_0 + m_1 y_1}{m_0 + m_1} \\   x_{cm} &= \frac{m_0 x_0 + m_1 x_1}{m_0 + m_1} \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{m_0 y_0 + m_1 (y_0 + c(t) \sin \theta)}{m_0 + m_1} \\   x_{cm} &= \frac{m_0 x_0 + m_1 (x_0 + c(t) \cos \theta)}{m_0 + m_1} \\   \end{aligned}}

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Let \displaystyle{M = m_0 + m_1}.

\displaystyle{ \begin{aligned}   y_{cm} &= \frac{M y_0 + m_1 c(t) \sin \theta}{M} \\   x_{cm} &= \frac{M x_0 + m_1 c(t) \cos \theta}{M} \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_0 &= y_{cm} - \frac{m_1}{M} c(t) \sin \theta \\   x_0 &= x_{cm} - \frac{m_1}{M} c(t) \cos \theta \\   \end{aligned}}

\displaystyle{ \begin{aligned}   y_1 &= y_{cm} + \frac{m_0}{M} c(t) \sin \theta \\   x_1 &= x_{cm} + \frac{m_0}{M} c(t) \cos \theta \\   \end{aligned}}

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The Lagrangian

\displaystyle{ \begin{aligned}   L &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2) + \lambda \left[ (x_1(t) - x_0(t))^2 + (y_1(t) - y_0(t))^2 - l^2 \right] \\     &= \frac{1}{2} m_0 (\dot x_0^2 + \dot y_0^2) + \frac{1}{2} m_1 (\dot x_1^2 + \dot y_1^2)   - \frac{F}{2l} \left[ (c(t))^2 - l^2 \right] \\   \end{aligned}}

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\displaystyle{ \begin{aligned}   \dot y_0 &= \dot y_{cm} - \frac{m_1}{M} \dot c(t) \sin \theta - \frac{m_1}{M} c(t) \dot \theta \cos \theta\\  \dot x_0 &= \dot x_{cm} - \frac{m_1}{M} \dot c(t) \cos \theta - \frac{m_1}{M} c(t) \dot \theta \sin \theta \\   \end{aligned}}

[guess]

— Me@2021-04-17 05:40:46 PM

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