Daily Archives: October 15, 2022

Ex 1.28 Kinetic energy contains terms that are linear

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Structure and Interpretation of Classical Mechanics

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An analogous result holds when the f_\alpha‘s depend explicitly on time.

a. Show that in this case the kinetic energy contains terms that are linear in the generalized velocities.

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\displaystyle{\begin{aligned} \mathbf{v_\alpha} &= \partial_0 f_\alpha (t,q) + \partial_1 f_\alpha (t,q) v \\ T(t,q,v) &= \frac{1}{2} \sum_\alpha m_\alpha v^2_\alpha \\ v_\alpha &= |\mathbf{v}_\alpha| \\ v &= \text{generalized velocity} \\ \mathbf{v}_\alpha &= \text{rectangular velocity} \\ \end{aligned}}

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\displaystyle{\begin{aligned} &T(t,q,v) \\ &= \frac{1}{2} \sum_\alpha m_\alpha |\partial_0 f_\alpha (t,q) + \partial_1 f_\alpha (t,q) v|^2 \\ &= \frac{1}{2} \sum_\alpha m_\alpha \left \{ [\partial_0 f_\alpha (t,q)]^2 + 2 \partial_0 f_\alpha (t,q) \partial_1 f_\alpha (t,q) v + [\partial_1 f_\alpha (t,q) v]^2 \right \} \\ \end{aligned}}

— Me@2022-10-15 11:17:59 AM

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