Ex 1.14 Lagrange equations for L’

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Structure and Interpretation of Classical Mechanics

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Show by direct calculation that the Lagrange equations for \displaystyle{L'} are satisfied if the Lagrange equations for \displaystyle{L} are satisfied.

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Equation (1.69):

\displaystyle{C \circ \Gamma[q'] = \Gamma[q]}

Equation (1.70):

\displaystyle{L' = L \circ C}

Equation (1.71):

\displaystyle{L' \circ \Gamma[q'] = L \circ C \circ \Gamma[q'] = L \circ \Gamma[q]}

The Lagrange equation:

\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\ \end{aligned}}

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\displaystyle{ \begin{aligned} &\partial_2 L \circ \Gamma[q] \\ &= \frac{\partial}{\partial v} L(t, x, v) \\ &= \frac{\partial}{\partial v} L'(t, x', v') \\ &= \frac{\partial}{\partial x'} L'(t, x', v') \frac{\partial x'}{\partial v} + \frac{\partial}{\partial v'} L'(t, x', v') \frac{\partial v'}{\partial v} \\ \end{aligned}}

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Since it is just a coordinate transformation \displaystyle{x = F(t, x')}, \displaystyle{x} has no explicitly dependent on \displaystyle{v'}. Similarly, if we consider the coordinate transformation \displaystyle{x' = G(t, x)}, \displaystyle{x'} has no explicitly dependent on \displaystyle{v}. So

\displaystyle{ \begin{aligned} \frac{\partial x'}{\partial v} &= 0 \\ \end{aligned}}

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\displaystyle{ \begin{aligned} &\partial_2 L \circ \Gamma[q] \\ &= \frac{\partial}{\partial v'} L'(t, x', v') \frac{\partial v'}{\partial v} \\ \end{aligned}}

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— Me@2020-12-28 04:03:24 PM

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2020.12.28 Monday (c) All rights reserved by ACHK