Ex 1.14 Lagrange equations for L’, 2

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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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\displaystyle{x = F(t, x')}

\displaystyle{x' = G(t, x)}

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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} \\ &= \frac{\partial}{\partial v'} L'(t, x', v') \frac{\partial v'}{\partial v} \\ \end{aligned}}

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\displaystyle{v' = \partial_0 G(t, x) + \partial_1 G(t,x) v}

\displaystyle{ \begin{aligned} \frac{\partial v'}{\partial v} &= \partial_1 G(t,x) = \frac{\partial x'}{\partial x} \\ \end{aligned}}

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\displaystyle{ \begin{aligned} &\partial_2 L \circ \Gamma[q] \\ &= \frac{\partial L'}{\partial v'} \frac{\partial x'}{\partial x} \\ \end{aligned}}

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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_1 L \circ \Gamma[q] \\ &= \partial_1 L' \circ \Gamma[q'] \\ &= \frac{\partial}{\partial x} L' (t, x', v') \\ &= \frac{\partial L'}{\partial x'} \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} \\ \end{aligned}}

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\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\ D \left( \frac{\partial L'}{\partial v'} \frac{\partial x'}{\partial x} \right) - \left( \frac{\partial L'}{\partial x'} \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} \right) &= 0 \\ D \left( \frac{\partial L'}{\partial v'} \right) \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} D \left( \frac{\partial x'}{\partial x} \right) - \left( \frac{\partial L'}{\partial x'} \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} \right) &= 0 \\ \end{aligned}}

\displaystyle{ \begin{aligned} \left[ D \left( \frac{\partial L'}{\partial v'} \right) - \frac{\partial L'}{\partial x'} \right] \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} D \left( \frac{\partial x'}{\partial x} \right) - \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} &= 0 \\ \end{aligned}}

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\displaystyle{ \begin{aligned} &D \left( \frac{\partial x'}{\partial x} \right) \\ &= \frac{d}{dt} \left( \frac{\partial x'}{\partial x} \right) \\ &= \frac{\partial}{\partial t} \left( \frac{\partial x'}{\partial x} \right) \frac{\partial t}{\partial t} + \frac{\partial}{\partial x'} \left( \frac{\partial x'}{\partial x} \right) \frac{\partial x'}{\partial t} + \frac{\partial}{\partial v'} \left( \frac{\partial x'}{\partial x} \right) \frac{\partial v'}{\partial t} \\ &= \frac{\partial}{\partial t} \left( \frac{\partial x'}{\partial x} \right) + \frac{\partial}{\partial x} \left( \frac{\partial x'}{\partial x'} \right) \frac{\partial x'}{\partial t} + \frac{\partial}{\partial x} \left( \frac{\partial x'}{\partial v'} \right) \frac{\partial v'}{\partial t} \\ &= \frac{\partial}{\partial t} \left( \frac{\partial x'}{\partial x} \right) + \frac{\partial}{\partial x} \left( 1 \right) \frac{\partial x'}{\partial t} + \frac{\partial}{\partial x} \left( 0 \right) \frac{\partial v'}{\partial t} \\ &= \frac{\partial}{\partial x} \left( \frac{\partial x'}{\partial t} \right) + 0 + 0 \\ &= \frac{\partial v'}{\partial x} \\ \end{aligned}}

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\displaystyle{ \begin{aligned} \left[ D \left( \frac{\partial L'}{\partial v'} \right) - \frac{\partial L'}{\partial x'} \right] \frac{\partial x'}{\partial x} + \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} - \frac{\partial L'}{\partial v'} \frac{\partial v'}{\partial x} &= 0 \\ \left[ D \left( \frac{\partial L'}{\partial v'} \right) - \frac{\partial L'}{\partial x'} \right] \frac{\partial x'}{\partial x} &= 0 \\ D \left( \frac{\partial L'}{\partial v'} \right) - \frac{\partial L'}{\partial x'} &= 0 \\ \end{aligned}}

— Me@2021-01-06 08:10:46 PM

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