1.7 Evolution of Dynamical State, 2.2

Structure and Interpretation of Classical Mechanics

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\displaystyle{ \begin{aligned}     \partial_1 L \circ \Gamma[q]     &= D ( \partial_2 L \circ \Gamma[q]) \\ \\    &= \partial_0 ( \partial_2 L \circ \Gamma[q]) Dt +  \partial_1 ( \partial_2 L \circ \Gamma[q]) Dq + \partial_2 ( \partial_2 L \circ \Gamma[q]) Dv \\ \\     &= \partial_0 \partial_2 L \circ \Gamma[q] +  ( \partial_1 \partial_2 L \circ \Gamma[q]) Dq + (\partial_2 \partial_2 L \circ \Gamma[q]) D^2 q \\ \\     \end{aligned}}

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\displaystyle{ \begin{aligned}      (\partial_2 \partial_2 L \circ \Gamma[q]) D^2 q     &=     \partial_1 L \circ \Gamma[q]     - \partial_0 \partial_2 L \circ \Gamma[q]     - (\partial_1 \partial_2 L \circ \Gamma[q]) Dq  \\ \\         D^2 q     &=     \left[ \partial_2 \partial_2 L \circ \Gamma[q] \right]^{-1}    \left\{ \partial_1 L \circ \Gamma[q]     - \partial_0 \partial_2 L \circ \Gamma[q]     - (\partial_1 \partial_2 L \circ \Gamma[q]) Dq  \right\} \\ \\      \end{aligned}}

where \displaystyle{\left[ \partial_2 \partial_2 L \circ \Gamma \right]} is a structure that can be represented by a symmetric square matrix, so we can compute its inverse.

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[guess]

The Lagrange equation:

\displaystyle{ \begin{aligned} D \left( \frac{\partial}{\partial \dot q_1} L \circ \Gamma[\begin{bmatrix} q_1 \\ q_2 \end{bmatrix}] \right) - \left(\frac{\partial}{\partial q_1} L \circ \Gamma[\begin{bmatrix} q_1 \\ q_2 \end{bmatrix}]]\right) &= 0 \\ \end{aligned}}

\displaystyle{ \begin{aligned} D \left( \frac{\partial}{\partial \dot q_2} L \circ \Gamma[\begin{bmatrix} q_1 \\ q_2 \end{bmatrix}]] \right) - \left(\frac{\partial}{\partial q_2} L \circ \Gamma[\begin{bmatrix} q_1 \\ q_2 \end{bmatrix}]]\right) &= 0 \\ \end{aligned}}

\displaystyle{ \begin{aligned}   \frac{d}{dt} \left( \frac{\partial}{\partial \dot q_1} L (t, q_1(t), \dot q_1(t), q_2(t), \dot q_2(t)) \right) - \frac{\partial}{\partial q_1} L (t, q_1(t), \dot q_1(t), q_2(t), \dot q_2(t))    &= 0       \end{aligned}}

\displaystyle{ \begin{aligned}   \frac{d}{dt} \left( \frac{\partial}{\partial \dot q_2} L (t, q_1(t), \dot q_1(t), q_2(t), \dot q_2(t)) \right) - \frac{\partial}{\partial q_2} L (t, q_1(t), \dot q_1(t), q_2(t), \dot q_2(t))    &= 0       \end{aligned}}

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\displaystyle{   \begin{aligned} D \left( \begin{bmatrix} \frac{\partial}{\partial \dot q_1}  \\ \frac{\partial}{\partial \dot q_2}  \end{bmatrix}       L \circ \Gamma[q_1, q_2] \right)     - \left(     \begin{bmatrix} \frac{\partial}{\partial q_1}  \\ \frac{\partial}{\partial  q_2}  \end{bmatrix}     L \circ \Gamma[q_1, q_2]\right) &= 0 \\       D \left(     \vec \partial_2    L \circ \Gamma[q_1, q_2] \right)     - \left(     \vec \partial_1     L \circ \Gamma[q_1, q_2]\right) &= 0 \\       \end{aligned}}

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\displaystyle{ \begin{aligned}         \vec \partial_1 L \circ \Gamma[q] &= D ( \vec \partial_2 L \circ \Gamma[q]) \\ \\         \begin{bmatrix} \frac{\partial}{\partial q_1}  \\ \frac{\partial}{\partial  q_2}  \end{bmatrix}     L \circ \Gamma[q_1, q_2]     &=     D \left( \begin{bmatrix} \frac{\partial}{\partial \dot q_1}  \\ \frac{\partial}{\partial \dot q_2}  \end{bmatrix}       L \circ \Gamma[q_1, q_2] \right)   \\  \\    \frac{\partial}{\partial q_1}      L \circ \Gamma[q_1, q_2]     &=     \partial_0 \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) Dt \\    &+ \partial_{q_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D q_1     + \partial_{v_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D v_1 \\    &+ \partial_{q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D q_2     + \partial_{v_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D v_2       \\       \frac{\partial}{\partial  q_2}      L \circ \Gamma[q_1, q_2]     &=     ...      \\      \end{aligned}}

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\displaystyle{ \begin{aligned}         \frac{\partial}{\partial q_1}      L \circ \Gamma[q_1, q_2]     &=     \partial_0 \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) Dt \\    &+ \frac{\partial}{\partial q_1}  \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D q_1       + \frac{\partial}{\partial q_2}  \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D q_2 \\    &+ \frac{\partial}{\partial \dot q_1}  \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D^2 q_1     + \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right) D^2 q_2       \\      \frac{\partial}{\partial  q_2}      L \circ \Gamma[q_1, q_2]     &=     \partial_0 \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) Dt \\    &+ \frac{\partial}{\partial q_1}  \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) D q_1       + \frac{\partial}{\partial q_2}  \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) D q_2 \\    &+ \frac{\partial}{\partial \dot q_1}  \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) D^2 q_1     + \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) D^2 q_2         \\           \end{aligned}}

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\displaystyle{\begin{aligned}    \vec \partial_1    L \circ \Gamma[q_1, q_2]     &=     \partial_0 \left(\vec \partial_2 L \circ \Gamma[q_1, q_2] \right) Dt \\    &+ \frac{\partial}{\partial q_1}  \left(    \vec \partial_2    L \circ \Gamma[q_1, q_2] \right) D q_1       + \frac{\partial}{\partial q_2}  \left(    \vec \partial_2     L \circ \Gamma[q_1, q_2] \right) D q_2 \\    &+ \frac{\partial}{\partial \dot q_1}  \left(    \vec \partial_2     L \circ \Gamma[q_1, q_2] \right) D^2 q_1     + \frac{\partial}{\partial \dot q_2} \left(    \vec \partial_2  L \circ \Gamma[q_1, q_2] \right) D^2 q_2       \\ \\    \end{aligned}}

\displaystyle{\begin{aligned}    \begin{bmatrix}     \frac{\partial}{\partial q_1} \\    \frac{\partial}{\partial q_2}    \end{bmatrix}    L \circ \Gamma[q_1, q_2]     &=     \partial_0 \left(  \begin{bmatrix}     \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}    \end{bmatrix}   L \circ \Gamma[q_1, q_2] \right) Dt \\    &+ \frac{\partial}{\partial q_1}  \left(    \begin{bmatrix}     \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}    \end{bmatrix}      L \circ \Gamma[q_1, q_2] \right) D q_1       + \frac{\partial}{\partial q_2}  \left(    \begin{bmatrix}     \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}    \end{bmatrix}       L \circ \Gamma[q_1, q_2] \right) D q_2 \\    &+ \frac{\partial}{\partial \dot q_1}  \left(    \begin{bmatrix}     \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}    \end{bmatrix}       L \circ \Gamma[q_1, q_2] \right) D^2 q_1     + \frac{\partial}{\partial \dot q_2} \left(    \begin{bmatrix}     \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}    \end{bmatrix}    L \circ \Gamma[q_1, q_2] \right) D^2 q_2       \\ \\    \end{aligned}}

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\displaystyle{ \begin{aligned}     \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D^2 q_1 + \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D^2 q_2    &= \frac{\partial}{\partial q_1} L \circ \Gamma[\vec q]     - \partial_0 \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) Dt \\     &- \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D q_1 - \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D q_2 \\      \\      \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D^2 q_1 + \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D^2 q_2      &= \frac{\partial}{\partial q_2} L \circ \Gamma[\vec q] - \partial_0 \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) Dt \\     &- \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D q_1 - \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D q_2 \\       \end{aligned}}

\displaystyle{ \begin{aligned}     &\begin{bmatrix}  \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) &    \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) \\       \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) &    \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right)    \end{bmatrix} \begin{bmatrix}  D^2 q_1 \\ D^2 q_2 \end{bmatrix} \\      &= \begin{bmatrix} \frac{\partial}{\partial q_1} L \circ \Gamma[\vec q]     - \partial_0 \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) Dt      - \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D q_1 - \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[\vec q] \right) D q_2 \\      \\    \frac{\partial}{\partial q_2} L \circ \Gamma[\vec q] - \partial_0 \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) Dt      - \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D q_1 - \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_2} L \circ \Gamma[\vec q] \right) D q_2 \\   \end{bmatrix}      \end{aligned}}

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\displaystyle{ \begin{aligned}     &\left(\begin{bmatrix}  \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_1} \right) &    \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_1} \right) \\       \frac{\partial}{\partial \dot q_1} \left(\frac{\partial}{\partial \dot q_2} \right) &    \frac{\partial}{\partial \dot q_2} \left(\frac{\partial}{\partial \dot q_2} \right)    \end{bmatrix} L \circ \Gamma[\vec q] \right)\begin{bmatrix}  D^2 q_1 \\ D^2 q_2 \end{bmatrix} \\      &=     \begin{bmatrix} \frac{\partial}{\partial q_1}       \\    \frac{\partial}{\partial q_2}   \end{bmatrix} L \circ \Gamma[\vec q]    -    \left( \partial_0    \begin{bmatrix}      \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}       \end{bmatrix} L \circ \Gamma[\vec q] \right) Dt    -    \left(  \begin{bmatrix}        \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_1}  \right)      & \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_1}  \right)  \\      \\         \frac{\partial}{\partial q_1} \left(\frac{\partial}{\partial \dot q_2} \right)     & \frac{\partial}{\partial q_2} \left(\frac{\partial}{\partial \dot q_2} \right)  \\   \end{bmatrix}    L \circ \Gamma[\vec q] \right)    \begin{bmatrix}  D q_1 \\ D q_2 \end{bmatrix} \\      \end{aligned}}

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\displaystyle{ \begin{aligned}     &\left(    \begin{bmatrix}  \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2} \\    \end{bmatrix}     \begin{bmatrix}  \frac{\partial}{\partial \dot q_1} &    \frac{\partial}{\partial \dot q_2} \\       \end{bmatrix}         L \circ \Gamma[\vec q] \right)\begin{bmatrix}  D^2 q_1 \\ D^2 q_2 \end{bmatrix} \\      &=     \begin{bmatrix} \frac{\partial}{\partial q_1}       \\    \frac{\partial}{\partial q_2}   \end{bmatrix} L \circ \Gamma[\vec q]    -    \left( \partial_0    \begin{bmatrix}      \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}       \end{bmatrix} L \circ \Gamma[\vec q] \right) Dt    -    \left(    \begin{bmatrix}        \frac{\partial}{\partial \dot q_1} \\    \frac{\partial}{\partial \dot q_2}     \end{bmatrix}      \begin{bmatrix}        \frac{\partial}{\partial q_1}   &    \frac{\partial}{\partial q_2}         \end{bmatrix}    L \circ \Gamma[\vec q] \right)    \begin{bmatrix}  D q_1 \\ D q_2 \end{bmatrix} \\      \end{aligned}}

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\displaystyle{ \begin{aligned}     &\left( \vec \partial_2 \vec \partial_2^T     L \circ \Gamma[\vec q] \right) D^2 \vec q \\    &= \vec \partial_1 L \circ \Gamma[\vec q]    - \left( \partial_0 \vec \partial_2 L \circ \Gamma[\vec q] \right)     - \left( \vec \partial_2 \vec \partial_1^T L \circ \Gamma[\vec q] \right) D \vec q \\      \end{aligned}}

[guess]

— Me@2022-07-07 05:20:38 PM

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2022.07.09 Saturday (c) All rights reserved by ACHK