1.7 Evolution of Dynamical State, 2.1

Structure and Interpretation of Classical Mechanics

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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]

Eq. (1.110):

\displaystyle{  \left(D \Gamma[q] \right)(t)   = \left( 1, Dq(t), D^2 q(t), ... \right)  = \begin{bmatrix} 1 \\ Dq(t) \\ D^2 q(t) \\ ... \\ \end{bmatrix} \\   }

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\displaystyle{ \Gamma[q](t) = \left( t, q(t), D q(t), D^2 q(t), ... \right)  = \begin{bmatrix} t \\ q(t) \\ D q(t) \\ D^2 q(t) \\ ... \\ \end{bmatrix} \\   }

\displaystyle{ \Gamma[q] = \begin{bmatrix} I \\ q \\ D q \\ D^2 q \\ ... \\ \end{bmatrix} \\   },

where I(t) = t.

\displaystyle{ \Gamma[q_1, q_2](t) = \left( t, \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix},   D \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix},   D^2 \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix},   ... \right) = \begin{bmatrix} t \\ \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix} \\   D \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix} \\   D^2 \begin{bmatrix} q_1(t) \\ q_2(t) \end{bmatrix} \\   ... \\ \end{bmatrix} \\ }

The Lagrange equation:

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

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

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

\displaystyle{ \begin{aligned} D \left( \frac{\partial}{\partial \dot q_2} L \circ \Gamma[q_1, q_2] \right) - \left(\frac{\partial}{\partial q_2} L \circ \Gamma[q_1, q_2]\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]     &=     D \left(\frac{\partial}{\partial \dot q_1} L \circ \Gamma[q_1, q_2] \right)   \\       \frac{\partial}{\partial  q_2}      L \circ \Gamma[q_1, q_2]     &=     D \left( \frac{\partial}{\partial \dot q_2}        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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This part is wrong.

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

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

This can be proven wrong by just checking the dimensions of the matrix products.

The source of the errors is the second step. It puts the column matrix \displaystyle{\vec \partial_2 = \begin{bmatrix} \frac{\partial}{\partial \dot q_1} & \frac{\partial}{\partial \dot q_2} \end{bmatrix}^T} into each equation. Instead, each row of \displaystyle{\vec \partial_2} should match only one of the two equations.

[guess]

— Me@2022-07-07 05:20:38 PM

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2022.07.09 Saturday (c) All rights reserved by ACHK