1.7 Evolution of Dynamical State, 2.1

Structure and Interpretation of Classical Mechanics

.

\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.

~~~

[guess]

Eq. (1.110):

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

.

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

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

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}}

.

\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}}

.

\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}}

.

\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}}

.

\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}}

.

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

.

.