Ex 1.28 Prequel 1

Structure and Interpretation of Classical Mechanics

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An analogous result holds when the f_\alpha‘s depend explicitly on time.

a. Show that in this case the kinetic energy contains terms that are linear in the generalized velocities.

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Eq. (1.141):

\displaystyle{  \textbf{v}_\alpha = \partial_0 f_\alpha (t,q) + \partial_1 f_\alpha (t, q) v  }

Eq. (1.142):

\displaystyle{T(t, q, v) = \frac{1}{2} \sum_{\alpha} m_\alpha v^2_\alpha}

Eq. (1.133):

\displaystyle{  \begin{aligned}  \mathcal{P}_i &= (\partial_2 L)_i \\   \end{aligned}}

where \displaystyle{\mathcal{P}} is called the generalized momentum.

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\displaystyle{  \begin{aligned}  \mathcal{P} \dot Q &= (\partial_2 L) \dot Q  \\   &= (\partial_2 (T-V)) \dot Q  \\   &= (\partial_2 T) \dot Q  \\   \end{aligned}}

where \displaystyle{\dot Q} is the velocity selector. And V has no velocity component?

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“… T is a homogeneous function of the generalized velocities of degree 2.”

\displaystyle{  \begin{aligned}  n &= 2 \\   (\partial_2 T) \dot Q &= ? \\   \end{aligned}}

\displaystyle{  \begin{aligned}  x Df(x) &= nf(x)  \\   \end{aligned}}

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What does the D actually mean?

\displaystyle{ \begin{aligned}    x Df(x) &= nf(x)  \\   x \frac{d}{dx} f(x) &= nf(x)  \\   \end{aligned}}

\displaystyle{ \begin{aligned}    \mathbf{v} D T (\mathbf{v}) &= 2 T (\mathbf{v}) ? \\     \end{aligned}}

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\displaystyle{ \begin{aligned}     \partial_{\vec 2}   = \frac{\partial}{\partial \vec{\dot q}}   &= \begin{bmatrix} \frac{\partial}{\partial \dot q_1} & \frac{\partial}{\partial \dot q_2} & ... \end{bmatrix} \\ \\      \mathcal{P} \dot Q &= (\partial_2 T) \dot Q \\ \\    \begin{bmatrix} \mathcal{P}_1 & \mathcal{P}_2 & ... \end{bmatrix}     \begin{bmatrix} \dot Q_1 \\ \dot Q_2 \\ \vdots \end{bmatrix}     &=    \begin{bmatrix} (\partial_2 T)_1 & (\partial_2 T)_2 & ... \end{bmatrix}     \begin{bmatrix} \dot Q_1 \\ \dot Q_2 \\ \vdots \end{bmatrix} \\     \\    &=    \begin{bmatrix} \frac{\partial T}{\partial \dot q_1} &  \frac{\partial T}{\partial \dot q_2}  & ... \end{bmatrix}     \begin{bmatrix} \dot Q_1 \\ \dot Q_2 \\ \vdots \end{bmatrix} \\     \\     &=    \frac{\partial T}{\partial \dot q_1} \dot Q_1 + \frac{\partial T}{\partial \dot q_2} \dot Q_2  + ...     \\       \end{aligned}}

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Euler’s Homogeneous Function Theorem:

If

\displaystyle{ \begin{aligned}      f(t x_1, t x_2, ...) &= t^n f( x_1, x_2 , ...),  \\     \end{aligned}}

then

\displaystyle{ \begin{aligned}      \sum_{i} x_i \frac{\partial f}{\partial x_i} &= n f(\mathbf{x}) \\    \end{aligned}}

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\displaystyle{T(t, q, v) = \frac{1}{2} \sum_{\alpha} m_\alpha v^2_\alpha}

So

\displaystyle{\begin{aligned}     T(t, q, uv)     &= \frac{1}{2} \sum_{\alpha} m_\alpha (uv)^2_\alpha \\    &= u^2 T(t, q, v) \\    \end{aligned}    }

Therefore,

\displaystyle{ \begin{aligned}     \frac{\partial T}{\partial \dot q_1} \dot Q_1 + \frac{\partial T}{\partial \dot q_2} \dot Q_2  + ...     &= 2 T \\     \end{aligned}}

— Me@2022.09.27 12:26:09 PM

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2022.09.27 Tuesday (c) All rights reserved by ACHK

Encyclopædia Britannica

The preceding illustrates, simply but clearly, one of the proclivities that are often associated with mathematical thought: relatively simple concepts (such as integers), initially based on very concrete operations (for example, counting), are found to be capable of assuming novel meanings and potential uses, extending far beyond the limits of the concept as originally defined.

— Encyclopædia Britannica

— 31.12.2002

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2022.09.27 Tuesday ACHK