# Ex 1.27 Identifying total time derivatives

Structure and Interpretation of Classical Mechanics

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From equation (1.112), we see that $\displaystyle{G}$ must be linear in the generalized velocities

$\displaystyle{ G(t, q, v) = G_0(t, q, v) + G_1(t, q, v) v }$

where neither $\displaystyle{G_1}$ nor $\displaystyle{G_0}$ depend on the generalized velocities: $\displaystyle{\partial_2 G_1 = \partial_2 G_0 = 0}$.

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So if $\displaystyle{G}$ is the total time derivative of $\displaystyle{F}$ then

$\displaystyle{ \partial_0 G_1 = \partial_1 G_0 }$

For each of the following functions, either show that it is not a total time derivative or produce a function from which it can be derived.

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[guess]

a. $\displaystyle{G(t, x, v_x) = m v_x}$

\displaystyle{ \begin{aligned} G_0 &= 0 \\ G_1 &= m \\ \\ \partial_0 G_1 &= 0 \\ \partial_1 G_0 &= 0 \\ \\ \end{aligned} }

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\displaystyle{ \begin{aligned} \partial_0 F &= 0 \\ F &= k_0(x, v_x) \\ \\ \partial_1 F &= m \\ F &= m x + k_1(t, v_x) \\ \end{aligned} }

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Let

\displaystyle{ \begin{aligned} k_0(x, v_x) &= mx \\ k_1(t, v_x) &= 0 \\ \end{aligned} }

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Then

\displaystyle{ \begin{aligned} F &= mx \\ \end{aligned} }

[guess]

— Me@2022-06-17 05:10:38 PM

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