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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2022.06.17 Friday (c) All rights reserved by ACHK