Ex 1.25 Properties of Dt, 2

Structure and Interpretation of Classical Mechanics

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Demonstrate that …

b. \displaystyle{D_t (c F) = c D_t F}

c. \displaystyle{D_t (F G) = F D_t G + (D_t F) G}

~~~

\displaystyle{  \begin{aligned}  &D_t (c F) \circ \Gamma[q] (t) \\  \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \partial_0 \left[c F(t, q, v, a, ...) \right]     + \partial_1 \left[c F(t, q, v, a, ...) \right] v(t)     + \partial_2 \left[c F(t, q, v, a, ...) \right] a(t) + ... \end{aligned}  }

\displaystyle{  \begin{aligned}      &= c \partial_0 F(t, q, v, a, ...) + c \partial_1 F(t, q, v, a, ...) v(t) + c \partial_2 F(t, q, v, a, ...) a(t) + ... \end{aligned}  }

\displaystyle{  \begin{aligned}      &= c \left[ \partial_0 F(t, q, v, a, ...) + \partial_1 F(t, q, v, a, ...) v(t) + \partial_2 F(t, q, v, a, ...) a(t) + ... \right] \end{aligned}  }

\displaystyle{  \begin{aligned}  &= c D_t F \circ \Gamma[q] (t) \\  \end{aligned}  }

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\displaystyle{  \begin{aligned}  &D_t (FG) \circ \Gamma[q] (t) \\  \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \partial_0 \left[ F(t, q, v, a, ...) G(t, q, v, a, ...) \right] \\   &+ \partial_1 \left[ F(t, q, v, a, ...) G(t, q, v, a, ...) \right] v(t) \\   &+ \partial_2 \left[ F(t, q, v, a, ...) G(t, q, v, a, ...) \right] a(t) + ... \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \left[ \partial_0 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) + F(t, q, v, a, ...) \partial_0 G(t, q, v, a, ...) \\   &+ \left\{ \left[ \partial_1 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) + F(t, q, v, a, ...) \partial_1 G(t, q, v, a, ...) \right\} v(t) \\   &+ \left\{ \left[ \partial_2 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) + F(t, q, v, a, ...) \partial_2 G(t, q, v, a, ...) \right\} a(t) + ... \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &=    \left[ \partial_0 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) \\    &+ \left[ \partial_1 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) v(t) \\   &+ \left[ \partial_2 F(t, q, v, a, ...) \right] G(t, q, v, a, ...) a(t) + ... \\ \\    &+ F(t, q, v, a, ...) \partial_0 G(t, q, v, a, ...) \\   &+ F(t, q, v, a, ...) \partial_1 G(t, q, v, a, ...) v(t) \\   &+ F(t, q, v, a, ...) \partial_2 G(t, q, v, a, ...) a(t) + ... \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &=    \left[ \partial_0 F(t, q, v, a, ...)   + \partial_1 F(t, q, v, a, ...) v(t)    + \partial_2 F(t, q, v, a, ...) a(t) + ... \right] G(t, q, v, a, ...) \\     &+ F(t, q, v, a, ...) \left[ \partial_0 G(t, q, v, a, ...) + \partial_1 G(t, q, v, a, ...) v(t) + \partial_2 G(t, q, v, a, ...) a(t) + ... \right] \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \left\{  D_t F \circ \Gamma[q] (t) \right\} G(t, q, v, a, ...) + F(t, q, v, a, ...) D_t G \circ \Gamma[q] (t) \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \left\{  D_t F \circ \Gamma[q] (t) \right\} G \circ \Gamma[q] (t) + F \circ \Gamma[q] (t) D_t G \circ \Gamma[q] (t) \\     \end{aligned}  }

The meaning of \displaystyle{\delta_\eta (fg)[q]} is

\displaystyle{\delta_\eta (f[q]g[q])}

\displaystyle{  \begin{aligned}  &D_t (FG) \circ \Gamma[q] (t) \\  \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \left\{  D_t F \circ \Gamma[q] (t) \right\} G \circ \Gamma[q] (t) + F \circ \Gamma[q] (t) D_t G \circ \Gamma[q] (t) \\     \end{aligned}  }

\displaystyle{  \begin{aligned}      &= \left[  (D_t F) G + F D_t G \right] \circ \Gamma[q] (t) \\     \end{aligned}  }

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\displaystyle{  \begin{aligned}      D_t (FG) \circ \Gamma[q] (t) &= \left[  (D_t F) G + F D_t G \right] \circ \Gamma[q] (t) \\ \\    D_t (FG) &= (D_t F) G + F D_t G \\     \end{aligned}  }

— Me@2022.05.12 07:02:30 PM

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2022.05.14 Saturday (c) All rights reserved by ACHK