1.9 Abstraction of Path Functions, 3.2

Structure and Interpretation of Classical Mechanics

…

$\displaystyle{\begin{aligned} f (t, q(t), v(t), \cdots, q^{(n)}(t)) &= f(\Gamma[q])(t) \\ \bar \Gamma (\bar f) (t, q(t), v(t), \cdots, q^{(n)}(t)) &= \bar f [q](t) \\ \end{aligned}}$

3. The key point is that since $f$ is also a function with the path tuple as input, we can regard $F$ as one possible $f$ . This allows us to use $\bar f (= f \circ \Gamma)$ to define $F \to C$ .

$\displaystyle{\begin{aligned} q &= F \circ (\Gamma[q']) \\ \bar f [q'] &= \Gamma[q] \\ &= \Gamma[F \circ \Gamma[q']] \end{aligned}}$

Note that it is not the definition of $\bar f[q']$ . Instead, it is just an instance of it.

(define (F->C F)
  (define (f-bar q-prime)
    (define q
      (compose F (Gamma q-prime)))
    (Gamma q))
  (Gamma-bar f-bar))

$\displaystyle{\begin{aligned} f &= \bar \Gamma (\bar f) \\ \end{aligned}}$

4. However, in the original definition,

(define ((F->C F) local)
  (->local (time local)
           (F local)
           (+ (((partial 0) F) local)
              (* (((partial 1) F) local)
                 (velocity local)))))

there are some partial differentiation operators. Where are they in the new definition?

Those partial differentiation operators are within the definition of the function osculating-path.

(define ((Gamma-bar f-bar) path-q-local-tuple)
  (let* ((tqva path-q-local-tuple)
         (t (time tqva))         
         (O-tqva (osculating-path tqva)))   
    ((f-bar O-tqva) t)))

— Me@2024-06-13 03:47:47 PM

Physics Town

continues

1.9 Abstraction of Path Functions, 3.2

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