1.9 Abstraction of Path Functions, 3.1

Structure and Interpretation of Classical Mechanics

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

The goal of this post is to explain why the code above can be replaced by the following code:

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

While the input of $\displaystyle{f}$ is a tuple $\displaystyle{(t, q, v, \cdots)}$ , the input of $\displaystyle{\bar f}$ is an abstract path $\displaystyle{q}$ .

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

Let us define $\bar \Gamma$ as

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

The difference between $\displaystyle{\begin{aligned} \Gamma \end{aligned}}$ and $\displaystyle{\begin{aligned} \bar \Gamma \end{aligned}}$ is that, in an abstract sense, $\Gamma$ transforms $f$ to $\bar f$ , while $\bar \Gamma$ does the opposite.

The explicit form of $\bar \Gamma$ is provided by

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

(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)))

Note that $\bar f$ is not defined yet because it can be any path-dependent function.

What is $\displaystyle{F}$ ?

Equation (1.68):

$\displaystyle{\begin{aligned} L' \circ \Gamma[q'] &= L \circ \Gamma [q] \\ \Gamma[q] &= C \circ \Gamma[q'] \\ L' &= L \circ C \\ \end{aligned}}$

While $F$ is a coordinate transformation, $C$ is the corresponding local-tuple transformation.

Equation (1.74):

$\displaystyle{\begin{aligned} \Gamma[q] &= C \circ \Gamma[q'] \\ (t, x, v, \cdots) &= (t, F(t,x'), \partial_0 F(t, x') + \partial_1 F(t, x') v', \cdots) \\ \end{aligned}}$