# Ex 1.8.2.5 Implementation of $delta$

Structure and Interpretation of Classical Mechanics

.

Verify that the operators $\displaystyle{D}$ (differentiation) and $\displaystyle{\delta}$ (variation) commute (Equation 1.27) using the scmutils software library:

$\displaystyle{D \delta_{\eta} f [q] = \delta_\eta g[q]}$ with $\displaystyle{g [q] = D ( f[q] )}$

~~~

(define (((delta eta) f) q)
(define (g epsilon)
(f (+ q (* epsilon eta))))
((D g) 0))

(define q (literal-function 'q (-> Real (UP Real))))

(define eta (literal-function 'eta (-> Real (UP Real))))


.

(define (f q)
(compose (literal-function 'f
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))

(define (g q)
(compose (literal-function 'g
(-> (UP Real (UP* Real) (UP* Real)) Real))
(Gamma q)))


.

(define (g q) (D (f q)))


.

(define LHS ( (D (((delta eta) f) q)) 't))

(define RHS ((((delta eta) g) q) 't))


.

(print-expression LHS)

(show-expression LHS)


.

$\displaystyle{\partial_1 \partial_1 f \left( \begin{bmatrix} t \\ q(t) \\ Dq(t) \end{bmatrix} \right) D q(t) \eta(t) + D^2 q \partial_2 \partial_2 f D \eta + D^2 q \partial_1 \partial_2 f \eta + ... }$

$\displaystyle{... \partial_1 \partial_2 f D \eta D q + D^2 \eta \partial_2 f + \partial_0 \partial_2 f D \eta + \partial_0 \partial_1 f \eta + D \eta \partial_1 f}$

.

(print-expression RHS)

(show-expression RHS)

(- LHS RHS)


— Me@2020-08-24 03:18:21 PM

.

.