SICM Exercise 1.05

September 4, 2026

Solution to exercise 1.5 of Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman and Jack Wisdom.

Solution process

🛈 Note

We can watch the progress of the minimization by modifying the procedure parametric-path-action to plot the path each time the action is computed. Try this:

(define win2 (frame 0.0 :pi/2 0.0 1.2))

(define ((parametric-path-action Lagrangian t0 q0 t1 q1) intermediate-qs)
  (let ((path (make-path t0 q0 t1 q1 intermediate-qs)))
    ;; display path
    (graphics-clear win2)
    (plot-function win2 path t0 t1 (/ (- t1 t0) 100))
    ;; compute action
    (Lagrangian-action Lagrangian path t0 t1)))

(define (Lagrangian-action L q t1 t2)
  (definite-integral (compose L (Gamma q)) t1 t2))

(define (find-path Lagrangian t0 q0 t1 q1 n)
  (let ((initial-qs (linear-interpolants q0 q1 n)))
    (let ((minimizing-qs
           (multidimensional-minimize
            (parametric-path-action Lagrangian t0 q0 t1 q1)
            initial-qs)))
      (make-path t0 q0 t1 q1 minimizing-qs))))

(define win2 (frame 0.0 :pi/2 0.0 1.2))

(define ((parametric-path-action Lagrangian t0 q0 t1 q1) intermediate-qs)
  (let ((path (make-path t0 q0 t1 q1 intermediate-qs)))
    ;; display path
    (graphics-clear win2)
    (plot-function win2 path t0 t1 (/ (- t1 t0) 100))
    ;; compute action
    (Lagrangian-action Lagrangian path t0 t1)))

(define ((L-harmonic m k) local)
  (let ((q (coordinate local))
        (v (velocity local)))
    (- (* 1/2 m (square v)) (* 1/2 k (square q)))))

(find-path (L-harmonic 1.0 1.0) 0.0 1.0 :pi/2 0.0 2)

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Authors:

Dobi József