Update 1_lagrangian_mechanics.md

[dpgao]

First, thank you for documenting your work online! I've been going through SICM lately, too, and I'm sure these material will be very helpful. I do think, however, that this 'optimisation' to reduce the number of dimensions is a bit quick. A juggling pin spinning around its central axis will behave differently from one which is stationary, just like how a spinning bicycle wheel behaves differently from a stationary one.

[dpgao]

The same goes for the answers to questions e and f. I think the top always has three degrees of freedom regardless of its shape.

[dpgao]

I took a go at Exercise 1.6 just now. I think your solution does not necessarily fix the endpoint velocities, because the two extra points you added are passed to Lagrange-interpolation-function together with the other intermediate points in the call to make-path, so the resulting path will still have undetermined endpoint velocities. One can still say that the endpoints are 'constrained', but only in a very weak sense.

Below is my code which constructs a piecewise path where the initial and final segments are forced to be straight lines defined by extra parameters t0*, q0*, t1*, and q1*.

(define (((parametric-path-action* win) L t0 q0 t0* q0* t1* q1* t1 q1) qs)
  (let* ((curve (make-path t0* q0* t1* q1* qs))
	 (path (lambda (t) (cond ((< t t0*) (+ q0 (* (- t t0) (/ (- q0* q0) (- t0* t0)))))
				 ((> t t1*) (+ q1 (* (- t t1) (/ (- q1* q1) (- t1* t1)))))
				 (else (curve t))))))
    ;; display path
    (graphics-clear win)
    (plot-function win path t0 t1 (/ (- t1 t0) 32))
    ;; compute action
    (Lagrangian-action L path t0 t1)))

(define (find-path L t0 q0 t0* q0* t1* q1* t1 q1 n)
  (let ((initial-qs (linear-interpolants q0* q1* n))
	(win (frame 0 :pi -1.2 1.2)))
    (let ((minimizing-qs
	   (multidimensional-minimize
	    ((parametric-path-action* win) L t0 q0 t0* q0* t1* q1* t1 q1)
	    initial-qs)))
      (make-path t0* q0* t1* q1* minimizing-qs))))

If I run the new make-path on a free particle

(find-path (L-free-particle 3.) 0. 1. 0.5 1.1 (- :pi 0.5) -1.1 :pi -1. 4)

I get

which shows that the optimiser does not care about jump discontinuities in the velocity.

I also ran this on the L-harmonic Lagrangian and obtained these curves:

[sritchie]

@dpgao , somehow I was not subscribed to notifications on my own repository, and missed this! Apologies, I feel terrible for letting this languish for month.

Your answer to 1.6 is way better than mine; you can probably see that I was flailing to justify what I was seeing, instead of understanding this problem.

This is awesome. Do you mind if I incorporate this change into the .org file, with credit to you, of course?

[dpgao]

This is awesome. Do you mind if I incorporate this change into the .org file, with credit to you, of course?

Not at all. You’re welcome!

[sritchie]

I'll pull this in soon as I tidy up this repository!