Ex 1.24 Constraint forces, 1.2

Ex 1.22 Driven pendulum, 3.2

Structure and Interpretation of Classical Mechanics

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Find the tension in an undriven planar pendulum.

~~~

[guess]


(define ((q->r x_s y_s l) local)
  (let* ((q (coordinate local))
         (t (time local))
         (theta (ref q 0))
         (c (ref q 1))
         (F (ref q 2)))
    (up (+ (x_s t) (* c (sin theta)))
        (- (y_s t) (* c (cos theta)))
        F)))

(let* ((xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)
              (literal-function 'c)
              (literal-function 'F))))
  (show-expression ((compose (q->r xs ys 'l) (Gamma q)) 't)))


(define (L-theta m l x_s y_s U)
  (compose
   (L-driven-free m l x_s y_s U) (F->C (q->r x_s y_s l))))

(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)
              (literal-function 'c)             
              (literal-function 'F)))
       (L (L-theta 'm 'l xs ys U)))
  (show-expression (L ((Gamma q) 't))))
(/
 (+ (* l m ((D x_s) t) (c t) ((D theta) t) (cos (theta t)))
    (* 1/2 l m (expt (c t) 2) (expt ((D theta) t) 2))
    (* l m (c t) ((D theta) t) (sin (theta t)) ((D y_s) t))
    (* g l m (c t) (cos (theta t)))
    (* l m ((D x_s) t) ((D c) t) (sin (theta t)))
    (* -1 l m ((D c) t) (cos (theta t)) ((D y_s) t))
    (* -1 g l m (y_s t))
    (* 1/2 l m (expt ((D x_s) t) 2))
    (* 1/2 l m (expt ((D c) t) 2))
    (* 1/2 l m (expt ((D y_s) t) 2))
    (* 1/2 (expt l 2) (F t))
    (* -1/2 (expt (c t) 2) (F t)))
 l)

(let* ((U (U-gravity 'g 'm))
       (xs (literal-function 'x_s))
       (ys (literal-function 'y_s))
       (q (up (literal-function 'theta)
              (literal-function 'c)
              (literal-function 'F)))
       (L (L-theta 'm 'l xs ys U)))
  (show-expression (((Lagrange-equations L) q) 't)))

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(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 'l))
       (q (up (literal-function 'theta)
              (lambda (t) 'l)
              (literal-function 'F)))
       (L (L-theta 'm 'l xs ys U)))
  (show-expression (((Lagrange-equations L) q) 't)))

.

\displaystyle{F(t) = l m \left[D \theta(t)\right]^2 + g m \cos \theta(t)}

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[guess]

— Me@2022-04-05 04:16:51 PM

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2022.04.05 Tuesday (c) All rights reserved by ACHK