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.
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[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)))
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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
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