Structure and Interpretation of Classical Mechanics

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Show that the Lagrangian (1.89) …

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[guess]

The Lagrangian (1.89):

Formally, we can reproduce Newton’s equations with the Lagrangian:

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(define (KE-particle m v) (* 1/2 m (square v))) (define ((extract-particle pieces) local i) (let* ((indices (apply up (iota pieces (* i pieces)))) (extract (lambda (tuple) (vector-map (lambda (i) (ref tuple i)) indices)))) (up (time local) (extract (coordinate local)) (extract (velocity local))))) (define (U-constraint q0 q1 F l) (* (/ F (* 2 l)) (- (square (- q1 q0)) (square l)))) (define ((U-gravity g m) q) (let* ((y (ref q 1))) (* m g y))) (define ((L-driven-free m l x_s y_s U) local) (let* ((extract (extract-particle 2)) (p (extract local 0)) (q (coordinate p)) (qdot (velocity p)) (F (ref (coordinate local) 2))) (- (KE-particle m qdot) (U q) (U-constraint (up (x_s (time local)) (y_s (time local))) q F l)))) (let* ((U (U-gravity 'g 'm)) (x_s (literal-function 'x_s)) (y_s (literal-function 'y_s)) (L (L-driven-free 'm 'l x_s y_s U)) (q-rect (up (literal-function 'x) (literal-function 'y) (literal-function 'F)))) (show-expression ((compose L (Gamma q-rect)) 't)))

[guess]

— Me@2022-01-13 01:19:34 PM

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2022.01.14 Friday (c) All rights reserved by ACHK

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