# Ex 1.21 The dumbbell, 2.2.1

Structure and Interpretation of Classical Mechanics

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b. Write the formal Lagrangian $\displaystyle{L(t;x_0, y_0, x_1, y_1, F; \dot x_0, \dot y_0, \dot x_1, \dot y_1, \dot F)}$

such that Lagrange’s equations will yield the Newton’s equations you derived in part a.

~~~

[guess]


(define (U-constraint q0 q1 F l)
(* (/ F (* 2 l))
(- (square (- q1 q0))
(square l))))

(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 q-rect
(up (literal-function 'x_0)
(literal-function 'y_0)
(literal-function 'x_1)
(literal-function 'y_1)
(literal-function 'F)))

(show-expression q-rect) (show-expression (q-rect 't)) (show-expression (Gamma q-rect)) (show-expression ((Gamma q-rect) 'w)) (show-expression ((Gamma q-rect) 't)) (show-expression (time ((Gamma q-rect) 't))) (show-expression (coordinate ((Gamma q-rect) 't))) [guess]

— based on /sicmutils/sicm-exercises

— Me@2021-04-27 05:03:59 PM

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