# 1990s, 12

— Me@2022-01-14 12:09:34 PM

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# Ex 1.22 Driven pendulum, 2.1

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:

$\displaystyle{ L(t;x, F; \dot x, \dot F)}$

$\displaystyle{= \sum_\alpha \frac{1}{2} m_\alpha \dot{\mathbf{x}_\alpha}^2 - V(t, x) - \sum_{\{ \alpha, \beta | \alpha < \beta, \alpha \leftrightarrow \beta \}} \frac{F_{\alpha \beta}}{2 l_{\alpha \beta}} \left[ (\mathbf{x}_\beta - \mathbf{x}_\alpha)^2 - l_{\alpha \beta}^2 \right] }$

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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)))



$\displaystyle{ L = \frac{1}{2} m \left[(Dx)^2 + (Dy)^2 \right] - mgy - \frac{F}{2l} \left[ (x-x_s)^2 + (y-y_s)^2 - l^2 \right] }$

[guess]

— Me@2022-01-13 01:19:34 PM

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