Structure and Interpretation of Classical Mechanics
.
Consider a pendulum of length attached to a support that is free to move horizontally, as shown in figure 1.4. Let the mass of the support be
and the mass of the pendulum be
. Formulate a Lagrangian and derive Lagrange’s equations for this system.
~~~
[guess]
(define ((F->C F) local) (->local (time local) (F local) (+ (((partial 0) F) local) (* (((partial 1) F) local) (velocity local))))) ; (define ((q->r l y1) local) (let ((q (coordinate local))) (let ((x1 (ref q 0)) (theta (ref q 1))) (let ((x2 (+ x1 (* l (sin theta)))) (y2 (- y1 (* l (cos theta))))) (up x1 y1 x2 y2))))) (show-expression ((q->r 'l 'y_1) (up 't (up 'x_1 'theta) (up 'xdot_1 'thetadot)))) ; (define (KE m vx vy) (* 1/2 m (+ (square vx) (square vy)))) (define ((T-rect m1 m2) local) (let ((q (coordinate local)) (v (velocity local))) (let ((x1dot (ref v 0)) (y1dot (ref v 1)) (x2dot (ref v 2)) (y2dot (ref v 3))) (+ (KE m1 x1dot y1dot) (KE m2 x2dot y2dot))))) (show-expression ((T-rect 'm_1 'm_2) (up 't (up 'x_1 'y_1 'x_2 'y_2) (up 'xdot_1 'ydot_1 'xdot_2 'ydot_2)))) ; (define ((U-rect g m1 m2) local) (let* ((q (coordinate local)) (y1 (ref q 1)) (y2 (ref q 3))) (* g (+ (* m1 y1) (* m2 y2))))) (show-expression ((U-rect 'g 'm_1 'm_2) (up 't (up 'x_1 'y_1 'x_2 'y_2) (up 'xdot_1 'ydot_1 'xdot_2 'ydot_2)))) ; (define (L-rect g m1 m2) (- (T-rect m1 m2) (U-rect g m1 m2))) (show-expression ((L-rect 'g 'm_1 'm_2) (up 't (up 'x_1 'y_1 'x_2 'y_2) (up 'xdot_1 'ydot_1 'xdot_2 'ydot_2)))) ; (define (L-l l y1 g m_1 m_2) (compose (L-rect g m_1 m_2) (F->C (q->r l y1)))) (show-expression ((L-l 'l 'y_1 'g 'm_1 'm_2) (->local 't (up 'x_1 'theta) (up 'xdot_1 'thetadot)))) ; (show-expression (((Lagrange-equations (L-l 'l 'y_1 'g 'm_1 'm_2)) (up (literal-function 'x_1) (literal-function 'theta))) 't)) ;
[guess]
— Me@2021-03-26 08:22:01 PM
.
.
2021.03.28 Sunday (c) All rights reserved by ACHK
You must be logged in to post a comment.