Structure and Interpretation of Classical Mechanics

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The two-bar linkage shown in figure 1.3 is constrained to move in the plane. It is composed of three small massive bodies interconnected by two massless rigid rods in a uniform gravitational field with vertical acceleration . The rods are pinned to the central body by a hinge that allows the linkage to fold. The system is arranged so that the hinge is completely free: the members can go through all configurations without collision. Formulate a Lagrangian that describes the system and find the Lagrange equations of motion. Use the computer to do this, because the equations are rather big.

~~~

[guess]

(define ((F->C F) local)
  (->local (time local)
           (F local)
           (+ (((partial 0) F) local)
              (* (((partial 1) F) local) 
                 (velocity local)))))
 
;

(define ((q->r l1 l2) local)
  (let ((q (coordinate local)))
    (let ((x2 (ref q 0))
	      (y2 (ref q 1))
	      (theta (ref q 2)) 
          (phi (ref q 3)))
      (let ((x1 (+ x2 (* l1 (cos theta))))
	        (y1 (+ y2 (* l1 (sin theta))))
	        (x3 (+ x2 (* l2 (cos phi))))
	        (y3 (+ y2 (* l2 (sin phi)))))
        (up x1 y1 x2 y2 x3 y3)))))

;

(show-expression
  ((q->r 'l_1 'l_2) 
     (up 't
	     (up 'x_2 'y_2 'theta 'phi)
	     (up 'xdot_2 'ydot_2 'thetadot 'phidot))))

;

(define (KE m vx vy)
  (* 1/2 m (+ (square vx) (square vy))))

(define ((T-rect m1 m2 m3) 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))
	      (x3dot (ref v 4))
	      (y3dot (ref v 5)))
      (+ (KE m1 x1dot y1dot)
  	     (KE m2 x2dot y2dot)
	     (KE m3 x3dot y3dot)))))

(show-expression
 ((T-rect 'm_1 'm_2 'm_3)
    (up 't
	    (up 'x_1 'y_1 'x_2 'y_2 'x_3 'y_3)
	    (up 'xdot_1 'ydot_1 'xdot_2 'ydot_2 'xdot_3 'ydot_3))))

;

[guess]

— Me@2021-03-12 05:37:27 PM

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