Ex 1.21 The dumbbell, 3.4.2

Structure and Interpretation of Classical Mechanics

e. Make a Lagrangian ( $\displaystyle{= T - V}$ ) for the system described with the irredundant generalized coordinates $\displaystyle{x_{cm}}$ , $\displaystyle{y_{cm}}$ , $\displaystyle{\theta}$ and compute the Lagrange equations from this Lagrangian. They should be the same equations as you derived for the same coordinates in part d.

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[guess]


(define ((L-cm m0 m1 l) local)
  (let* ((q (coordinate local))
	 (v (velocity local))
	 (x_cm (ref q 0))
	 (y_cm (ref q 1))
	 (theta (ref q 2))

	 (x_cm_dot (ref v 0))
	 (y_cm_dot (ref v 1))
	 (theta_dot (ref v 2))
	 
	 (M (+ m0 m1))
	 (mu (/ 1 (+ (/ 1 m0) (/ 1 m1)))))

    (+ (* (/ 1 2) M (+ (square x_cm_dot) (square y_cm_dot)))
       (* (/ 1 2) mu (square l) (square theta_dot)))))

$\displaystyle{ L_{cm} = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} \mu l^2 \dot \theta^2 }$


(show-expression
 (((Lagrange-equations
    (L-cm 'm_0 'm_1 'l))

   (up (literal-function 'x_cm)
       (literal-function 'y_cm)
       (literal-function 'theta)))
  't))