Ex 1.17 Bead on a helical wire

Structure and Interpretation of Classical Mechanics

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A bead of mass m is constrained to move on a frictionless helical wire. The helix is oriented so that its axis is horizontal. The diameter of the helix is d and its pitch (turns per unit length) is h. The system is in a uniform gravitational field with vertical acceleration g. Formulate a Lagrangian that describes the system and find the Lagrange equations of motion.

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[guess]

The coordinates of the bead is

\displaystyle{\left( \frac{d}{2} \cos \theta, \frac{d}{2} \sin \theta, \frac{\theta}{2 \pi} h \right)},

where the \displaystyle{x}-direction is horizontal, the \displaystyle{y}-direction points upwards, and the \displaystyle{z}-direction is along the axis of the helix.

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Lagrangian \displaystyle{L = T - V}, where the kinetic energy, \displaystyle{T = \frac{1}{2} m \left( \dot x^2 + \dot y^2 + \dot z^2 \right)} and the potential energy, \displaystyle{V = m g y}.

\displaystyle{ (\dot x, \dot y, \dot z) = \left( \frac{-d}{2} (\sin \theta) \dot \theta, \frac{d}{2} (\cos \theta) \dot \theta, \frac{h}{2 \pi} \dot \theta \right)}

\displaystyle{L = \frac{1}{2} m \left( \frac{d^2}{4} + \frac{h^2}{4 \pi^2} \right) \dot \theta^2 - \frac{d}{2} m g \sin \theta}

\displaystyle{\frac{\partial L}{\partial \dot \theta} = m \left( \frac{d^2}{4} + \frac{h^2}{4 \pi^2} \right) \dot \theta}
\displaystyle{\frac{\partial L}{\partial \theta} = - \frac{d}{2} m g \cos \theta}

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The Lagrange equation:

\displaystyle{ \begin{aligned} D ( \partial_2 L \circ \Gamma[q]) - (\partial_1 L \circ \Gamma[q]) &= 0 \\   \frac{d}{dt} \frac{\partial L}{\partial \dot \theta} - \frac{\partial L}{\partial \theta} &= 0 \\    m \left( \frac{d^2}{4} + \frac{h^2}{4 \pi^2} \right) \ddot{\theta} + \frac{d}{2} m g \cos \theta &= 0 \\   \end{aligned}}

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(define ((T-hl d h m g) local)
  (let ((t (time local))
        (thetadot (velocity local)))    
    (* 1/8 m (square thetadot) 'H)))

;; (+ (square d) 
;; (/ (square h) (square 'pi))))))

(show-expression
  ((T-hl 'd 'h 'm 'g)
    (->local 't
             'theta
             'thetadot)))

(define ((V-hl d h m g) local)
  (let ((t (time local))
        (theta (coordinate local)))
    (let ((y (* 1/2 d (sin theta))))
      (* m g y))))


(show-expression
  ((V-hl 'd 'h 'm 'g)
    (->local 't
             'theta
             'thetadot)))

(define L-hl (- T-hl V-hl))

(show-expression
  ((L-hl 'd 'h 'm 'g)
    (->local 't
             'theta
             'thetadot)))

(show-expression
 (((Lagrange-equations
    (L-hl 'd 'h 'm 'g))
   (literal-function 'theta))
  't))

[guess]

— Me@2021-02-05 04:23:02 PM

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2021.02.06 Saturday (c) All rights reserved by ACHK