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))
(* 1/8 m (square thetadot) 'H)))

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

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

(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

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

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

(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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