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

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where the -direction is horizontal, the -direction points upwards, and the -direction is along the axis of the helix.

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Lagrangian , where the kinetic energy, and the potential energy, .

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The Lagrange equation:

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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