Explanation Given: Equation a z + b θ = c includes the circles z = const , straight lines θ = const . Proof: Let the equation of the cylinder be r = C , where C is constant We want to make following integral stationary: ∫ d s In cylindrical coordinates, we know that: d s = ( d r ) 2 + r 2 ( d θ ) 2 + ( d z ) 2 = 0 + C 2 ( d θ ) 2 + ( d z ) 2 = C 2 θ ' 2 + 1 d z because on the cylinder r = C ⇒ d r = 0 So, ∫ d s = ∫ C 2 θ ' 2 + 1 d z So, F = C 2 θ ' 2 + 1 Now, ∂ F ∂ θ = 0........ ( 1 ) And, ∂ F ∂ θ ' = ∂ ∂ θ ' = ( C 2 θ ' 2 + 1 ) = 2 C 2 θ ' 2 C 2 θ ' 2 + 1 = C 2 θ ' C 2 θ ' 2 + 1 .......... ( 2 ) Following is the Euler-Lagrange equation: d d r ∂ F ∂ θ ' − ∂ F ∂ θ = 0 Now put the values from equation ( 1 ) and ( 2 ) in this equation, we get; d d z ( C 2 θ ' C 2 θ ' 2 + 1 ) − 0 = 0 d d z ( C 2 θ ' C 2 θ ' 2 + 1 </

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Chapter 9.3, Problem 16P

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To show: the geodesics on a circular cylinder are helices az+bθ=c .

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Chapter 9.3, Problem 15P

Chapter 9.3, Problem 17P

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Chapter 9 Solutions

Mathematical Methods in the Physical Sciences

Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.1 - The speed of light in a medium of index of...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...Ch. 9.2 - Write and solve the Euler equations to make the...

Knowledge Booster

the geodesics on a circular cylinder are helices a z + b θ = c .

Solution Summary: The author explains that the geodesics on a circular cylinder are helices and straight lines.

To show: the geodesics on a circular cylinder are helices az+bθ=c .

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Chapter 9 Solutions