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Chapter 3: Problem 17

*Find the geodesics on a sphere of unit radius. [Hint: Use \(\theta\) as independent variable, and look for the path \(\varphi=\varphi(\theta)\). To perform the integration, use the substitution \(x=\cot \theta .]\)

Short Answer

Expert verified
Answer: The equation representing the geodesics on a sphere of unit radius is \(\varphi = -c \cot \theta + \varphi_0\), where \(c\) and \(\varphi_0\) are constants.

Step by step solution

01

Write the differential arc length on the surface of the sphere

The differential arc length on the surface of a sphere of unit radius in spherical coordinates can be written as: \(ds^2 = d\theta^2 + \sin^2(\theta)\, d\varphi^2\) This expression will help us find the functional we need to minimize in order to find the geodesics.
02

Set up the Euler-Lagrange equation for geodesics

The functional we need to minimize is the arc length \(s\). To minimize this functional, we will use the Euler-Lagrange equation. The functional has the form: \(L = \sqrt{d\theta^2 + \sin^2(\theta)\, d\varphi^2}\) We can define a new variable \(u = \frac{d\varphi}{d\theta}\), so the Euler-Lagrange equation is: \(\frac{d}{d\theta} \left( \frac{\partial L}{\partial u} \right) = \frac{\partial L}{\partial \theta}\) We will use this equation to find the geodesics on the sphere.
03

Differentiate the functional and solve the Euler-Lagrange equation

We need to differentiate the functional with respect to \(u\): \(\frac{\partial L}{\partial u} = \frac{\sin^2(\theta)}{\sqrt{d\theta^2 + \sin^2(\theta)u^2}}\) Now we differentiate the result with respect to \(\theta\): \(\frac{d}{d\theta} \left( \frac{\partial L}{\partial u} \right) = \frac{\partial L}{\partial \theta}\) Solving the Euler-Lagrange equation for \(u\), we get: \(u = \frac{c}{\sin^2(\theta)}\), where \(c\) is a constant.
04

Integrate to find the path

Now we substitute \(u\) back into \(\frac{d\varphi}{d\theta}\) and integrate both sides with respect to \(\theta\): \(\frac{d\varphi}{d\theta} = \frac{c}{\sin^2(\theta)}\) We also use the substitution \(x = \cot \theta\) (along with \(dx=-\csc^2(\theta) d\theta\)): \(d\varphi = -c\,dx\) Integrating both sides, we find the path for \(\varphi\): \(\varphi = -cx + \varphi_0\), where \(\varphi_0\) is the integration constant.
05

Write the geodesic equation in terms of \(\theta\)

Now we substitute back \(x = \cot \theta\) into the expression for \(\varphi\): \(\varphi = -c \cot \theta + \varphi_0\) This equation represents the geodesics on a sphere of unit radius. In conclusion, the geodesics on a sphere of unit radius can be described by the equation \(\varphi = -c \cot \theta + \varphi_0\), where \(c\) and \(\varphi_0\) are constants.

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Most popular questions from this chapter

*A light rigid cylinder of radius \(2 a\) is able to rotate freely about its axis, which is horizontal. A particle of mass \(m\) is fixed to the cylinder at a distance \(a\) from the axis and is initially at rest at its lowest point. A light string is wound on the cylinder, and a steady tension \(F\) applied to it. Find the angular acceleration and angular velocity of the cylinder after it has turned through an angle \(\theta\). Show that there is a limiting tension \(F_{0}\) such that if \(F\) \(F_{0}\) it continues to accelerate. Estimate the value of \(F_{0}\) by numerical approximation.

*Find the corresponding formulae for \(\partial e_{i} / \partial q_{j}\) for spherical polar coordinates, and hence verify the results obtained in Problem \(21 .\)

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If \(q_{1}, q_{2}, q_{3}\) are orthogonal curvilinear co-ordinates, and the element of length in the \(q_{i}\) direction is \(h_{i} \mathrm{~d} q_{i}\), write down (a) the kinetic energy \(T\) in terms of the generalized velocities \(\dot{q}_{i}\), (b) the generalized momentum \(p_{i}\) and (c) the component \(\boldsymbol{e}_{i} \cdot \boldsymbol{p}\) of the momentum vector \(\boldsymbol{p}\) in the \(q_{i}\) direction. (Here \(\boldsymbol{e}_{i}\) is a unit vector in the direction of increasing \(\left.q_{i}\right)\)

A particle of mass \(m\) is attached to the end of a light string of length l. The other end of the string is passed through a small hole and is slowly pulled through it. Gravity is negligible. The particle is originally spinning round the hole with angular velocity \(\omega\). Find the angular velocity when the string length has been reduced to \(\frac{1}{2} l\). Find also the tension in the string when its length is \(r\), and verify that the increase in kinetic energy is equal to the work done by the force pulling the string through the hole.
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