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Chapter 4: Problem 7

Let \((\rho, \theta)\) be a system of geodesic polar coordinates \((E=1, F=0)\) on a surface, and let \(\gamma(\rho(s), \theta(s))\) be a geodesic that makes an angle \(\varphi(s)\) with the curves \(\theta=\) const. For definiteness, the curves \(\theta=\) const. are oriented in the sense of increasing \(\rho\) 's and \(\varphi\) is measured from \(\theta=\) const. to \(\gamma\) in the orientation given by the parametrization \((\rho, \theta)\). Show that $$ \frac{d \varphi}{d s}+(\sqrt{G})_{\rho} \frac{d \theta}{d s}=0 $$

Short Answer

Expert verified
\(\frac{d \varphi}{d s} + (\sqrt{G})_{\rho} \frac{d \theta}{d s} = 0\)

Step by step solution

01

- Recall the geodesic equations

The geodesic equations in polar coordinates \(\(\rho\), \(\theta\)\) are given by \[\frac{d^2 \rho}{ds^2} + \frac{1}{2} G_{\rho} \left(\frac{d \theta}{ds}\right)^2 = 0\] and \[\frac{d^2 \theta}{ds^2} + \frac{2}{G} G_{\rho} \frac{d \rho}{ds} \frac{d \theta}{ds} = 0\], where G is the metric component for \(\theta\).
02

- Define angle \(\varphi\) with respect to \(\theta = \text{const.}\)

The angle \(\varphi(s)\) is measured between the geodesic \(\gamma\) and the constant \(\theta\) curves. This relation gives \[\tan(\varphi) = \frac{\frac{d\theta}{ds}}{\frac{d\rho}{ds}}.\]
03

- Differentiate the tangent relation

We differentiate the relation \[\tan(\varphi) = \frac{\frac{d\theta}{ds}}{\frac{d\rho}{ds}}\] with respect to \(s\). Applying the chain rule, we have \[\sec^2(\varphi) \frac{d \varphi}{d s} = \frac{d}{d s} \left( \frac{\frac{d \theta}{d s}}{\frac{d \rho}{d s}} \right).\]
04

- Apply product and quotient rules

The derivative becomes \[\frac{d}{d s} \left( \frac{\frac{d \theta}{d s}}{\frac{d \rho}{d s}} \right) = \frac{\frac{d^2 \theta}{d s^2} \frac{d \rho}{d s} - \frac{d \theta}{d s} \frac{d^2 \rho}{d s^2}}{\left( \frac{d \rho}{d s} \right)^2}.\] Substituting from the geodesic equations, we get \[\sec^2(\varphi) \frac{d \varphi}{d s} = \frac{-\frac{1}{2} G_{\rho} \left( \frac{d \theta}{d s} \right)^2}{\left( \frac{d \rho}{d s} \right)^2}.\]
05

- Simplify using trigonometric identities

Rewriting the expression, we find \[\frac{d \varphi}{d s} = -\frac{1}{2} (\frac{G_{\rho}}{G}) \frac{d \theta}{d s}.\] Since \(G = \sqrt{G_{\theta\theta}} = \sqrt{G}\), we simplify to \[\frac{d \varphi}{d s} + \left( \sqrt{G} \right)_{\rho} \frac{d \theta}{d s} = 0.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geodesic equations
Geodesic equations are essential in understanding the shortest path between two points on a surface. They describe how a curve, called a geodesic, behaves when it is locally the shortest path. In the context of polar coordinates, the geodesic equations can be written using the parameterizations \(\(\rho(s), \theta(s)\)\). These equations can be written as follows:
\[\frac{d^2 \rho}{ds^2} + \frac{1}{2} G_{\rho} \left(\frac{d \theta}{ds}\right)^2 = 0\] and
\[\frac{d^2 \theta}{ds^2} + \frac{2}{G} G_{\rho} \frac{d \rho}{ds} \frac{d \theta}{ds} = 0\].
Here, \(s\)\ represents the arc-length parameter along the geodesic. \(G\)\ is the metric component that represents the geometry of the surface related to the angle coordinate \(\theta\). Simplifying these equations helps in understanding the intrinsic geometry of the surface. They also show the relationship between the changes in angles and radial distances as one moves along the geodesic.
polar coordinates
Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. They are often written as \( ( \; \rho, \theta \; ) \), where \(\rho\)\ is the radial distance and \(\theta\)\ is the angular coordinate.
In this context, geodesic polar coordinates help describe positions and movements on curved surfaces. These coordinates simplify many calculations because they align with the symmetry of the problem.

Using polar coordinates, the curves \(\theta = \text{const.}\) represent circular paths centered on the reference point. The parameter \(\rho\)\ increases along these radial paths, acting like concentric circles. The angle \(\varphi(s)\) is measured between a geodesic and these radial paths, aiding in understanding the geometry and dynamics of the motion along the surface.
metric component
The metric component, denoted as \(G\),\ is a function that describes the geometry of a surface in a coordinate system. Specifically, for polar coordinates, it relates to the angular portion of the geometry.
The role of the metric component is crucial when dealing with distances and angles on a curved surface. It helps convert these measurements appropriately between coordinate systems.
Given the geodesic equation: \[\frac{d^2 \rho}{ds^2} + \frac{1}{2} G_{\rho} \left(\frac{d \theta}{ds}\right)^2 = 0\],\ the term \(G_{\rho}\) denotes the partial derivative of \(G\) with respect to \(\rho\), signifying how \(G\) changes as one moves radially.
  • The metric component also appears in the term \[\frac{d^2 \theta}{ds^2} + \frac{2}{G} G_{\rho} \frac{d \rho}{ds} \frac{d \theta}{ds} = 0\],\ illustrating its influence on the angle evolution along the geodesic.

    • The significance of the metric component in equations is its impact on capturing the curvature and spatial relationships of the surface efficiently. This makes solving geodesic equations more manageable by incorporating the surface's inherent geometric properties.

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Most popular questions from this chapter

Prove that an orientable compact surface \(S \subset R^{3}\) has a differentiable vector field without singular points if and only if \(S\) is homeomorphic to a torus.

Let \(\alpha: I-R^{3}\) be a parametrized regular curve. For each \(t \in I\), let \(P(t) \subset\) \(R^{3}\) be a plane through \(\alpha(t)\) which contains \(\alpha^{\prime}(t)\). When the unit normal vector \(N(t)\) of \(P(t)\) is a differentiable function of \(t\) and \(N^{\prime}(t) \neq 0, t \in I\), we say that the map \(t \rightarrow\\{\alpha(t), N(t)\\}\) is a differentiable family of tangent planes. Given such a family, we determine a parametrized surface (cf. Def. 2, Sec. \(2-3)\) by $$ \mathbf{x}(t, v)=\alpha(t)+v \frac{N(t) \wedge N^{\prime}(t)}{\left|N^{\prime}(t)\right|} $$ The parametrized surface \(\mathbf{x}\) is called the envelope of the family \(\\{\alpha(t), N(t)\\}\) (cf. Example 4, Sec. 3-5). a. Let \(S\) be an oriented surface and let \(\gamma: I \rightarrow S\) be a geodesic parametrized by arc length with \(k(s) \neq 0\) and \(\tau(s) \neq 0, s \in I\). Let \(N(s)\) be the unit normal vector of \(S\) along \(\gamma\). Prove that the envelope of the family of tangent planes \(\\{\gamma(s), N(s)\\}\) is regular in a neighborhood of \(\gamma\), has Gaussian curvature \(K \equiv 0\), and is tangent to \(S\) along \(\gamma\). (Thus, we have obtained a surface locally isometric to the plane which contains \(\gamma\) as a geodesic.) b. Let \(\alpha: I \rightarrow R^{3}\) be a curve parametrized by arc length with \(k(s) \neq 0\) and \(\tau(s) \neq 0, s \in I\), and let \(\\{\alpha(s), n(s)\\}\) be the family of its rectifying planes. Prove that the envelope of this family is regular in a neighborhood of \(\alpha\), has Gaussian curvature \(K=0\), and contains \(\alpha\) as a geodesic. (Thus, every curve is a geodesic in the envelope of its rectifying planes; since this envelope is locally isometric to the plane, this justifies the name rectifying plane.)

Let \(y\) and \(w\) be differentiable vector fields on an open set \(U \subset S .\) Let \(p \in S\) and let \(\alpha: I \rightarrow U\) be a curve such that \(\alpha(0)=p, \alpha^{\prime}(0)=y\). Denote by \(P_{\alpha, t}: T_{\alpha(0)}(S) \rightarrow T_{\alpha(t)}(S)\) the parallel transport along \(\alpha\) from \(\alpha(0)\) to \(\alpha(t), t \in I\). Prove that $$ \left(D_{y} w\right)(p)=\left.\frac{d}{d t}\left(P_{\alpha, t}^{-1}(w(\alpha(t)))\right)\right|_{t=0} $$ where the second member is the velocity vector of the curve \(P_{\alpha, t}^{-1}(w(\alpha(t)))\) in \(T_{p}(S)\) at \(t=0\). (Thus, the notion of covariant derivative can be derived from the notion of parallel transport.)

Let \(S\) be a surface of constant Gaussian curvature. Choose points \(p_{1}, p_{1}^{\prime} \in S\) and let \(V, V^{\prime}\) be convex neighborhoods of \(p_{1}, p_{1}^{\prime}\), respectively. Choose geodesic triangles \(p_{1}, p_{2}, p_{3}\) in \(V\) (geodesic means that the sides \(\widetilde{p_{1} p_{2}, \widetilde{p_{2} p_{3}}, \widetilde{p_{3}} p_{1} \text { are }}\) geodesic arcs) in \(v\) and \(p_{1}^{\prime}, p_{2}^{\prime}, p_{3}^{\prime}\) in \(V^{\prime}\) in such a way that $$ \begin{aligned} &l\left(p_{1}, p_{2}\right)=l\left(p_{1}^{\prime}, p_{2}^{\prime}\right) \\ &l\left(p_{2}, p_{3}\right)=l\left(p_{2}^{\prime}, p_{3}^{\prime}\right) \\ &l\left(p_{3}, p_{1}\right)=l\left(p_{3}^{\prime}, p_{1}^{\prime}\right) \end{aligned} $$ (here \(l\) denotes the length of a geodesic arc). Show that there exists an isometry \(\theta: V \rightarrow V^{\prime}\) which maps the first triangle onto the second. (This is the local version, for surfaces of constant curvature, of the theorem of high school geometry that any two triangles in the plane with equal corresponding sides are congruent.)

Let \(S \subset R^{3}\) be a regular, compact, connected, orientable surface which is not homeomorphic to a sphere. Prove that there are points on \(S\) where the Gaussian curvature is positive, negative, and zero.
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