Problem 23 Hyperbolic coordinates \(u, v, \... [FREE SOLUTION] (2024)

Chapter 10: Problem 23

Hyperbolic coordinates $u, v, \phi$ are defined in terms of Cartesiancoordinates by $$ x=\cosh u \cos v \cos \phi, \quad y=\cosh u \cos v \sin \phi, \quad z=\sinh u\sin v $$ Sketch the coordinate curves in the $\phi=0$ plane, showing that far from theorigin they become concentric circles and radial lines. In particular,identify the curves $u=0, v=0, v=\pi / 2$ and $v=\pi .$ Calculate the tangentvectors at a general point, show that they are mutually orthogonal and deducethat the appropriate scale factors are $$ h_{u}=h_{v}=\left(\cosh ^{2} u-\cos ^{2} v\right)^{1 / 2}, \quadh_{\phi}=\cosh u \cos v $$ Find the most general function $\varphi(u)$ of $u$ only that satisfiesLaplace's equation $\nabla^{2} \psi=0$

Short Answer

Expert verified

For hyperbolic coordinates, solve the coordinate transformation, identify curves, verify orthogonality, determine scale factors ($h_u\cos v, h_v, h_phi$), then solve Laplace's solution.

Step by step solution

Understand the Coordinate Transformation

Given the hyperbolic coordinates and their relationship with Cartesian coordinates: $x = \cosh u \cos v \cos \phi$, $y = \cosh u \cos v \sin \phi$, $z = \sinh u \sin v$, start by analyzing each coordinate in the plane where \phi=0\.

Simplify Coordinates for $\phi = 0$

In the plane where \phi = 0\, the coordinates reduce to: $x = \cosh u \cos v$, $y = 0$, $z = \sinh u \sin v$. So, we need to sketch the coordinate curves in this simplified plane.

Sketching Coordinate Curves

For the curves in \phi = 0\ plane: 1. When \u = 0\: The equations simplify to $x = \cos v$ and $z = 0$, representing a line segment on the x-axis.2. When \v = 0\: The equations simplify to $x = \cosh u$ and $z = 0$, forming a hyperbola along the x-axis.3. When \v = \pi / 2\: The equations simplify to $x = 0$ and $z = \sinh u$, representing a line along the z-axis.4. When \v = \pi\: The equations simplify to $x = -\cosh u$ and $z = 0$, forming a hyperbola along the negative x-axis.

Showing Curves at Distance

Far from the origin, hyperbolic functions \cosh u\ and \sinh u\ behave exponentially, and $v$ is almost linear, forming radial and circular patterns. Identify how hyperbolas and lines transform.

Calculate Tangent Vectors

Calculate partial derivatives to find tangent vectors:$\frac{\partial x}{\partial u} = \sinh u \cos v \cos \phi$, $\frac{\partial y}{\partial u} = \sinh u \cos v \sin \phi$, $\frac{\partial z}{\partial u} = \cosh u \sin v$. Similarly for \frac{\partial}{\partial v}\.

Show Orthogonality

Demonstrate orthogonality by calculating dot products of partial derivatives and showing they are zero. For example, $\frac{\partial x}{\partial u} \cdot \frac{\partial x}{\partial v}\ = 0$.

Determine Scale Factors

Obtain length of tangents using Euclidean norms: $ h_u = (\frac{\partial x}{\partial u})^2 + (\frac{\partial y}{\partial u})^2 + (\frac{\partial z}{\partial u})^2 = (\cosh^2 u - \cos^2 v)^{1/2}$, repeat for \h_v\ and \h_{\frac{u \cos v}{\phi}}\.

Solve Laplace's Equation for $abla^2 psi = 0$

Substitute given scales into Laplace for coordinate system: $\frac{1}{h_u^2} \frac{\partial}{\partial u}\left(h_\phi \frac{\partial \varphi}{\partial u} \right) = 0$. Solve general solution in terms of $u$.

Key Concepts

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

Coordinate Transformation

Coordinate transformations are crucial in converting one set of coordinates to another, allowing us to explore geometric and physical properties in different coordinate systems. In this exercise, we deal with hyperbolic coordinates linked to Cartesian coordinates through the relations:

$x = \cosh(u) \cos(v) \cos(\phi)$
$y = \cosh(u) \cos(v) \sin(\phi)$
$z = \sinh(u) \sin(v)$

Visualizing these structures in the plane where $ \phi = 0 $ simplifies our sketching. When $ \phi = 0 $, the coordinates reduce to:

$x = \cosh u \cos v$
$y=0$
$z = \sinh u \sin v$

. This visualization helps identify how these hyperbolic structures project onto simpler, more familiar shapes far from the origin.

Tangent Vectors

Tangent vectors at a point in space show the direction and rate of change of coordinates at that point. Tangent vectors for the hyperbolic coordinates are calculated using partial derivatives of the Cartesian coordinates. For instance, the partial derivatives with respect to $u$ and $v$ are:

$\frac{\partial x}{\partial u} = \sinh u \cos v \cos \phi$
$\frac{\partial y}{\partial u} = \sinh u \cos v \sin \phi$
$\frac{\partial z}{\partial u} = \cosh u \sin v$

Similarly, calculating the partial derivatives with respect to $v$ will give us another set of tangent vectors. These vectors are essential for studying the behavior and properties of spaces defined by our coordinate system.

Laplace's Equation

Laplace's Equation, $abla^2 \varphi = 0$, is a second-order partial differential equation widely used in physics and engineering. It describes the behavior of scalar fields, such as electric potential and heat distribution. To solve Laplace's equation in hyperbolic coordinates, we substitute the scale factors into the transformed Laplacian. Starting with: \[\frac{1}{h_u^2} \frac{\partial}{\partial u}\left( h_\phi \frac{\partial \varphi}{\partial u} \right) = 0\] where appropriate scale factors $h_u$ and $h_\phi$ depend on $u$ and $v$. Solving the equation for any function $\varphi(u)$, we ensure it satisfies the condition $abla^2 \varphi = 0$.

Orthogonality of Vectors

Orthogonality means that vectors are perpendicular to each other. Demonstrating this property for hyperbolic coordinates involves calculating the dot products of tangent vectors. For example:

$\frac{\partial x}{\partial u} \cdot \frac{\partial x}{\partial v} = 0$

If the dot products of all corresponding tangent vectors are zero, the vectors are mutually orthogonal. This orthogonality helps simplify calculations, such as finding lengths and angles, in the coordinate system.

Hyperbolic Functions

Hyperbolic functions, like $\cosh(u)$ and $\sinh(u)$, behave similarly to trigonometric functions but describe hyperbolic rather than circular geometry. In our coordinates, these functions define the relationship between hyperbolic and Cartesian coordinates:

$ \cosh(u) $ represents the hyperbolic cosine, which influences the stretching along the $u$-axis.
$ \sinh(u)$ represents the hyperbolic sine, contributing to the stretching along the $z$-axis.

As $u \rightarrow \infty$, $ \cosh(u) $ and $ \sinh(u) $ increase exponentially, leading to the formation of radial and circular patterns far from the origin. Understanding these functions is key to analyzing hyperbolic spaces and their properties.

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