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Mobile App Chapter 9: Problem 16
Show that the geodesics on a circular cylinder (with elements parallel to the z axis) are helices \(a z+b \theta=c\), where \(a, b, c\) are constants depending on the given endpoints.
Short Answer
Expert verified
The geodesics on a cylinder are helices described by \ \( az + b \theta = c \).
Step by step solution
01
- Understanding the Problem
Consider a circular cylinder oriented along the z-axis. Geodesics are the curves representing the shortest path between two points on the surface. For a cylinder, we need to show that the geodesic path satisfies the equation of a helix.
02
- Parameterize the Cylinder
Parameterize the surface of the cylinder using cylindrical coordinates. Assume the radius of the cylinder is 1 (for simplicity). Then, the position vector on the surface can be written as \ \(\mathbf{r}(z, \theta) = (\cos(\theta), \sin(\theta), z) \).
03
- Calculating the Metric
Calculate the metric tensor for the surface of the cylinder in these coordinates. The line element \ \( ds \) is given by \ \( ds^2 = d\mathbf{r} \cdot d\mathbf{r} \). Substituting the parameterization, we have \ \(ds^2 = d\theta^2 + dz^2 \).
04
- Euler-Lagrange Equations
Use the metric to form the Lagrangian \ \( L = \sqrt{\dot{\theta}^2 + \dot{z}^2} \). The Euler-Lagrange equations for \ \(\theta \) and \ \(z \) are derived from \ \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 \) and \ \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{z}} \right) - \frac{\partial L}{\partial z} = 0 \).
05
- Solving Euler-Lagrange Equations
Solve the resulting differential equations. Since \ \( L \) does not explicitly depend on \ \(\theta \) or \ \(z \), both conjugate momenta \ \( \frac{\partial L}{\partial \dot{\theta}} = \frac{\dot{\theta}}{\sqrt{\dot{\theta}^2 + \dot{z}^2}} \) and \ \( \frac{\partial L}{\partial \dot{z}} = \frac{\dot{z}}{\sqrt{\dot{\theta}^2 + \dot{z}^2}} \) are constants.
06
- Forming the Helix Equation
Express the constants of motion in terms of a, b, and c. Integrate the equations to find \ \( a z + b \theta = c \), which describes a helical path.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geodesic Equations
When discussing geodesics, we refer to the shortest path between two points on a surface. For a circular cylinder, these paths need to be analyzed using specific equations known as geodesic equations. These equations emerge from the calculus of variations and employ sophisticated mathematical tools to determine the shortest distance.
Geodesic equations can be derived from the Euler-Lagrange equations. This involves first expressing the metric of the surface, which in our case is the cylinder, and then applying these equations to find the shortest path. It's a fundamental concept in differential geometry, significant in many fields such as physics, engineering, and computer graphics.
Geodesic equations can be derived from the Euler-Lagrange equations. This involves first expressing the metric of the surface, which in our case is the cylinder, and then applying these equations to find the shortest path. It's a fundamental concept in differential geometry, significant in many fields such as physics, engineering, and computer graphics.
Cylindrical Coordinates
Cylindrical coordinates are used to simplify the description of points in a three-dimensional space, especially when dealing with cylindrical shapes like our problem. They consist of three components: radius (r), angle (θ), and height (z).
In our problem, the radius is fixed and assumed to be 1 for simplicity, so we only need to consider θ and z. Cylindrical coordinates are particularly useful because they closely match the symmetry of the cylinder, making our calculations more straightforward.
The position on the cylinder can be expressed as \(\textbf{r}(z, \theta) = (\text{cos}(\theta), \text{sin}(\theta), z)\). This parameterization helps us translate the problem into mathematical terms that are easier to manage.
In our problem, the radius is fixed and assumed to be 1 for simplicity, so we only need to consider θ and z. Cylindrical coordinates are particularly useful because they closely match the symmetry of the cylinder, making our calculations more straightforward.
The position on the cylinder can be expressed as \(\textbf{r}(z, \theta) = (\text{cos}(\theta), \text{sin}(\theta), z)\). This parameterization helps us translate the problem into mathematical terms that are easier to manage.
Euler-Lagrange Equations
To find the geodesics or shortest paths on the cylinder, we use the Euler-Lagrange equations. These equations are derived from the principle of least action in physics and are a cornerstone of the calculus of variations.
The Euler-Lagrange equations allow us to find the extrema (minimum or maximum) of functionals, which, in this case, represent the distance on the cylinder. For our problem, the Lagrangian \(L = \sqrt{\dot{\theta}^2 + \dot{z}^2}\) was used.
From here, we derive the differential equations that must be satisfied, leading to the geodesic paths we seek. These equations take the form:
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{θ}} \right) - \frac{\partial L}{\partial θ} = 0 \]
and
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{z}} \right) - \frac{\partial L}{\partial z} = 0 \].
Solving these provides the necessary conditions for our required geodesic path.
The Euler-Lagrange equations allow us to find the extrema (minimum or maximum) of functionals, which, in this case, represent the distance on the cylinder. For our problem, the Lagrangian \(L = \sqrt{\dot{\theta}^2 + \dot{z}^2}\) was used.
From here, we derive the differential equations that must be satisfied, leading to the geodesic paths we seek. These equations take the form:
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{θ}} \right) - \frac{\partial L}{\partial θ} = 0 \]
and
\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{z}} \right) - \frac{\partial L}{\partial z} = 0 \].
Solving these provides the necessary conditions for our required geodesic path.
Helical Paths
Helical paths are a type of three-dimensional curve that resemble a spring or the threading on a screw. These curves are characterized by their constant pitch, which is the distance between turns along the cylinder's axis.
In our cylinder problem, the geodesics turn out to be helices. This is because the natural shortest path along the cylinder's surface, when traced out, forms a helical shape. Mathematically, this is shown using the equation \( a z + b θ = c \), where a, b, and c are constants.
This equation encapsulates the relationship between vertical movement along the z-axis and the angular movement around the cylinder. The constants a, b, and c are determined by the specific endpoints of the path or the boundary conditions.
Understanding helical paths is not just an academic exercise; it has real-world applications in areas such as spiral staircases, helical gears, and even DNA's double helix structure.
In our cylinder problem, the geodesics turn out to be helices. This is because the natural shortest path along the cylinder's surface, when traced out, forms a helical shape. Mathematically, this is shown using the equation \( a z + b θ = c \), where a, b, and c are constants.
This equation encapsulates the relationship between vertical movement along the z-axis and the angular movement around the cylinder. The constants a, b, and c are determined by the specific endpoints of the path or the boundary conditions.
Understanding helical paths is not just an academic exercise; it has real-world applications in areas such as spiral staircases, helical gears, and even DNA's double helix structure.
