Question

Find the geodesics on the cone \(x^{2}+y^{2}=z^{2} .\) Hint: Use cylindrical coordinates.

Short answer

The geodesics on the cone are spiral curves.

Step-by-step solution

- Convert to cylindrical coordinates

In cylindrical coordinates, the given cone equation can be expressed as: \[ x = r \cos\theta \] \[ y = r \sin\theta \] \[ z = r \] Hence, the cone equation transforms to: \[ r^{2} \cos^{2}\theta + r^{2} \sin^{2}\theta = r^{2} \Rightarrow z=r \] This is the new equation describing the cone.

- Set up the Lagrangian

To find the geodesics, use the line element in cylindrical coordinates, which is given by: \[ ds^{2} = dr^{2} + r^{2} d\theta^{2} + dz^{2} \] Substituting the cone equation, where \( z = r \), we get: \[ dz = dr \] Hence, the line element becomes: \[ ds^{2} = dr^{2} + r^{2} d\theta^{2} + dr^{2} = 2dr^{2} + r^{2} d\theta^{2} \]

- Use the Lagrangian for geodesics

The Lagrangian is then: \[ L = \sqrt{2 \left(\frac{dr}{d\tau}\right)^{2} + r^{2} \left(\frac{d\theta}{d\tau}\right)^{2}} \] Applying the Euler-Lagrange equation, which is: \[ \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{q_{i}}} \right) - \frac{\partial L}{\partial q_{i}} = 0 \] Where \( q_{i} \) are the generalized coordinates, namely \( r \) and \( \theta \).

- Solve Euler-Lagrange equations for

\( r \)-coordinate: \[ \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} = 0 \] Substitute and compute: use \( \dot{r} = \frac{dr}{d\tau} \) With \( L = \sqrt{2 \dot{r}^{2} + r^{2} \dot{\theta}^{2}} \), obtain: \[ \frac{d}{d\tau} \left( \frac{2 \dot{r}}{L} \right) - \frac{r \dot{\theta}^{2}}{L} = 0 \] Simplify and solve to find: \[ \ddot{r} - r \dot{\theta}^{2} = 0 \]

- Solve Euler-Lagrange equations for

\( \theta \)-coordinate: \[ \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 \] Compute: \[ \frac{d}{d\tau} \left( \frac{r^{2} \dot{\theta}}{L} \right) = 0 \] This equality implies that: \[ \frac{r^{2} \dot{\theta}}{L} = k \] Where \( k \) is a constant.

- Combine the solutions

\( r\ddot{\theta} + 2\dot{r}\dot{\theta} = 0 \) -> equation of geodesics can be integrated successively: Use known trigonometric identities to express \( \theta \) and \( r \) for specific intervals.

- Final solution

Combining these with boundary conditions if present provides the geodesics, the geodesic equation on the cone also understood as: path trace such as spiral curve.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a way of representing points in a three-dimensional space using a combination of radial distance, angle, and height. This system is especially handy for problems involving symmetry around an axis, such as those involving cones, cylinders, and circles. In cylindrical coordinates, a point \((x, y, z)\) is described by three parameters: the radial distance \(r\) from the z-axis, the angle \(\theta\) around the z-axis, and the height \(z\) along the z-axis.

The conversion formulas from Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\) are:

• \( x = r \cos\theta \)
• \( y = r \sin\theta \)
• \( z = z \)

We used these formulas to transform the cone equation \(x^2 + y^2 = z^2\) into cylindrical coordinates, simplifying our calculations.

Lagrangian Mechanics

Lagrangian mechanics is a formulation of classical mechanics that makes use of the concept of energy. It's particularly useful for dealing with complex systems and allows us to derive the equations of motion of a system using the principle of least action.

The Lagrangian \(L\) of a system is defined as the difference between the kinetic energy \(T\) and the potential energy \(V\):

• \( L = T - V \)

For our problem on the cone, the kinetic energy can be expressed in terms of the radial distance \(r\) and angular position \(\theta\), leading to the Lagrangian:

\[ L = \sqrt{2 \left(\frac{dr}{d\tau}\right)^{2} + r^{2} \left(\frac{d\theta}{d\tau}\right)^{2}} \]
This formulation is essential to apply the Euler-Lagrange equation later to find the geodesic paths on the cone.

Euler-Lagrange Equation

The Euler-Lagrange equation is a fundamental equation in the calculus of variations used to find functions that optimize a certain functional. In physics, it's used to derive the equations of motion from the Lagrangian of the system.

The Euler-Lagrange equation is given by:
\[ \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{q_{i}}} \right) - \frac{\partial L}{\partial q_{i}} = 0 \]

• Here, \(q_{i}\) are the generalized coordinates, and \(\dot{q_{i}}\) are their time derivatives.

In our problem, we apply it to each coordinate \(r\) and \(\theta\). For the coordinate \(r\), we get:
\[ \frac{d}{d\tau} \left( \frac{2 \dot{r}}{L} \right) - \frac{r \dot{\theta}^{2}}{L} = 0 \]
This simplifies to:
\[ \ddot{r} - r \dot{\theta}^{2} = 0 \]
For the coordinate \(\theta\), we obtain:
\[ \frac{d}{d\tau} \left( \frac{r^{2} \dot{\theta}}{L} \right) = 0 \]
This implies:
\[ \frac{r^{2} \dot{\theta}}{L} = k \]
where \(k\) is a constant.

Geodesic Equation

The geodesic equation describes the shortest path between two points in a given space. In general relativity and differential geometry, understanding geodesics is fundamental for understanding the curvature of space.

For our specific problem involving a cone and using cylindrical coordinates, the geodesic paths are derived by solving the Euler-Lagrange equations obtained from our Lagrangian.

Upon solving, we get two key equations:

• \[ \ddot{r} - r \dot{\theta}^{2} = 0 \]
• \[ \frac{r^{2} \dot{\theta}}{L} = k \]

By integrating and combining these solutions according to the boundary conditions, we get the final geodesic paths, which can often look like spiral curves on the surface of the cone.
Understanding these paths helps in various practical applications, including navigation, computer graphics, and physics simulations.