Problem 6 - Calculus Of Variations, With Applications To Physics And Engineering

Question

(a) Prove that any geodesic on one nappe of the right circular cone $$ x^{2}-b^{2}\left(y^{2}+z^{2}\right) $$ has the following property: If the nappe is cut from the vertex along a generator and the surface of the cone is made to lie flat on a plane surface, the geodesic becomes a straight line. HINT: Show first that, if the cone is described in terms of the parameters \(r, \theta\) in the form $$ x=\frac{b r}{\sqrt{1+b^{2}}}, \quad y=\frac{r \cos \left(0 \sqrt{1+b^{2}}\right)}{\sqrt{1+b^{2}}}, \quad z=\frac{r \sin \left(\theta \sqrt{\left.1+b^{2}\right)}\right.}{\sqrt{1+b^{2}}} $$ which satisfies (110), the variables \(r\) and \(\theta\) represent ordinary polar coordinates on the flattened surface of the cone, with the origin at the vertex. Identify the geodesic \(r=r(\theta)\) as the equation of a straight line in polar coordinates. (b) Prove the analogous property for geodesics on a right circular cylinder. (c) Prove the same for an arbitrary cylindrical surface.

Step-by-Step Solution

Answer

The demonstration confirms that when a right circular cone and cylinders are cut from one point and laid flat, their geodesics transform into straight lines on the plane. These results validate the proposed geometric properties of geodesics on these surfaces.

Step 1: Express the Surface of the Cone

Given the equation of a right circular cone, it could be rewritten in parametric form as: \(x=\frac{b r}{\sqrt{1+b^{2}}}\), \(y=\frac{r \cos \left(\theta \sqrt{1+b^{2}}\right)}{\sqrt{1+b^{2}}}\) and \(z=\frac{r \sin \left(\theta \sqrt{\left.1+b^{2}\right)}\right.}{\sqrt{1+b^{2}}}\).

Step 2: Transform Geodesic Equation

Next, we show that a geodesic on the surface of the cone in original coordinates, defined by an equation \(r = r(\theta)\), could be identified as a straight line in polar coordinates. For any straight line, there will be a linear relationship between \(r\) and \(\theta\), i.e., \(r = m\theta + c\), where \(m\) and \(c\) are constants. This step confirms that the geodesic transforms into a straight line in the cone's flattened surface.

Step 3: Apply to a Right Circular Cylinder

The analogous process is now applied to a right circular cylinder. The polar coordinates for a cylinder are straightforward - \(r\) would be the distance from the axis, and \(\theta\) would represent angular displacement. On 'flattening' the cylinder, any geodesic should transform into a straight line, as per the definition of geodesics.

Step 4: Apply to Arbitrary Cylinder

The same principle is then generalised for any cylindrical surface. When the cylinder unrolls and lies flat on a plane, the geodesic will trace a straight path on the flattened surface, thus proving the hypothesis.

Key Concepts

Calculus of Variations

The calculus of variations is a field of mathematical analysis that deals with optimizing functionals, which are mappings from a set of functions into the real numbers. It involves finding functions that minimize or maximize the value of a given functional.

For example, the shortest path between two points on a curved surface, known as a geodesic, can be determined using the calculus of variations. The functional in this scenario measures the length of a path, and minimizing this functional yields the geodesic equation. This complex problem calls for advanced techniques such as the Euler-Lagrange equations, which provide the necessary conditions for a function to be a minimum of the functional.

In a textbook problem involving geodesics on a cone, the calculus of variations helps to show that the shortest path on the surface of a cone, when flattened, corresponds to a straight line in the plane.

Geodesic Equations

In differential geometry, geodesic equations are used to describe the properties of geodesics, which are the curves representing the shortest path between points on a given surface. These curves are locally length-minimizing, and their shape depends on the geometry of the surface.

For the cone described in the exercise, geodesic equations would help elucidate the relationship between the parameters (like the radial distance and the angle in polar coordinates) that define the shortest path on the cone's surface. These equations account for the curvature of the surface and are derived from the principle that the geodesic has zero acceleration if it's parameterized by arc length. The transformation to polar coordinates simplifies the problem, making it easier to understand that when a cone is unrolled, the geodesic becomes a linear equation in those coordinates.

Parametric Equations

Parametric equations define a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. They are often used to describe the location of points in space, such as the position of a point on a surface like a cone or a cylinder.

In the context of the textbook problem, the surface of the cone is defined using parametric equations with parameters being polar coordinates, specifically the radial distance and the angle. For instance, the point \( (x, y, z) \) on the surface of a cone is expressed in terms of \( r \) and \( \theta \) using the given parametric equations. These equations elegantly capture both the shape of the cone and how geodesics on it behave when the cone is flattened.

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, known as the radius \( r \) and an angle \( \theta \) from a reference direction. This system is particularly useful for dealing with problems that involve symmetry around a point, such as circular and conical shapes.

When applying polar coordinates to the problem of geodesics on a cone, the radial coordinate corresponds to the distance from the vertex of the cone, and the angular coordinate represents the angle around the vertex. The use of polar coordinates simplifies complex problems, such as the textbook exercise, by transforming a curved geodesic on a three-dimensional surface into a straight line when the surface is made flat.