Find the geodesics on a sphere. Hints: Use spherical coordinates with constant \(r=a\). Choose your integration variable so that you can write a first integral of the Euler equation. For the second integration, make the change of variable \(w=\cot \theta .\) To recognize your result as a great circle, find, in terms of spherical coordinates \(\theta\) and \(\phi\), the equation of intersection of the sphere with a plane through the origin.

Understanding spherical coordinates is essential for studying geodesics on a sphere. These coordinates \( (r, \theta, \phi) \) extend the 2D polar coordinates into three dimensions. Here, \( r \) represents the radius or distance from the origin, \( \theta \) is the polar angle (or colatitude, measured from the z-axis), and \( \phi \) is the azimuthal angle (longitude, measured in the x-y plane from the x-axis).
For our problem, the radius \( r \) is constant and equal to \( a \), simplifying our focus to paths on a spherical surface. This leads to parameterization formulas:
\[ x = a \sin\theta \cos\phi, \ y = a \sin\theta \sin\phi, \ z = a \cos\theta. \]
These equations help us map any point on the sphere's surface by just knowing \( \theta \) and \( \phi \). They will be instrumental in deriving further equations and recognising the geodesic paths.
Understanding this coordinate system makes it easier to visualize movements and surfaces, which aids in applying further mathematical concepts like the Euler-Lagrange Equation.

The Euler-Lagrange equation is a fundamental equation in the calculus of variations. It provides a way to find the path that minimizes (or maximizes) a given quantity, like the geodesic distance on a sphere.
For our problem, consider the arc length \( ds \) on a sphere, simplified to:
\[ ds^2 = a^2 (d\theta^2 + \sin^2\theta d\phi^2). \]
To find the geodesics, the integral of the arc length \(\text{S} = \int ds\) is minimized. Applying the Euler-Lagrange equation:
\[ \frac{d}{d\phi} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 \]
Here, \( L \) typically represents the Lagrangian, which is the integrand in our arc length formula. Solving this differential equation helps in finding the path \( \theta(\phi) \) that represents the geodesics on the sphere.
In our scenario, this process leads to a reduced first integral, and later, substitution and simplification give us the geodesic equation. Understanding and applying the Euler-Lagrange equation can help derive equations that describe many physical phenomena, not just geodesics.

In spherical geometry, a great circle is the largest possible circle that can be drawn on a sphere. It's the shortest path between two points on the surface.
A plane that goes through the center of the sphere and the sphere's surface intersects to create a great circle. Mathematically, this can be expressed using spherical coordinates.
From our derived equations, substituting the variable change and back-integration results in the general solution:
\[ \cos \phi \sin \theta = C_3. \]
This expression can be recognized as the equation for a great circle. It represents the intersection of the sphere with a plane passing through the origin.
Interpreting this result, imagine a huge circle drawn on Earth's surface, such as the equator or any line of longitude. These are great circles, and they naturally arise as the geodesic paths — the shortest paths on a sphere. Therefore, finding geodesics on a sphere essentially boils down to identifying the great circles, which helps in navigation and understanding spherical geometry.