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Geodesics

Source lecture(s): PHY622 Lec1

Intuition

A geodesic is the shortest path between two points on a curved surface.

Formal Definition

A curve that locally extremizes arc length on a manifold.

Mathematical Formulation

Arc length: \(L=\int\sqrt{g_{ij}\dot{x}^i\dot{x}^j}\,dt\). Geodesic equation: \(\ddot{x}^k + \Gamma^k_{ij}\dot{x}^i\dot{x}^j = 0\).

Derivation

Apply Euler-Lagrange to the arc-length functional in generalized coordinates; Christoffel symbols \(\Gamma\) encode the metric derivatives.

Worked Example

On a sphere of radius \(R\), great circles are geodesics.

Common Mistakes

  • Assuming geodesics are unique between any two points.
  • Ignoring conjugate points.

Quiz

Q1. Are great circles on Earth geodesics?

Answer

Yes—locally shortest paths.