Geodesic Equation

Source lecture(s): PHY622 Lec1

Physical Meaning

Locally shortest/longest curves on a curved manifold.

Variable Definitions

Symbol	Meaning
\(\Gamma\)	Christoffel symbols
\(x^i\)	generalized coordinates

Assumptions

Metric-compatible, torsion-free connection.

Derivation

Apply EL to arc-length functional in curvilinear coordinates.

Applications

General relativity (null geodesics), computer graphics (shortest paths on meshes).

Connections to Other Equations

• euler-lagrange-eq
• minimal-surfaces