Definition:Geodesic Equation
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Definition
Let $M$ be an $n$-dimensional smooth manifold with or without boundary.
Let $I \subseteq \R$ be a real interval.
Let $\gamma : I \to M$ be a smooth curve.
Let $\tuple {x^i}$ be smooth coordinates.
Let the component functions of $\gamma$ be written as:
- $\map \gamma t = \tuple {\map {x^1} t, \ldots, \map {x^n} t}$
Let $\nabla$ be the connection on $M$.
Let $\set {\Gamma^k_{i j} }$ be connection coefficients of $\nabla$.
Then the following system of second-order ordinary differential equations is called the geodesic equation:
- $\dfrac {\d^2 \map {x^k} t} {\d t^2} + \dfrac {\d \map {x^i} t} {\d t} \dfrac {\d \map {x^j} t} {\d t} \map {\Gamma^k_{i j} } {\map \gamma t} = 0$
where $k \in \N_{>0}: k \le n$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Geodesics