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Tensor[GeodesicEquations] - calculate the geodesic equations for a symmetric linear connection on the tangent bundle
Calling Sequences
GeodesicEquations (C, Gamma, t)
Parameters
C - a list of functions of a single variable, defining the components of a curve on a manifold M with respect to a given system of coordinates
Gamma - a connection on the tangent bundle to a manifold M
t - the curve parameter
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Description
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Let M be a manifold and let nabla be a symmetric linear connection on the tangent bundle of M. If C is a curve in M with tangent vector T, then the geodesic equations for C with respect to the connection nabla is the system of second order ODEs defined by nabla_T(T) = 0.
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The procedure GeodesicEquations(C, Gamma, t) returns the vector nabla_T(T).
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form GeodesicEquations(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-GeodesicEquations.
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Examples
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Example 1.
First create a 2 dimensional manifold M and define a connection on the tangent space of M.
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![DGsetup([x, y], M)](/support/helpjp/helpview.aspx?si=5624/file05875/math84.png)
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| (2.1) |
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| (2.2) |
To determine the geodesic equations for nabla we first define a curve on M by specifying a list of functions of a single variable t.
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![C := [x(t), y(t)]](/support/helpjp/helpview.aspx?si=5624/file05875/math108.png)
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![C := [x(t), y(t)]](/support/helpjp/helpview.aspx?si=5624/file05875/math111.png)
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The program GeodesicEquations returns a vector whose components are the components of the geodesic equations.
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To solve these geodesic equations use DGinfo to obtain the coefficients of V as a list. Pass the result to dsolve to solve this system of 2 second order ODEs.
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| (2.5) |
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| (2.6) |
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