calculate the geodesic equations for a symmetric linear connection on the tangent bundle - Maple Programming Help

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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

Description

 • 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 ${\nabla }_{T}T=0$.
 • The procedure GeodesicEquations(C, Gamma, t) returns the vector ${\nabla }_{T}T$.
 • 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.

Examples

 > with(DifferentialGeometry): with(Tensor):

Example 1.

First create a 2-dimensional manifold $M$ and define a connection on the tangent space of $M$.

 > DGsetup([x, y], M);
 ${\mathrm{frame name: M}}$ (2.1)
 M > Gamma := Connection(a*(D_x &t dx &t dy + D_x &t dy &t dx) + b*y*D_x &t dy &t dy);
 ${\mathrm{Γ}}{:=}{a}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{a}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{b}{}{y}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)

To determine the geodesic equations for this connection we first define a curve on $M$ by specifying a list of functions of a single variable $t$.

 M > C := [x(t), y(t)];
 ${C}{:=}\left[{x}{}\left({t}\right){,}{y}{}\left({t}\right)\right]$ (2.3)

The program GeodesicEquations returns a vector whose components are the components of the geodesic equations.

 M > V := GeodesicEquations(C, Gamma, t);
 ${V}{:=}\left({\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{2}}{}{b}{}{y}{}\left({t}\right){+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{x}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right){}{a}{+}\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{x}{}\left({t}\right)\right)\right){}{\mathrm{D_x}}{+}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)\right){}{\mathrm{D_y}}$ (2.4)

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. See also DGsolve.

 M > DE := Tools:-DGinfo(V, "CoefficientSet");
 ${\mathrm{DE}}{:=}\left\{{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{2}}{}{b}{}{y}{}\left({t}\right){+}{2}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{x}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right){}{a}{+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{x}{}\left({t}\right){,}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{y}{}\left({t}\right)\right\}$ (2.5)
 M > simplify(dsolve(DE, explicit));
 $\left\{{x}{}\left({t}\right){=}{-}\frac{{1}}{{4}}{}\frac{{2}{}{b}{}{{\mathrm{_C3}}}^{{2}}{}{a}{}{\mathrm{_C4}}{}{t}{-}{b}{}{{\mathrm{_C3}}}^{{2}}{}{t}{+}{b}{}{{\mathrm{_C3}}}^{{3}}{}{a}{}{{t}}^{{2}}{+}{2}{}{{ⅇ}}^{{-}{2}{}{a}{}\left({\mathrm{_C3}}{}{t}{+}{\mathrm{_C4}}\right)}{}{a}{}{\mathrm{_C1}}{-}{4}{}{\mathrm{_C2}}{}{\mathrm{_C3}}{}{{a}}^{{2}}}{{\mathrm{_C3}}{}{{a}}^{{2}}}{,}{y}{}\left({t}\right){=}{\mathrm{_C3}}{}{t}{+}{\mathrm{_C4}}\right\}$ (2.6)