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.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{Gamma}≔\mathrm{Connection}\left(a\left(\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}+\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)+by\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}\right)$

$\mathrm{\Γ}:=a\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\+a\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\+by\mathrm{D\_x}\mathrm{dy}\mathrm{dy}$

$C≔\left[x\left(t\right)\,y\left(t\right)\right]$

$C:=\left[x\left(t\right)\,y\left(t\right)\right]$

$V≔\mathrm{GeodesicEquations}\left(C\,\mathrm{Gamma}\,t\right)$

$V:=\left({\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^{2}by\left(t\right)\+2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)a\+\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\right)\mathrm{D\_x}\+\left(\frac{\ⅆ}{\ⅆt}\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\right)\mathrm{D\_y}$

$\mathrm{DE}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(V\,''CoefficientSet''\right)$

$\mathrm{DE}:=\left\{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^{2}by\left(t\right)\+2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)a\+\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\,\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right\}$

$\mathrm{simplify}\left(\mathrm{dsolve}\left(\mathrm{DE}\,\mathrm{explicit}\right)\right)$

$\left\{x\left(t\right)\=-\frac{1}{4}\frac{2b{\mathrm{\_C3}}^{2}a\mathrm{\_C4}t-b{\mathrm{\_C3}}^{2}t\+b{\mathrm{\_C3}}^{3}a{t}^2\+2{\ⅇ}^{-2a\left(\mathrm{\_C3}t\+\mathrm{\_C4}\right)}a\mathrm{\_C1}-4\mathrm{\_C2}\mathrm{\_C3}{a}^2}{\mathrm{\_C3}{a}^2}\,y\left(t\right)\=\mathrm{\_C3}t\+\mathrm{\_C4}\right\}$