Let $M$ be a manifold and let $\nabla$ be a linear connection on the tangent bundle of $M$ or a connection on a vector bundle $E\to M$. If $C$ is a curve in $M$ with tangent vector $T$, then the parallel transport equations for a vector field $Y$ along $C$ are the linear, first order ODEs defined by ${\nabla }_{T}Y\=0$.

The procedure ParallelTransportEquations(C, Y, $\mathbf{\Gamma }$, t) returns the vector ${\nabla }_{T}Y$.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form ParallelTransportEquations(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-ParallelTransportEquations.

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

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

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

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

$\mathrm{\Γ}:=-\mathrm{D\_x}\mathrm{dx}\mathrm{dy}+\mathrm{D\_y}\mathrm{dy}\mathrm{dx}$

$C≔\left[\cos \left(t\right)\,\sin \left(t\right)\right]$

$C:=\left[\cos \left(t\right)\,\sin \left(t\right)\right]$

$Y≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D\_x}+B\left(t\right)\mathrm{D\_y}\right)$

$Y:=A\left(t\right)\mathrm{D\_x}+B\left(t\right)\mathrm{D\_y}$

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

$V:=-\left(A\left(t\right)\cos \left(t\right)-\stackrel{\mathbf{.}}{A}\left(t\right)\right)\mathrm{D\_x}-\left(B\left(t\right)\sin \left(t\right)-\stackrel{\mathbf{.}}{B}\left(t\right)\right)\mathrm{D\_y}$

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

$\mathrm{DE}:=\left\{-A\left(t\right)\cos \left(t\right)\+\frac{\ⅆ}{\ⅆt}A\left(t\right)\,-B\left(t\right)\sin \left(t\right)\+\frac{\ⅆ}{\ⅆt}B\left(t\right)\right\}$

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

$\mathrm{soln}:=\left\{A\left(t\right)\=\mathrm{\_C2}{\ⅇ}^{\sin \left(t\right)}\,B\left(t\right)\=\mathrm{\_C1}{\ⅇ}^{-\cos \left(t\right)}\right\}$

$\mathrm{Y\_t}≔\mathrm{eval}\left(Y\,\mathrm{soln}\right)$

$\mathrm{Y\_t}:=\mathrm{\_C2}{\ⅇ}^{\sin \left(t\right)}\mathrm{D\_x}+\mathrm{\_C1}{\ⅇ}^{-\cos \left(t\right)}\mathrm{D\_y}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\right]\,E\right)\:$

$\mathrm{Gamma}≔\mathrm{Connection}\left(\left(\mathrm{D\_v}\&t\mathrm{dv}\right)\&t\mathrm{dy}-\left(\mathrm{D\_u}\&t\mathrm{dv}\right)\&t\mathrm{dx}\right)$

$\mathrm{\Γ}:=-\mathrm{D\_u}\mathrm{dv}\mathrm{dx}+\mathrm{D\_v}\mathrm{dv}\mathrm{dy}$

$C≔\left[t\,t\right]$

$C:=\left[t\,t\right]$

$Y≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D\_u}+B\left(t\right)\mathrm{D\_v}\right)$

$Y:=A\left(t\right)\mathrm{D\_u}+B\left(t\right)\mathrm{D\_v}$

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

$V:=-\left(B\left(t\right)-\stackrel{\mathbf{.}}{A}\left(t\right)\right)\mathrm{D\_u}+\left(B\left(t\right)+\stackrel{\mathbf{.}}{B}\left(t\right)\right)\mathrm{D\_v}$

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

$\mathrm{DE}:=\left\{-B\left(t\right)\+\frac{\ⅆ}{\ⅆt}A\left(t\right)\,B\left(t\right)\+\frac{\ⅆ}{\ⅆt}B\left(t\right)\right\}$

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

$\mathrm{soln}:=\left\{A\left(t\right)\=-\mathrm{\_C2}{\ⅇ}^{-t}\+\mathrm{\_C1}\,B\left(t\right)\=\mathrm{\_C2}{\ⅇ}^{-t}\right\}$

$\mathrm{Y\_t}≔\mathrm{eval}\left(Y\,\mathrm{soln}\right)$

$\mathrm{Y\_t}:=\left(-\mathrm{\_C2}{\ⅇ}^{-t}+\mathrm{\_C1}\right)\mathrm{D\_u}+\mathrm{\_C2}{\ⅇ}^{-t}\mathrm{D\_v}$