The function geodesic_eqns(coord, Tau, Cf2) generates (but does not solve) the Euler-Lagrange equations of the geodesics for a metric with Christoffel symbols of the second kind Cf2 and coordinate variables coord.  The equations are written in terms of the coordinate variable names as functions of the given parameter Tau. They are returned in the format of a list of equations.

Cf2 should be indexed using the cf2 indexing function provided by the tensor package.  It can be computed using the Christoffel2 routine.

$\mathrm{with}\left(\mathrm{tensor}\right)\:$

$\mathrm{coord}≔\left[u\,v\right]$

$\mathrm{coord}≔\left[u\,v\right]$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,\mathrm{sparse}\,1..2\,1..2\,\left[\left(1\,1\right)=\frac{1}{v^2}\,\left(2\,2\right)=\frac{1}{v^2}\right]\right)\:$

$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{g\_compts}\right)\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cc}\frac{1}{v^2}& 0\\ 0& \frac{1}{v^2}\end{array}\right]\right]\right)$

$\mathrm{ginv}≔\mathrm{invert}\left(g\,'\mathrm{detg}'\right)\:$

$\mathrm{d1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$$\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g}\,\mathrm{coord}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right)\:$

$\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{ginv}\,\mathrm{Cf1}\right)\:$

$\mathrm{displayGR}\left(\mathrm{Christoffel2}\,\mathrm{Cf2}\right)$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; Second\; Kind}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{\{1,12\}}=-\frac{1}{v}$

$\mathrm{\{2,11\}}=\frac{1}{v}$

$\mathrm{\{2,22\}}=-\frac{1}{v}$

$\mathrm{eqns}≔\mathrm{geodesic\_eqns}\left(\mathrm{coord}\,t\,\mathrm{Cf2}\right)$

$\mathrm{eqns}≔\left\{\frac{{\ⅆ}^2}{\ⅆt^2}u\left(t\right)-\frac{2\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}v\left(t\right)\right)}{v}=0\,\frac{{\ⅆ}^2}{\ⅆt^2}v\left(t\right)+\frac{{\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)}^2}{v}-\frac{{\left(\frac{\ⅆ}{\ⅆt}v\left(t\right)\right)}^2}{v}=0\right\}$

$\mathrm{coord}≔\left[x\,y\,z\right]$

$\mathrm{coord}≔\left[x\,y\,z\right]$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,\mathrm{sparse}\,1..3\,1..3\,\left[\left(1\,1\right)=1\,\left(2\,2\right)=1\,\left(3\,3\right)=1\right]\right)\:$

$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{g\_compts}\right)\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\right]\right)$

$\mathrm{ginv}≔\mathrm{invert}\left(g\,'\mathrm{detg}'\right)\:$

$\mathrm{d1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$$\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g}\,\mathrm{coord}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right)\:$

$\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{ginv}\,\mathrm{Cf1}\right)\:$

$\mathrm{displayGR}\left(\mathrm{Christoffel2}\,\mathrm{Cf2}\right)$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; Second\; Kind}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{None}$

$\mathrm{eqns}≔\mathrm{geodesic\_eqns}\left(\mathrm{coord}\,t\,\mathrm{Cf2}\right)$

$\mathrm{eqns}≔\left\{\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)=0\,\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)=0\,\frac{{\ⅆ}^2}{\ⅆt^2}z\left(t\right)=0\right\}$

$\mathrm{map}\left(\mathrm{eval}\,\mathrm{subs}\left(x\left(t\right)=at+b\,y\left(t\right)=ct+e\,z\left(t\right)=ft+h\,\mathrm{eqns}\right)\right)$

$\left\{0=0\right\}$

$\mathrm{coord}≔\left[r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{coord}≔\left[r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,\mathrm{sparse}\,1..3\,1..3\,\left[\left(1\,1\right)=1\,\left(2\,2\right)=r^2\,\left(3\,3\right)=r^2{\sin \left(\mathrm{\theta }\right)}^2\right]\right)\:$

$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{g\_compts}\right)\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}1& 0& 0\\ 0& r^2& 0\\ 0& 0& r^2{\sin \left(\mathrm{\theta }\right)}^2\end{array}\right]\right]\right)$

$\mathrm{ginv}≔\mathrm{invert}\left(g\,'\mathrm{detg}'\right)\:$

$\mathrm{d1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$$\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g}\,\mathrm{coord}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right)\:$

$\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{ginv}\,\mathrm{Cf1}\right)\:$

$\mathrm{displayGR}\left(\mathrm{Christoffel2}\,\mathrm{Cf2}\right)$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; Second\; Kind}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{\{1,22\}}=-r$

$\mathrm{\{1,33\}}=-r{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{\{2,12\}}=\frac{1}{r}$

$\mathrm{\{2,33\}}=-\frac{\sin \left(2\mathrm{\theta }\right)}{2}$

$\mathrm{\{3,13\}}=\frac{1}{r}$

$\mathrm{\{3,23\}}=\cot \left(\mathrm{\theta }\right)$

$\mathrm{eqns}≔\mathrm{geodesic\_eqns}\left(\mathrm{coord}\,t\,\mathrm{Cf2}\right)$

$\mathrm{eqns}≔\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{\phi }\left(t\right)+\frac{2\left(\frac{\ⅆ}{\ⅆt}r\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{\phi }\left(t\right)\right)}{r}+2\cot \left(\mathrm{\theta }\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{\theta }\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{\phi }\left(t\right)\right)=0\,\frac{{\ⅆ}^2}{\ⅆt^2}r\left(t\right)-r{\left(\frac{\ⅆ}{\ⅆt}\mathrm{\theta }\left(t\right)\right)}^2-r{\sin \left(\mathrm{\theta }\right)}^2{\left(\frac{\ⅆ}{\ⅆt}\mathrm{\phi }\left(t\right)\right)}^2=0\,\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{\theta }\left(t\right)+\frac{2\left(\frac{\ⅆ}{\ⅆt}r\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{\theta }\left(t\right)\right)}{r}-\frac{\sin \left(2\mathrm{\theta }\right){\left(\frac{\ⅆ}{\ⅆt}\mathrm{\phi }\left(t\right)\right)}^2}{2}=0\right\}$