User:Z E U S/geodesicsequation

$$ {d^2 x^\mu \over ds^2} + \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\ = 0 $$

$$ \ddot x^\lambda + \Gamma^\lambda {}_{\mu \nu} \dot x^\mu \dot x^\nu = 0\ .$$

$$\begin{cases}{\frac {d^{2}}{d{\tau}^{2}}}t -1/2\,{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{0 0}}   \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{{\it g_{3 3}}  }}+{\frac { \left( {\frac {\partial }{\partial x}}{\it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}t  }{{\it g_{3 3}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{1 1}}   \right)  \left( {\frac {d}{d\tau}}y   \right) ^{2}}{{\it g_{3 3}}  }}+{\frac { \left( {\frac {\partial }{\partial y}}{\it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}y   \right) {\frac {d}{d\tau}}t  }{{\it g_{3 3}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{2 2}}   \right)  \left( {\frac {d}{d\tau}}z   \right) ^{2}}{{\it g_{3 3}}  }}+{\frac { \left( {\frac {\partial }{\partial z}}{\it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}z   \right) {\frac {d}{d\tau}}t  }{{\it g_{3 3}}  }}+1/2\,{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}t   \right) ^{2}}{{\it g_{3 3}}  }}=0 \\ {\frac {d^{2}}{d{\tau}^{2}}}x +1/2\,{\frac { \left( {\frac {\partial }{\partial x}}{\it g_{0 0}}  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{{\it g_{0 0}}  }}+{\frac { \left( { \frac {\partial }{\partial y}}{\it g_{0 0}}   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}y }{{\it g_{0 0}}  }}+{\frac { \left( {\frac {\partial }{\partial z}}{\it g_{0 0}}   \right)  \left( {\frac {d}{d\tau}}x \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{{\it g_{0 0}} }}+{\frac { \left( {\frac {\partial }{ \partial t}}{\it g_{0 0}}   \right)  \left( {\frac { d}{d\tau}}x   \right) {\frac {d}{d\tau}}t \left( \tau \right) }{{\it g_{0 0}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial x}}{\it g_{1 1}}  \right)  \left( {\frac {d}{d\tau}}y   \right) ^{2}}{{\it g_{0 0}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial x}}{\it g_{2 2}}  \right)  \left( {\frac {d}{d\tau}}z   \right) ^{2}}{{\it g_{0 0}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial x}}{\it g_{3 3}}  \right)  \left( {\frac {d}{d\tau}}t   \right) ^{2}}{{\it g_{0 0}}  }}=0 \\ {\frac {d^{2}}{d{ \tau}^{2}}}y -1/2\,{\frac { \left( {\frac { \partial }{\partial y}}{\it g_{0 0}}   \right) \left( {\frac {d}{d\tau}}x  \right) ^{2}}{{\it g_{1 1}} }}+{\frac { \left( {\frac {\partial }{ \partial x}}{\it g_{1 1}}   \right)  \left( {\frac { d}{d\tau}}x   \right) {\frac {d}{d\tau}}y \left( \tau \right) }{{\it g_{1 1}}  }}+1/2\,{\frac { \left( {\frac {\partial }{\partial y}}{\it g_{1 1}}  \right)  \left( {\frac {d}{d\tau}}y   \right) ^{2}}{{\it g_{1 1}}  }}+{\frac { \left( { \frac {\partial }{\partial z}}{\it g_{1 1}}   \right)  \left( {\frac {d}{d\tau}}y   \right) { \frac {d}{d\tau}}z }{{\it g_{1 1}}  }}+{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{1 1}}   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{{\it g_{1 1}} }}-1/2\,{\frac { \left( {\frac {\partial } {\partial y}}{\it g_{2 2}}   \right)  \left( {\frac {d}{d\tau}}z   \right) ^{2}}{{\it g_{1 1}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial y}}{ \it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}t   \right) ^{2}}{{\it g_{1 1}}  } }=0 \\ {\frac {d^{2}}{d{\tau}^{2}}}z -1/2\,{\frac { \left( {\frac {\partial }{\partial z}}{\it g_{0 0}}  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{{\it g_{2 2}}  }}+{\frac { \left( { \frac {\partial }{\partial x}}{\it g_{2 2}}   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}z }{{\it g_{2 2}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial z}}{\it g_{1 1}}   \right)  \left( {\frac {d}{d\tau}}y   \right) ^{2}}{{\it g_{2 2}}  } }+{\frac { \left( {\frac {\partial }{\partial y}}{\it g_{2 2}}  \right)  \left( {\frac {d}{d\tau}}y   \right) {\frac {d}{d\tau}}z  }{{\it g_{2 2}}  }}+1/2\,{\frac { \left( {\frac {\partial }{\partial z}}{ \it g_{2 2}}   \right)  \left( {\frac {d}{d\tau}}z   \right) ^{2}}{{\it g_{2 2}}  } }+{\frac { \left( {\frac {\partial }{\partial t}}{\it g_{2 2}}  \right)  \left( {\frac {d}{d\tau}}z   \right) {\frac {d}{d\tau}}t  }{{\it g_{2 2}}  }}-1/2\,{\frac { \left( {\frac {\partial }{\partial z}}{ \it g_{3 3}}   \right)  \left( {\frac {d}{d\tau}}t   \right) ^{2}}{{\it g_{2 2}}  } }=0 \end{cases}$$

$$ \left\{ {\frac {d^{2}}{d{\tau}^{2}}}t -1/2\,{ \frac { \left( {\frac {\partial }{\partial t}}g_  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  }}-{\frac { \left( { \frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}y }{g_  }}+{\frac { \left( {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}y   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}z   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}+1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}t   \right) ^{2}}{g_  }}=0,{\frac {d^{2}}{d{\tau}^{2}}}z  -1/2\, {\frac { \left( {\frac {\partial }{\partial z}}g_  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  }}-{\frac { \left( { \frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}y }{g_  }}+{\frac { \left( {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}y   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  }}+1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}z   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}t   \right) ^{2}}{g_  }}=0,{\frac {d^{2}}{d{\tau}^{2}}}x  +1/2\, {\frac { \left( g_ {\frac {\partial }{ \partial x}}g_  -2\,g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) ^{2}}{g_  g _  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial y}}g_  -g_  {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}y \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}+{ \frac { \left( g_ {\frac {\partial }{ \partial z}}g_  -g_  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}z  }{g_  g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial t}}g_  -g_  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}t \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}-1/2 \,{\frac { \left( -2\,g_ {\frac { \partial }{\partial y}}g_  +g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial z}}g_  -g_  {\frac { \partial }{\partial z}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) {\frac {d}{d \tau}}z }{g_  g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  { \frac {\partial }{\partial t}}g_  -g_{{12} }  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  g_  - \left( g_{ {12}}   \right) ^{2}}}-1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}z  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}-1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}t  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}=0,{\frac {d^{2}}{d{\tau}^{ 2}}}y -1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -2\,g_  {\frac {\partial }{ \partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-{ \frac { \left( g_ {\frac {\partial }{ \partial y}}g_  -g_  {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}y  }{g_  g_  - \left( g_   \right) ^{2}}}-{\frac { \left( g_  {\frac {\partial }{\partial z}}g_  -g_  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}z \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}-{ \frac { \left( g_ {\frac {\partial }{ \partial t}}g_  -g_  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}t  }{g_  g_  - \left( g_   \right) ^{2}}}+1/2\,{\frac { \left( -2\,g_  {\frac {\partial }{\partial y}}g_  +g_  {\frac { \partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-{\frac { \left( g_  { \frac {\partial }{\partial z}}g_  -g_{{11} }  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  g_  - \left( g_{ {12}}   \right) ^{2}}}-{\frac { \left( g_  {\frac {\partial }{\partial t}}g_  -g_  {\frac { \partial }{\partial t}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) {\frac {d}{d \tau}}t }{g_  g_  - \left( g_   \right) ^{2}}}+1/2\,{\frac { \left( g_  { \frac {\partial }{\partial x}}g_  -g_{{11} }  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-1/2\,{\frac { \left( -g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}t \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}=0 \right\}

$$

$$

{\frac {d^{2}}{d{\tau}^{2}}}t -1/2\,{ \frac { \left( {\frac {\partial }{\partial t}}g_  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  }}-{\frac { \left( { \frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}y }{g_  }}+{\frac { \left( {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}y   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}z   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}+1/2\,{\frac { \left( {\frac {\partial }{ \partial t}}g_   \right)  \left( {\frac {d }{d\tau}}t   \right) ^{2}}{g_  }}=0,

$$

$$

{\frac {d^{2}}{d{\tau}^{2}}}z -1/2\, {\frac { \left( {\frac {\partial }{\partial z}}g_  \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  }}-{\frac { \left( { \frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) { \frac {d}{d\tau}}y }{g_  }}+{\frac { \left( {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}y   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  }}+1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}z   \right) ^{2}}{g_  }}+{\frac { \left( {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  }}-1/2\,{\frac { \left( {\frac {\partial }{ \partial z}}g_   \right)  \left( {\frac {d }{d\tau}}t   \right) ^{2}}{g_  }}=0,

$$

$$

{\frac {d^{2}}{d{\tau}^{2}}}x +1/2\, {\frac { \left( g_ {\frac {\partial }{ \partial x}}g_  -2\,g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) ^{2}}{g_  g _  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial y}}g_  -g_  {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}y \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}+{ \frac { \left( g_ {\frac {\partial }{ \partial z}}g_  -g_  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}z  }{g_  g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial t}}g_  -g_  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}t \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}-1/2 \,{\frac { \left( -2\,g_ {\frac { \partial }{\partial y}}g_  +g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  {\frac {\partial }{\partial z}}g_  -g_  {\frac { \partial }{\partial z}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) {\frac {d}{d \tau}}z }{g_  g_  - \left( g_   \right) ^{2}}}+{\frac { \left( g_  { \frac {\partial }{\partial t}}g_  -g_{{12} }  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}t  }{g_  g_  - \left( g_{ {12}}   \right) ^{2}}}-1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}z  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}-1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -g_  {\frac { \partial }{\partial y}}g_   \right) \left( {\frac {d}{d\tau}}t  \right) ^{2}}{g_{{11} } g_  - \left( g_   \right) ^{2}}}=0,

$$

$$

{\frac {d^{2}}{d{\tau}^{ 2}}}y -1/2\,{\frac { \left( g_  {\frac {\partial }{\partial x}}g_  -2\,g_  {\frac {\partial }{ \partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-{ \frac { \left( g_ {\frac {\partial }{ \partial y}}g_  -g_  {\frac {\partial }{\partial x}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}y  }{g_  g_  - \left( g_   \right) ^{2}}}-{\frac { \left( g_  {\frac {\partial }{\partial z}}g_  -g_  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}} x   \right) {\frac {d}{d\tau}}z \left( \tau \right) }{g_  g_  - \left( g_   \right) ^{2}}}-{ \frac { \left( g_ {\frac {\partial }{ \partial t}}g_  -g_  {\frac {\partial }{\partial t}}g_   \right)  \left( {\frac {d}{d\tau}}x   \right) {\frac {d}{d\tau}}t  }{g_  g_  - \left( g_   \right) ^{2}}}+1/2\,{\frac { \left( -2\,g_  {\frac {\partial }{\partial y}}g_  +g_  {\frac { \partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-{\frac { \left( g_  { \frac {\partial }{\partial z}}g_  -g_{{11} }  {\frac {\partial }{\partial z}}g_   \right)  \left( {\frac {d}{d\tau}}y \left( \tau \right) \right) {\frac {d}{d\tau}}z  }{g_  g_  - \left( g_{ {12}}   \right) ^{2}}}-{\frac { \left( g_  {\frac {\partial }{\partial t}}g_  -g_  {\frac { \partial }{\partial t}}g_   \right) \left( {\frac {d}{d\tau}}y  \right) {\frac {d}{d \tau}}t }{g_  g_  - \left( g_   \right) ^{2}}}+1/2\,{\frac { \left( g_  { \frac {\partial }{\partial x}}g_  -g_{{11} }  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}z \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}-1/2\,{\frac { \left( -g_  {\frac {\partial }{\partial x}}g_  +g_  {\frac {\partial }{\partial y}}g_   \right)  \left( {\frac {d}{d\tau}}t \left( \tau \right) \right) ^{2}}{g_  g_  - \left( g_   \right) ^{2}}}=0

$$