The Elements of the Calculus of Variations

In §29.2 we defined the geodesics x i = x i (t) relative to a given connexion Γ jk i by means of the equation $${{\nabla }_{T}}\left( T \right) = 0$$ , where T i = dx i /dt is the velocity vector of the curve; in other words the geodesics are the solutions of the equations $$\frac{{{d^2}{x^i}}}{{d{t^2}}} + \Gamma _{jk}^i\frac{{d{x^j}}}{{dt}}\frac{{d{x^k}}}{{dt}} = 0$$ .