# Synge's World function generates geodesic flow

Let $$(Q,g)$$ be a Riemannian manifold and let $$q_0,q_1\in Q$$ be two points that are joined by a unique geodesic $$\gamma$$ (this holds in particular if $$q_1$$ belongs to a normal neighborhood of $$q_0$$). Suppose that $$q_0=\gamma(t_0)$$ and $$q_1=\gamma(t_1)$$, then we define Synge's World function as $$$$\label{Synge world function formula} \sigma(q_0,q_1):=\dfrac{1}{2}(t_1-t_0)\int_{t_0}^{t_1}g_{ij}\dot{\gamma}^i\dot{\gamma}^jdt,\tag{1}$$$$ where $$\gamma^i$$ are the components of $$\gamma$$ in a chart that contains $$q_0$$ and $$q_T$$.
Consider the Hamiltonian $$$$\label{geodesic Hamiltonian formula} \begin{array}{rcl} H:T^*Q & \rightarrow&\mathbb{R} \\ (q,p) & \mapsto&H(q,p):=\frac{1}{2}g^{ij}(q)p_ip_j. \end{array}\tag{2}$$$$ It can be shown that its flow $$\phi^t_H$$ is the geodesic flow, which means that the integral curves of the Hamiltonian vector field $$X_H$$ are the geodesic.
I've read that Synge's World function can be thought as a generating function for the geodesic flow, which means that $$$$\label{eq 1} p_0=-\dfrac{\partial\sigma}{\partial q_0}\qquad p_1=\dfrac{\partial\sigma}{\partial q_1},\tag{3}$$$$ where $$$$(q_1,p_1)=\phi^{t_1-t_0}_{H}(q_0,p_0).\tag{4}$$$$ How to prove the relations (3)?

• The Lagrangian associated to the geodesic Hamiltonian is $L(q,\dot{q})=\frac{1}{2}g_{ij}\dot{q}^i\dot{q}^j$? – Giulio Binosi Sep 22 '20 at 13:32
• $\uparrow$ Yes. – Qmechanic Sep 22 '20 at 14:10