Hadamard's dynamical system

The motion studied is that of a free particle sliding frictionlessly on the surface, namely, one having the Hamiltonian

${\displaystyle H(p,q)={\frac {1}{2m}}p_{i}p_{j}g^{ij}(q)}$

where m is the mass of the particle, ${\displaystyle q^{i}}$, ${\displaystyle i=1,2}$ are the coordinates on the manifold, ${\displaystyle p_{i}}$ are the conjugate momenta:

${\displaystyle p_{i}=mg_{ij}{\frac {dq^{j}}{dt}}}$

and

${\displaystyle ds^{2}=g_{ij}(q)dq^{i}dq^{j}\,}$

is the metric tensor on the manifold. Because this is the free-particle Hamiltonian, the solution to the Hamilton–Jacobi equations of motion are simply given by the geodesics on the manifold.

Hadamard was able to show that all geodesics are unstable, in that they all diverge exponentially from one another, as ${\displaystyle e^{\lambda t}}$ with positive Lyapunov exponent

${\displaystyle \lambda ={\sqrt {\frac {2E}{mR^{2}}}}}$

with E the energy of a trajectory, and ${\displaystyle K=-1/R^{2}}$ being the constant negative curvature of the surface.