We study the motion of a spinning test particle in Schwarzschild spacetime , analyzing the Poincaré map and the Lyapunov exponent .
We find chaotic behavior for a particle with spin higher than some critical value ( e.g . S _ { cr } \sim 0.64 \mu M for the total angular momentum J = 4 \mu M ) , where \mu and M are the masses of a particle and of a black hole , respectively .
The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods , which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena .

The “ effective potential ” analysis enables us to classify the particle orbits into four types as follows .
When the total angular momentum J is large , some orbits are bounded and the “ effective potential ” s are classified into two types : ( B1 ) one saddle point ( unstable circular orbit ) and one minimal point ( stable circular orbit ) on the equatorial plane exist for small spin ; and ( B2 ) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin .
When J is small , no bound orbits exist and the potentials are classified into another two types : ( U1 ) no extremal point is found for small spin ; and ( U2 ) one saddle point appears on the equatorial plane , which is unstable in the direction perpendicular to the equatorial plane , for large spin .
The types ( B1 ) and ( U1 ) are the same as those for a spinless particle , but the potentials ( B2 ) and ( U2 ) are new types caused by spin-orbit coupling .
The chaotic behavior is found only in the type ( B2 ) potential .
The “ heteroclinic orbit ” , which could cause chaos , is also observed in type ( B2 ) .