  In Schwarzschild spacetime, the timelike geodesic equations, which define
particle orbits, have a well-known formulation as a dynamical system in
coordinates adapted to the timelike hypersurface containing the geodesic. For
equatorial geodesics, the resulting dynamical system is shown to possess a
conserved angular quantity and two conserved temporal quantities, whose
properties and physical meaning are analogues of the conserved
Laplace-Runge-Lenz vector, and its variant known as Hamilton's vector, in
Newtonian gravity. When a particle orbit is projected into the spatial
equatorial plane, the angular quantity yields the coordinate angle at which the
orbit has either a turning point (where the radial velocity is zero) or a
centripetal point (where the radial acceleration is zero). This is the same
property as the angle of the respective Laplace-Runge-Lenz and Hamilton vectors
in the plane of motion in Newtonian gravity. The temporal quantities yield the
coordinate time and the proper time at which those points are reached on the
orbit. In general, for orbits that have a single turning point, the three
quantities are globally constant; for orbits that possess more than one turning
point, the temporal quantities are just locally constant as they jump at every
successive turning point, while the angular quantity similarly jumps only if an
orbit is precessing. This is analogous to the properties of a generalized
Laplace-Runge-Lenz vector and generalized Hamilton's vector which are known to
exist for precessing orbits in post-Newtonian gravity. The angular conserved
quantity can be used to define a direct analogue of these vectors at spatial
infinity.
