We solve the geodesic deviation equations for the orbital motions in the Schwarzschild metric which are close to a circular orbit. It turns out that in this particular case the equations reduce to a linear system, which after diagonalization describes just a collection of harmonic oscillators, with two characteristic frequencies. The new geodesic obtained by adding this solution to the circular one, describes not only the linear approximation of Kepler's laws, but gives also the right value of the perihelion advance (in the limit of almost circular orbits). We derive also the equations for higher-order deviations and show how these equations lead to better approximations, including the non-linear effects. The approximate orbital solutions are then inserted into the quadrupole formula to estimate the gravitational radiation from non-circular orbits.

We discuss the covariant formulation of the dynamics of particles with Abelian and non-Abelian gauge charges in external fields. Using this formulation we develop an algorithm for the construction of constants of motion, which makes use of a generalization of the concept of Killing vectors and tensors in differential geometry. We apply the formalism to the motion of classical charges in Abelian and non-Abelian monopole fields.

The systematic derivation of constants of the motion, based on Killing tensors and the gauge covariant approach, is outlined. Quantum dots are shown to support second-, third- and fourth-rank Killing tensors.

We study in detail the equations of the geodesic deviation in multidimensional theories of Kaluza-Klein type. We show that their 4-dimensional space-time projections are identical with the equations obtained by direct variation of the usual geodesic equation in the presence of the Lorentz force, provided that the fifth component of the deviation vector satisfies an extra constraint derived here.

We study the motion of neutral and charged spinning bodies in curved spacetime in the test-particle limit. We construct equations of motion using a closed covariant Poisson-Dirac bracket formulation that allows for different choices of the Hamiltonian. We derive conditions for the existence of constants of motion and apply the formalism to the case of spherically symmetric spacetimes. We show that the periastron of a spinning body in a stable orbit in a Schwarzschild or Reissner-Nordstrøm background not only precesses but also varies radially. By analyzing the stability conditions for circular motion we find the innermost stable circular orbit (ISCO) as a function of spin. It turns out that there is an absolute lower limit on the ISCOs for increasing prograde spin. Finally we establish that the equations of motion can also be derived from the Einstein equations using an appropriate energy-momentum tensor for spinning particles.