We consider the N-body problem given by quasihomogeneous force functions of the form (C_1)/r^a + (C_2)/r^b (C_1, C_2, a, b constants and a, b positive with a less than or equal to b) and address the fundamentals of homographic solutions. Generalizing techniques of the classical N-body problem, we prove necessary and sufficient conditions for a homographic solution to be either homothetic, or relative equilibrium. We further prove an analogue of the Lagrange-Pizzetti theorem based on our techniques. We also study the central configurations for quasihomogeneous force functions and settle the classification and properties of simultaneous and extraneous central configurations. In the last part of the thesis, we combine these findings with the Lagrange-Pizzetti theorem to show the link between homographic solutions and central configurations, to prove the existence of homographic solutions and to give algorithms for their construction. / Graduate

Indiana University-Purdue University Indianapolis (IUPUI) / We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.