Zhu, YichunThe theory of averaging, originated by Laplace and Lagrange, has beenapplied in its long history in many fields as, for example, celestial mechanics, oscillation theory and radio physics, and for a long period it has been used without a rigorous mathematical justification. A further development in the theory of averaging, which is of great interest in applications, concerns the case of random perturbations of dynamical systems. This dissertation, discussing some applications of the averaging principle, includes two parts. In Chapter 1 and Chpater 2, we first study the Smoluchowski-Kramers approximation of a system with a finite number of degrees of freedom, in the presence of a varying magnetic field, where the planar motion of a charged particle of small mass µ is governed by Newton’s law. A small friction is added to the magnetic field to regularize the problem so that the small mass approximation can be studied. When µ ! 0, a noise-drift term arises in the limit process due to the dependence of the strength of the magnetic field on the state of the particle and the system can be viewed as a small random perturbation of a Hamiltonian system in a large time scale and the Hamiltonian turns out to be the strength of the magnetic field. Therefore, by generalzing the Freidlin-Wentzell theory of averaging for Hamiltonian systems, we obtain a description of the slow motion of the charged particle when the mass is small. In the second part(Chapter 3), we study the validity of an averaging principle for a class of slow-fast systems of stochastic partial di↵erential equations. These systems are characterized by having weak regularity assumptions for the nonlinearity in the slow equation. Due to the weakness of these conditions, we are only able to prove the existence of martingale solutions and characterize the distributions of limiting points for the slow motions as solutions of a suitable averaged equation. After the publication of [33] and [31], there have been many papers in recent years that have stud- ied the validity of averaging principles for various types of slow-fast systems of SPDEs, but these previous papers have all assumed well-posedness in appropriate functional spaces, whereas the systems considered in this paper are so irregular that it is not possible to have the existence and uniqueness of solutions, not even in the martingale sense, for either the slow-fast system or the limiting averaged equation. In Chapter 4, we proved the existence of martingale solution for the stochastic PDE with rough nonlinear coeenDissertationMathematicsAveraging principleStochastic PDE