By Walter Craig

Physical legislation are for the main half expressed when it comes to differential equations, and common sessions of those are within the kind of conservation legislation or of difficulties of the calculus of adaptations for an motion useful. those difficulties can regularly be posed as Hamiltonian platforms, no matter if dynamical structures on finite dimensional section area as in classical mechanics, or partial differential equations (PDE) that are obviously of infinitely many levels of freedom. This quantity is the gathered and prolonged notes from the lectures on Hamiltonian dynamical platforms and their functions that got on the NATO complex research Institute in Montreal in 2007. Many elements of the fashionable thought of the topic have been coated at this occasion, together with low dimensional difficulties in addition to the idea of Hamiltonian structures in endless dimensional section area; those are defined extensive during this quantity. functions also are offered to a number of vital parts of study, together with difficulties in classical mechanics, continuum mechanics, and partial differential equations. those lecture notes hide many parts of contemporary mathematical development during this box, together with the recent choreographies of many physique orbits, the advance of rigorous averaging tools which provide desire for lifelike very long time balance effects, the improvement of KAM idea for partial differential equations in a single and in better dimensions, and the hot advancements within the lengthy remarkable challenge of Arnold diffusion. it is also different contributions to celestial mechanics, to regulate conception, to partial differential equations of fluid dynamics, and to the idea of adiabatic invariants. particularly the final numerous years has noticeable significant development at the difficulties of KAM thought and Arnold diffusion; as a result, this quantity comprises lectures on contemporary advancements of KAM concept in endless dimensional part area, and outlines of Arnold diffusion utilizing variational equipment in addition to geometrical ways to the space challenge. the themes in query contain through necessity essentially the most technical points of study coming from a few diversified fields. earlier than the current quantity, there has no longer been one textual content nor one process examine within which complicated scholars or skilled researchers from different parts can receive an outline and heritage to go into this learn region. This quantity bargains this, in an extraordinary sequence of prolonged lectures encompassing this vast spectrum of themes in PDE and dynamical systems.