Co rre ct ed Pr oo f V Variational Integrators

Introduction 12 Geometric numerical integrators are numerical meth13 ods that preserve the geometric structure of a continu14 ous dynamical system (see, e.g., [8, 11], and references 15 therein), and variational integrators provide a system16 atic framework for constructing numerical integrators 17 that preserve the symplectic structure and momen18 tum, of Lagrangian and Hamiltonian systems, while 19 exhibiting good energy stability for exponentially long 20 times. 21 In many problems, the underlying geometric struc22 ture affects the qualitative behavior of solutions, and 23 as such, numerical methods that preserve the geometry 24 of a problem typically yield more qualitatively accu25 rate simulations. This qualitative property of geometric 26 integrators can be better understood by viewing a 27 numerical method as a discrete dynamical system that 28 approximates the flow map of the continuous system 29 (see, e.g., [1, 21]), as opposed to the traditional view 30 that a numerical method approximates individual tra31 jectories. In particular, this viewpoint allows questions 32 about long-time stability to be addressed, which would 33 otherwise be difficult to answer. 34