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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

Numerical integration. --- Hamiltonian systems. --- Differential equations --- Numerical solutions. --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Biomathematics. --- Physics. --- Numerical Analysis. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Methods in Physics. --- Numerical and Computational Physics. --- Mathematical and Computational Biology. --- 517.91 Differential equations --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Global analysis (Mathematics). --- Mathematical physics. --- Numerical and Computational Physics, Simulation. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis --- Mathematics --- Biology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis --- Hamiltonian systems --- Differential equations - Numerical solutions --- 517.91 --- Numerical integration --- 519.62 --- 681.3*G17 --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Numerical solutions