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The geometric description of linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
A symplectic form is nondegenerate if V⊥ = {0}. Symplectic forms are bilinear forms which are described by skew-symmetric Gram matrices A. The form is nondegenerate if and only if det(A) ≠ 0. Also, the same proof as in the symmetric case shows that the dual W⊥ = {x|(x, W) = 0} = {x|(W, x) = 0}
Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Problem 10.55. In general, a manifold may admit several different symplectic forms, resulting in different phase space flows for the same Hamiltonian function. Show that the flow lines in a two-dimensional manifold are the same regardless of the symplectic form imposed, but that the velocity with which they are being followed differs.
Practical perspectives on symplectic accelerated optimization
Published in Optimization Methods and Software, 2023
Valentin Duruisseaux, Melvin Leok
Symplectic integrators form a class of geometric numerical integrators of interest since, when applied to Hamiltonian systems, they yield discrete approximations of the flow that preserve the symplectic 2-form. The preservation of the symplectic 2-form results in the preservation of many qualitative aspects of the underlying dynamical system. In particular, the numerical solution of a Hamiltonian system obtained using a constant time-step symplectic integrator is exponentially-near to the exact solution of a nearby Hamiltonian system for an exponentially-long time [14,43]. It explains why symplectic integrators exhibit good energy conservation with essentially no accumulation of errors in time, when applied to Hamiltonian systems, and why symplectic methods are best suited to integrate Hamiltonian systems. We refer the reader to [45] for a brief recent overview of geometric numerical integration, and to [17,43,53] for a more comprehensive presentation of structure-preserving integration techniques.