zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Approximate first integrals for discrete Hamiltonian systems. (English) Zbl 0894.65036
For readers interested in numerical methods: By applying the formalism developed for continuous Hamiltonian ordinary differential equations (ODEs) derived from a fixed end-points variational problem, some criteria are proposed for choosing an appropriate integration stepsize. It is shown that the resulting `polygonal’ solution will be close to the continuous one. For readers interested in the relations between continuous and discrete formulations: The argumentation is based on a well-known fact: A Hamiltonian ODE system $L= 0$ with periodic coefficients can be reduced to an autonomous recurrence system $L_n= 0$ by the Poincaré method of sections. Since for $L= 0$ the (nonautonomous) Hamiltonian $H=\text{const.}$ is not an integral of motion, the discretized Hamiltonian $H_n=\text{const.}$ is also not one. It is equally well known that even a second-order autonomous $L_n= 0$ can describe chaotic dynamics. The associated $H_n=\text{const.}$ provides then no information on the structure of the phase space. The authors assume implicitly that this structure is `locally orderly’. The variable stepsizes are not related to `intermediate iterates’ (fractional values of $n$).

65L10Boundary value problems for ODE (numerical methods)
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
37D45Strange attractors, chaotic dynamics
65L50Mesh generation and refinement (ODE)
37-99Dynamic systems and ergodic theory (MSC2000)
65L12Finite difference methods for ODE (numerical methods)