Numerische Mathematik

, Volume 103, Issue 4, pp 575–590 | Cite as

An Algebraic Approach to Invariant Preserving Integators: The Case of Quadratic and Hamiltonian Invariants

  • Philippe ChartierEmail author
  • Erwan Faou
  • Ander Murua
Original article


In this article, conditions for the preservation of quadratic and Hamiltonian invariants by numerical methods which can be written as B-series are derived in a purely algebraical way. The existence of a modified invariant is also investigated and turns out to be equivalent, up to a conjugation, to the preservation of the exact invariant. A striking corollary is that a symplectic method is formally conjugate to a method that preserves the Hamitonian exactly. Another surprising consequence is that the underlying one-step method of a symmetric multistep scheme is formally conjugate to a symplectic P-series when applied to Newton’s equations of motion.


Hamiltonian System Hopf Algebra Rooted Tree Hamiltonian Function Algebraic Approach 
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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.IPSOINRIARennesFrance
  2. 2.Konputazio Zientziak eta A. A. saila, Informatika FakultateaUniversity of the Basque CountryDonostia/San SebastiànSpain

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