We recall the question of geometric integrators in the context of Poisson geometry and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and compared to traditional methods.
Gay-BalmazFHolmDDPutkaradzeV, et al. Exact geometric theory of dendronized polymer dynamics. Adv Appl Math2012; 48(4): 535–574.
2.
CSLiu. A Lie–Poisson bracket formulation of plasticity and the computations based on the Lie-group SO(n). Int J Solids Struct2013; 50(13): 2033–2049.
3.
KouloukasTEQuispelGRWVanhaeckeP. Liouville integrability and superintegrability of a generalized Lotka–Volterra system and its Kahan discretization. J Phys A: Math Theor2016; 49(22): 225201.
CrainicMFernandesRMărcutI. Lectures on Poisson geometry. Providence, RI: American Mathematical Society, 2021.
16.
KarasevMV. Analogues of the objects of Lie group theory for nonlinear Poisson brackets. Math USSR Izv1987; 28(3): 497.
17.
GeZ. Generating functions, Hamilton-Jacobi equations and symplectic groupoids on Poisson manifolds. Indiana U Math J1990; 39(3): 859–876.
18.
FerraroSde LeónMMarreroJC, et al. On the geometry of the Hamilton–Jacobi equation and generating functions. Arch Ration Mech Anal2017; 226(1): 243–302.
19.
CosseratO. Theory and construction of structure preserving integrators in Poisson geometry. PhD Thesis, La Rochelle University, La Rochelle, 2023. https://hal.science/tel-04279466/
20.
CrainicMMărcutI. On the existence of symplectic realizations. J Symplect Geom2011; 9(4): 435–444.
21.
WeinsteinA. Symplectic manifolds and their Lagrangian submanifolds. Adv Math1971; 6(3): 329–346.
BenettinGGiorgilliA. On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys1994; 74(5–6): 1117–1143.
24.
CosseratOFalaizeASalnikovV. On automatic generation of Poisson numerical methods of higher order, in preparation.
