On Computational Poisson Geometry II: Numerical Methods

Abstract

We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1-forms, gauge transformations, and normal forms of Lie-Poisson structures on ${\mathbf{R}}^3$. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

References in corpus (28)

• Deformation quantization of Poisson manifolds, I
• A Variational Perspective on Accelerated Methods in Optimization
• Hamiltonian Neural Networks
• Learning to Simulate Complex Physics with Graph Networks
• Discovering Symbolic Models from Deep Learning with Inductive Biases
• Sampling Can Be Faster Than Optimization
• Predicting the long-term stability of compact multiplanet systems
• Newton vs the machine: solving the chaotic three-body problem using deep neural networks
• Hamiltonian Graph Networks with ODE Integrators
• Normalizing Flows on Tori and Spheres
• Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
• Equivariant Hamiltonian Flows
• Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems
• Brackets
• Neural Canonical Transformation with Symplectic Flows
• A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator
• Poisson brackets with prescribed Casimirs
• Computer-assisted estimates for Birkhoff normal forms
• Linear Poisson structures on R^4
• Poisson structures on smooth 4-manifolds
• Plasma in monopole background is not twisted Poisson
• On Bott-Morse Foliations and their Poisson Structures in Dimension 3
• Poisson structures on wrinkled fibrations
• Hamiltonian systems: symbolical, numerical and graphical study
• On Computational Poisson Geometry I: Symbolic Foundations
• Classification of the classical $SL(2,\mathbb{R})$ gauge transformations in the rigid body
• The Poisson cohomology of ${\mathfrak{sl}}_2^{\ast }(\mathbb{R})$
• Unimodularity Criteria for Poisson Structures on Foliated Manifolds

Cited by in corpus (2)

• On Computational Poisson Geometry I: Symbolic Foundations
• Areas on the space of smooth probability density functions on $S^2$