Everything about Poisson bracket
Operation in Hamiltonian mechanics
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Poisson bracketA 5-minute orientation
- Skim a recent paper: Nonlinear Feedback, Double-bracket Dissipation and Port Control of Lie–Poisson SystemsInfluential work since 2022
- Pick up a classic book: Lectures on Poisson GeometryThe most-reprinted book on the subject
- Read the landmark paper: Poisson manifolds and the schouten bracketCited 63 times
- Go sideways: Hamiltonian mechanicsA closely related topic
- Go sideways: Hamiltonian vector fieldA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🏆 Landmark papers (most cited)
Poisson manifolds and the schouten bracket
THE POISSON BRACKET FOR POISSON FORMS IN MULTISYMPLECTIC FIELD THEORY
Poisson bracket formulation of incompressible flow equations in continuum mechanics
The Schouten - Nijenhuis bracket, cohomology and generalized Poisson structures
Quasi-morphisms and the Poisson Bracket
Gyrokinetic energy conservation and Poisson-bracket formulation
Truncated Nambu-Poisson bracket and entropy formula for multiple membranes
Liouville field theory: IST and Poisson bracket structure
🆕 Influential since 2022
Nonlinear Feedback, Double-bracket Dissipation and Port Control of Lie–Poisson Systems
Two‐dimensional electronic spectrum simulation of simple photosynthetic complex models with semi‐classical Poisson bracket mapping equation
The Poisson bracket invariant on surfaces
Canonical form of a deformed Poisson bracket spacetime
The Poisson Bracket Invariant for Open Covers Consisting of Topological Disks on Surfaces
📚 Books worth reading
Lectures on Poisson GeometryMarius Crainic, Rui Loja Fernandes · 2021- 📕Poisson geometryJ. Grabowski, Paweł Urbański · 2000
Nonlinear Poisson bracketsM. V. Karasev · 1993
Quantization, Poisson brackets and beyondTheodore Voronov · 2002
Thermodynamics of flowing systemsAntony N. Beris · 1994
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian…
- Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named…
- Poisson manifold In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold…
- Symplectic manifold In differential geometry, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic…
- Lagrange bracket Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange from 1808 to 1810 for the purposes…
- Canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations…
- Liouville–Arnold theorem The Liouville–Arnold theorem is a result in classical mechanics which says, roughly speaking, that seemingly complicated systems can be described as combination…
- Symplectic integrator In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric…