Generalizing Poisson Algebra with Geometry
© The Physical Society of Japan
This article is on
Generalization of Hamiltonian mechanics to a three-dimensional phase space
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2021, 063A01 (2021).
Using a differential geometric interpretation of Hamiltonian mechanics, a generalized Poisson bracket formulation is developed for a three-dimensional phase space characterized by a triplet of canonical variables.
The Hamiltonian formulation of classical mechanics is an elegant formalism. It is characterized by the Hamiltonian function, which represents the total energy of the system, and a Poisson bracket acting on it to give the Hamilton’s equation of motion. From a geometrical perspective, these equations can be interpreted as the flow of a vector field in phase space, or the space of all possible states of the physical system, such that it always conserves the phase space volume, representing the conservation of energy.
Typically, the phase space is “two-dimensional,” in the sense that it is characterized by the positions and momenta only. But, can the formalism be generalized for a higher dimensional phase space? The physicist Yoichiro Nambu proposed such a formalism for a three-dimensional phase space with “Nambu brackets” replacing the Poisson bracket. However, constructing an algebraic framework analogous to the Poisson bracket proved to be difficult because the Jacobi identity, representing the closure property of Poisson bracket, could not be generalized for Nambu brackets.
In a recent research article, Prof. Naoki Sato from the University of Tokyo managed to evade this problem. Starting from a differential geometric approach, he constructed a framework for a three-dimensional phase space with generalized Poisson brackets characterized by anti-symmetric, third-order, contravariant tensors. His approach not only led to a generalized Jacobi identity that conserved the phase space volume, but also showed that it was a weaker condition than that represented by the substitute for Jacobi identity for Nambu brackets.
Such a generalization could have profound implications for theoretical physics, providing new ways of understanding the laws of physics and possibly leading to real-world applications.
Generalization of Hamiltonian mechanics to a three-dimensional phase space
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2021, 063A01 (2021).
Share this topic
Fields
Related Articles
-
A New Method for Finding Bound States in the Continuum
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
Nuclear physics
2024-10-1
This study presents a general theory for constructing potentials supporting bound states in the continuum, offering a method for identifying such states in real quantum systems.
-
General Quasi-Joint Probabilities on Finite-State Quantum Systems
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
2024-8-15
This study investigates the properties of general quasi-joint probability distributions in finite-state quantum systems, revealing the Kirkwood-Dirac distribution as among the most favorable. This highlights the importance of complex distributions in understanding quantum probability.
-
Solving a Stochastic Differential Equation is Solving a Mean-Field Quantum Spin System
Statistical physics and thermodynamics
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
Magnetic properties in condensed matter
2024-5-16
The replica method maps matrix-valued geometric Brownian motion to a mean-field quantum spin system. This correspondence makes it possible to obtain an exact solution for matrix-valued geometric Brownian motion.
-
Quantum Mechanics of One-Dimensional Three-Body Contact Interactions
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
Theoretical Particle Physics
2024-2-13
The quantum mechanical description of topologically nontrivial three-body contact interactions in one dimension is not well understood. This study explores the Hamiltonian description of these interactions using the path-integral formalism.
-
Exploring Recent Advances in the Physics of Biofluid Locomotion
Measurement, instrumentation, and techniques
Cross-disciplinary physics and related areas of science and technology
Electromagnetism, optics, acoustics, heat transfer, and classical and fluid mechanics
Statistical physics and thermodynamics
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
Structure and mechanical and thermal properties in condensed matter
2023-12-8
This Special Topics Edition of the JPSJ describes the latest advances in the field of biofluid locomotion, shedding light on the underlying physics behind the movement of organisms that swim and fly.