Generalizing Poisson Algebra with Geometry

2021-9-6

JPS Hot Topics 1, 052

https://doi.org/10.7566/JPSHT.1.052

© The Physical Society of Japan

Generalization of Hamiltonian mechanics to a three-dimensional phase space
(PTEP Editors' Choice)

Naoki Sato
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)

Naoki Sato
Prog. Theor. Exp. Phys. 2021, 063A01 (2021).