在1954年，科学界庆祝了Poincare的诞辰100周年纪念日，在那个时候，Poincare在数学家们当中的名声并没有达到最高峰，而希尔伯特的精神则主导了很多数学家的思想，Poincare在物理学家们当中的名声也没有达到最高峰，因为在那时候，物理学在本质上是和量子理论相关的。

Henri Poincare，为科学服务的一生

Dynamical systems described by equations of motion with the first-order time derivative (dissipative) terms of even and odd powers, and coefficients varying either in time or in space, are considered. Methods to obtain standard and non-standard Lagrangians are presented and used to identify classes of equations of motion that admit a Lagrangian description. It is shown that there are two general classes of equations that have standard Lagrangians and one special class of equations that can only be derived from non-standard Lagrangians. In addition, each general class has a subset of equations with non-standard Lagrangians. Conditions required for the existence of standard and non-standard Lagrangians are derived and a relationship between these two types of Lagrangians is introduced. By obtaining Lagrangians for several dynamical systems and some basic equations of mathematical physics, it is demonstrated that the presented methods can be applied to a broad range of physical problems.

https://iopscience.iop.org/article/10.1088/1751-8113/41/5/055205/pdf

Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.

Topics in Noncommutative Geometry Inspired Physics

Equations of motion describing dissipative dynamical systems with coefficients varying either in time or in space are considered. To identify the equations that admit a Lagrangian description, two classes of non-standard Lagrangians are introduced and general conditions required for the existence of these Lagrangians are determined. The conditions are used to obtain some non-standard Lagrangians and derive equations of motion resulting from these Lagrangians.

General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems

Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.

Introduction of chaotic dynamical systems

The model of the position-dependent noncommutativity in quantum mechanics is proposed. We start with given commutation relations between the operators of coordinates $[{\stackrel{^}{x}}^{i},{\stackrel{^}{x}}^{j}]={\ensuremath{\omega}}^{ij}(\stackrel{^}{x})$, and construct the complete algebra of commutation relations, including the operators of momenta. The constructed algebra is a deformation of a standard Heisenberg algebra and obeys the Jacobi identity. The key point of our construction is a proposed first-order Lagrangian, which after quantization reproduces the desired commutation relations. Also we study the possibility to localize the noncommutativity.

Position-dependent noncommutativity in quantum mechanics

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.

/pdf/bootstrapping-non-commutative-gauge-theories-from-l-infty-1q7hfg84hk.pdf

Bootstrapping Non-commutative Gauge Theories from L$_\infty$ algebras

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which is based on Weyl symmetrically ordered operator products. By using a polydifferential representation for the deformed coordinates, $\hat{x}^{j}$ , we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth-order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.

Star products made (somewhat) easier

We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations , where θ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as ωij(x) = eijkxkf(xixi). In this case, the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The corresponding nonlocality is calculated: , where m is a magnetic quantum number.

A hydrogen atom on curved noncommutative space

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. By using a polydifferential representation for deformed coordinates $\hat x^j$ we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.

Vladislav G. Kupriyanov

Papers

Position-dependent noncommutativity in quantum mechanics

Bootstrapping Non-commutative Gauge Theories from L$_\infty$ algebras

Star products made (somewhat) easier

A hydrogen atom on curved noncommutative space

Star products made (somewhat) easier