Many Body Symmetrical Dynamical Systems

On the Stability of Lagrange Relative Equilibrium in The Planar Three-body Problem

Since the strong degeneracies present in the N-body problem, even in the basic case of the planar three-body problem, nobody inspects the problem of nonlinear stability of Lagrange relative equilibrium. We introduce a new coordinate system to reduce degeneracies according to intrinsic symmetrical characteristic of the N-body problem, then we prove that Lagrange relative equilibrium is stable in the sense of measure, provided it is spectrally stable and except six special resonant cases. Indeed, under this condition, there are abundant KAM invariant tori or quasi-periodic solutions near Lagrang…

Affinely-rigid body and oscillatory dynamical models on GL(2,R)

Discussed is a model of the two-dimensional affinely-rigid body with the double dynamical isotropy. We investigate the systems with potential energies for which the variables can be separated. The special stress is laid on the model of the harmonic oscillator potential and certain anharmonic alternatives. Some explicit solutions are found on the classical, quasiclassical (Bohr-Sommerfeld) and quantum level.

New Concept for Studying the Classical and Quantum Three-Body Problem: Fundamental Irreversibility and Time's Arrow of Dynamical Systems

The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a local coordinate system allows us to detect new hidden symmetries of the internal motion of a dynamical system, which allows us to reduce the three-body problem to the 6\emph{th} order system. A new approach makes the system of geodesic equations with respect to the evolution parameter of a dynamical system (\emph{internal time}) \emph{fundamentally irreversibl…

The Universe from a Single Particle

We explore the emergence of many-body physics from quantum mechanics via spontaneous symmetry breaking. To this end, we study potentials which are functionals on the space of Hamiltonians enjoying an unstable critical point corresponding to a random quantum mechanical system (the Gaussian unitary ensemble), but also less symmetrical local minima corresponding to interacting systems at the level of operators.

Fundamental irreversibility of the classical three-body problem. New approaches and ideas in the study of dynamical systems

The three-body general problem is formulated as a problem of geodesic trajectories flows on the Riemannian manifold. It is proved that a curved space with local coordinate system allows to detect new hidden symmetries of the internal motion of a dynamical system and reduce the three-body problem to the system of 6\emph{th} order. It is shown that the equivalence of the initial Newtonian three-body problem and the developed representation provides coordinate transformations in combination with the underdetermined system of algebraic equations. The latter makes a system of geodesic equations rel…

An Explicit Bound for Dynamical Localisation in an Interacting Many-Body System

We characterise and study dynamical localisation of a finite interacting quantum many-body system. We present explicit bounds on the disorder strength required for the onset of localisation of the dynamics of arbitrary ensemble of sites of the XYZ spin-1/2 model. We obtain these results using a novel form of the fractional moment criterion, which we establish, together with a generalisation of the self-avoiding walk representation of the system Green's functions, called path-sums. These techniques are not specific to the XYZ model and hold in a much more general setting. We further present bou…

A dynamical systems approach for the shape matching of polytopes along rigid-body motions

We present a dynamical systems approach for geometric matchings in an ensemble of polytopes along rigid-body motions. Each polytope can be characterized by a vertex set and edge or faces determined by vertices, and polygons and simplexes correspond to a polytope. For a geometric matching, we propose a system of dynamical system for the evolution of centroids and rotations of polytopes to match the vertices under rigid-body motions which can be decomposed as a composition of translation and rotations. Our proposed dynamical system acts on the product space $({\mathbb R}^d \times SO(d))^N$. The …

Many-body systems with SU(1,1) dynamical symmetry: from dynamics to thermodynamics based on the trace formula

For a quantum (many-body) system with dynamical symmetry described by a given Lie group, we study the trace of exponential operators with complex coefficients in one of the irreducible subspaces in terms of the boson realization. By using this approach, for compact groups, we obtain the result of the trace that is consistent with the well-known Weyl character formula. For non-compact groups (with SU(1,1) as an application), convergent condition of the trace is also obtained. This approach may be a powerful tool to study the thermodynamics of quantum (many-body) systems in equilibrium state or …

Algebraic-Dynamical Theory for Quantum Many-body Hamiltonians: A Formalized Approach To Strongly Interacting Systems

Non-commutative algebras and entanglement are two of the most important hallmarks of many-body quantum systems. Dynamical perturbation methods are the most widely used approaches for quantum many-body systems. While study of entanglement-based numerical methods are booming recently, the traditional dynamical perturbation methods have not benefited from study of quantum entanglement. In this work, we formulate an algebraic-dynamical theory (ADT) by combining the power of quantum algebras and dynamical methods in which quantum entanglement naturally emerges as the organizing principle. We start …

平面上无滑动滚动凸体非完整问题的仿射推广

Affine generalizations of the nonholonomic problem of a convex body rolling without slipping on the plane

本研究提出一类仿射推广的凸体平面无滑动滚动非完整问题示例，分析其首次积分、光滑不变测度与可积性等动力学特性，重点考察动平衡球体和回转体的情形。

On reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval

The enclosure method was originally introduced for inverse problems of concerning non-destructive evaluation governed by elliptic equations. It was developed as one of useful approaches in inverse problems and applied for various equations. In this paper, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficient is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is …

Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Non-Relativistic N-Body Problem

After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a {\it spin frame} for the N-body system. Then, the existence of appropriate non-sy…