scholarly journals The Existence of Integrals of Dynamical Systems Linear in the Velocities

1961 ◽  
Vol 44 ◽  
pp. 13-16 ◽  
Author(s):  
C. W. Kilmister
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Quadratic Polynomial ◽  
Summation Convention ◽  
Image Position ◽  
Generalised Coordinates

A dynamical system means here a system specified by generalised coordinates qα(α = 1, 2, …, n) and a Lagrangian L which is a quadratic polynomial in the generalised velocities, say(with a summation convention).

scholarly journals Note on the hodograph of non-holonomic dynamical systems

1940 ◽  
Vol 6 (3) ◽  
pp. 176-180
Author(s):  
B. Spain
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Kinetic Energy ◽  
Image Position

Consider a non-holonomic dynamical system specified by the N coordinates qi, and kinetic energy defined bywhere amn are functions of qi and the dot denotes differentiation with respect to the time t.

scholarly journals On Kilmister's Conditions for the Existence of Linear Integrals of Dynamical Systems

1965 ◽  
Vol 14 (3) ◽  
pp. 243-244 ◽  
Author(s):  
R. H. Boyer
Keyword(s):  
Dynamical Systems ◽  
Singular Integral ◽  
First Integral ◽  
Killing Field ◽  
Summation Convention ◽  
Covariant Derivation ◽  
Image Position ◽  
Generally Covariant ◽  
Linear Integrals

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.

scholarly journals Integrals of dynamical systems linear in the velocities

1971 ◽  
Vol 17 (3) ◽  
pp. 241-244
Author(s):  
C. D. Collinson
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Integral Conditions ◽  
Two Degrees Of Freedom ◽  
Generalized Coordinates ◽  
Image Position ◽  
Linear Integrals

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangianrepeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).

Smooth derivations on abelian C*-dynamical systems

1987 ◽  
Vol 42 (2) ◽  
pp. 247-264 ◽  
Author(s):  
Derek W. Robinson
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Parameter Group ◽  
Group Of Automorphisms ◽  
Lipschitz Algebra ◽  
Image Position

AbstractLet (A, R, σ) be an abelian C*-dynamical system. Denote the generator of σ by δ0 and define A∞ = ∩n>1D (δ0n). Further define the Lipschitz algebra .If δ is a *-derivation from A∞ into A½, then it follows that δ is closable, and its closure generates a strongly continuous one-parameter group of *-automorphisms of A. Related results for local dissipations are also discussed.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

scholarly journals The Dynamics of Musical Participation

2021 ◽  
pp. 102986492098831
Author(s):  
Andrea Schiavio ◽  
Pieter-Jan Maes ◽  
Dylan van der Schyff
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Dynamical Systems Theory ◽  
Coordination Dynamics ◽  
Musical Experience ◽  
The Core ◽  
Interactive Dynamics ◽  
Interdisciplinary Scholarship ◽  
Core Features

In this paper we argue that our comprehension of musical participation—the complex network of interactive dynamics involved in collaborative musical experience—can benefit from an analysis inspired by the existing frameworks of dynamical systems theory and coordination dynamics. These approaches can offer novel theoretical tools to help music researchers describe a number of central aspects of joint musical experience in greater detail, such as prediction, adaptivity, social cohesion, reciprocity, and reward. While most musicians involved in collective forms of musicking already have some familiarity with these terms and their associated experiences, we currently lack an analytical vocabulary to approach them in a more targeted way. To fill this gap, we adopt insights from these frameworks to suggest that musical participation may be advantageously characterized as an open, non-equilibrium, dynamical system. In particular, we suggest that research informed by dynamical systems theory might stimulate new interdisciplinary scholarship at the crossroads of musicology, psychology, philosophy, and cognitive (neuro)science, pointing toward new understandings of the core features of musical participation.

Induced transformations of recurrent a.m.s. dynamical systems

2015 ◽  
Vol 15 (02) ◽  
pp. 1550010
Author(s):  
Sheng Huang ◽  
Mikael Skoglund
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Process ◽  
Ergodic Theorem ◽  
Finite Measure ◽  
Stationary Random Process ◽  
Finite State ◽  
Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

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