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).

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.

scholarly journals Entropy and irreversibility in dynamical systems

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120349 ◽  
Author(s):  
Oliver Penrose
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic System ◽  
Degrees Of Freedom ◽  
Usual Method ◽  
Chaotic Dynamical System ◽  
Two Degrees Of Freedom ◽  
Arnold’S Cat Map ◽  
Non Equilibrium

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann’s principle , does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold’s ‘cat’ map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

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 Centre manifolds of forced dynamical systems

1991 ◽  
Vol 32 (4) ◽  
pp. 401-436 ◽  
Author(s):  
S. M. Cox ◽  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Rational Approach ◽  
Centre Manifold ◽  
Formal Scheme ◽  
Dimensional Models ◽  
Centre Manifolds ◽  
Low Dimensional ◽  
Sivashinsky Equation

AbstractCentre manifolds arise in a rational approach to the problem of forming low-dimensional models of dynamical systems with many degrees of freedom. When a dynamical system with a centre manifold is subject to a small forcing, F, there are two effects: to displace the centre manifold; and to alter the evolution thereon. We propose a formal scheme for calculating the centre manifold of such a forced dynamical system. Our formalism permits the calculation of these effects, with errors of order |F|2. We find that the displacement of the manifold allows a reparameterisation of its description, and we describe two “natural” ways in which this can be carried out. We give three examples: an introductory example; a five-mode model of the atmosphere to display the quasi-geostrophic approximation; and the forced Kuramoto-Sivashinsky equation.

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 On the Number of Isolating Integrals in Systems with Three Degrees of Freedom

1971 ◽  
Vol 10 ◽  
pp. 110-117
Author(s):  
Claude Froeschle
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Weak Coupling ◽  
Degrees Of Freedom ◽  
Model Problem ◽  
Energy Integral ◽  
Dimensional Mapping ◽  
Three Degrees Of Freedom

AbstractDynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations: We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).

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