scholarly journals Splitting and composition methods for explicit time dependence in separable dynamical systems

2010 ◽  
Vol 235 (3) ◽  
pp. 646-659 ◽  
Author(s):  
Sergio Blanes ◽  
Fasma Diele ◽  
Carmela Marangi ◽  
Stefania Ragni
Keyword(s):  
Dynamical Systems ◽  
Time Dependence ◽  
Composition Methods ◽  
Explicit Time Dependence

scholarly journals New Results on the Coagulation Equation

1987 ◽  
Vol 115 ◽  
pp. 547-547
Author(s):  
Pierre Bastien ◽  
Claude Lejeune
Keyword(s):  
Mass Spectrum ◽  
Time Dependence ◽  
Star Clusters ◽  
Mass Function ◽  
Mass Dependence ◽  
Coagulation Rate ◽  
Precise Form ◽  
Explicit Time Dependence ◽  
Coagulation Equation ◽  
Free Parameters

In attempting to reproduce the initial stellar mass function, we solved analytically the coagulation equation with an explicit time dependence in the coagulation rate in order to simulate the gravitational collapse of the fragments upon themselves as they move within the progenitor cloud. Two separate cases have been studied, with and without a mass dependence in the coagulation rate. The solution show that (1) inclusion of self-gravitation is very important and can change the results to the point of preventing coalescence to work altogether, depending on the values of the two free parameters, (2) the precise form of the mass dependence of the coagulation rate is not of prime importance in most situations of astrophysical interest, and (3) coagulation alone is not sufficient to yield a realistic mass spectrum and fragmentation must also be taken into account. Coagulation is more efficient for massive fragments and fragmentation for the smaller ones. These results are applied to different regions: star clusters, associations, and starburst regions.

The Jacobi Energy Function

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Conservation Laws ◽  
Total Energy ◽  
Time Dependence ◽  
Energy Function ◽  
Tangent Bundle ◽  
Implicit Time ◽  
Translational Invariance ◽  
Explicit Time Dependence ◽  
Generalised Coordinates ◽  
Lagrange Formalism

This chapter focuses on the Jacobi energy function, considering how the Lagrange formalism treats the energy of the system. This discussion leads nicely to conservation laws and symmetries, which are the focus of the next chapter. The Jacobi energy function associated with a Lagrangian is defined as a function on the tangent bundle. The chapter also discuss explicit vs implicit time dependence, and shows how time translational invariance ensures the generalised coordinates are inertial, meaning that the energy function is the total energy of the system. In addition, it examines the energy function using non-inertial coordinates and explicit time dependence.

scholarly journals RESTORING TIME DEPENDENCE INTO QUANTUM COSMOLOGY

2012 ◽  
Vol 21 (11) ◽  
pp. 1242011 ◽  
Author(s):  
AHARON DAVIDSON ◽  
BEN YELLIN
Keyword(s):  
Quantum Theory ◽  
Time Dependence ◽  
Quantum Cosmology ◽  
Scale Factor ◽  
Time Dependent ◽  
Hamiltonian Constraint ◽  
Canonical Variables ◽  
Explicit Time Dependence ◽  
Dirac Brackets ◽  
Lapse Function

Mini superspace cosmology treats the scale factor a(t), the lapse function n(t) and an optional dilation field ϕ(t) as canonical variables. While pre-fixing n(t) means losing the Hamiltonian constraint, pre-fixing a(t) is serendipitously harmless at this level. This suggests an alternative to the Hartle–Hawking approach, where the pre-fixed a(t) and its derivatives are treated as explicit functions of time, leaving n(t) and a now mandatory ϕ(t) to serve as canonical variables. The naive gauge pre-fix a(t) = const . is clearly forbidden, causing evolution to freeze altogether; so pre-fixing the scale factor, say a(t) = t, necessarily introduces explicit time dependence into the Lagrangian. Invoking Dirac's prescription for dealing with constraints, we construct the corresponding mini superspace time-dependent total Hamiltonian and calculate the Dirac brackets, characterized by {n, ϕ}D ≠ 0, which are promoted to commutation relations in the quantum theory.

scholarly journals Symplectic Integrators to Stochastic Hamiltonian Dynamical Systems Derived from Composition Methods

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Tetsuya Misawa
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Operator Splitting ◽  
Symplectic Integrators ◽  
Hamiltonian Vector Fields ◽  
Operator Splitting Methods ◽  
Composition Methods ◽  
Symplectic Map ◽  
Hamiltonian Dynamical Systems ◽  
Symplectic Schemes

“Symplectic” schemes for stochastic Hamiltonian dynamical systems are formulated through “composition methods (or operator splitting methods)” proposed by Misawa (2001). In the proposed methods, a symplectic map, which is given by the solution of a stochastic Hamiltonian system, is approximated by composition of the stochastic flows derived from simpler Hamiltonian vector fields. The global error orders of the numerical schemes derived from the stochastic composition methods are provided. To examine the superiority of the new schemes, some illustrative numerical simulations on the basis of the proposed schemes are carried out for a stochastic harmonic oscillator system.

A Conservation Theorem for Simple Nonholonomic Systems

1983 ◽  
Vol 50 (3) ◽  
pp. 647-651 ◽  
Author(s):  
T. R. Kane ◽  
A. K. Banerjee
Keyword(s):  
Time Dependence ◽  
Nonholonomic System ◽  
Nonholonomic Systems ◽  
Holonomic System ◽  
Explicit Time Dependence ◽  
Conservation Theorem

When the Hamiltonian of a holonomic system is free of explicit time dependence it remains constant throughout all motions of the system. In this paper, it is shown how, given a homogeneous simple nonholonomic system S, one can form a function E that remains constant throughout all motions of S, providing the forces acting on S fulfill certain requirements. An illustrative example is examined in detail.

A UNIFIED DESCRIPTION OF TIME DEPENDENCE OF INFORMATION ENTROPY PRODUCTION AND FLUX IN THERMAL BROADBAND NOISE-DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 23 (18) ◽  
pp. 3789-3802 ◽  
Author(s):  
PRADIP MAJEE ◽  
GURUPADA GOSWAMI ◽  
DEBASHIS BARIK ◽  
BIDHAN CHANDRA BAG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Time Dependence ◽  
Entropy Production ◽  
Information Entropy ◽  
Closed System ◽  
Broadband Noise ◽  
Stationary States ◽  
Entropy Flux ◽  
Unified Description

In this paper we have studied the dynamics of thermal broadband noise-driven dynamical system in terms of information entropy at both the nonstationary and stationary states. Here, a unified description of fluctuating force is considered in a thermodynamically closed system. Based on the Fokker–Planck description of stochastic processes and the entropy balance equation, we have calculated the time-dependence of the information entropy production and entropy flux in the presence and absence of nonequilibrium constraint. Our calculation considers how the time evolution of these quantities is affected if the characteristic of noise changes from white to red or green and red to green in a unified scheme.

scholarly journals Time-dependent metric for the two-dimensional, non-Hermitian coupled oscillator

2019 ◽  
Vol 35 (08) ◽  
pp. 2050041 ◽  
Author(s):  
Andreas Fring ◽  
Thomas Frith
Keyword(s):  
Harmonic Oscillator ◽  
Time Dependence ◽  
Time Dependent ◽  
Two Dimensional ◽  
Coupling Term ◽  
Coupled Oscillator ◽  
Explicit Time Dependence ◽  
Independent Model

We provide a time-dependent Dyson map and metric for the two-dimensional harmonic oscillator with a non-Hermitian ixy coupling term. This particular time-independent model exhibits spontaneously broken [Formula: see text]-symmetry and becomes unphysical in the broken regime, with the spectrum becoming partially complex. By introducing an explicit time dependence into the Dyson map, we provide a time-dependent metric that renders the model consistent across the unbroken and broken regimes.

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