The intrinsic properties of rank and nullity of the Lagrange bracket in the one dimensional calculus of variations

1975 ◽  
Vol 279 (1290) ◽  
pp. 487-545 ◽  
Keyword(s):  
Differential Form ◽  
Equations Of Motion ◽  
Continuous System ◽  
Variational Problems ◽  
Intrinsic Properties ◽  
Degenerate Problem ◽  
One Dimensional ◽  
Full Development ◽  
The One ◽  
The Second Variation

This paper establishes the existence of symplectic structure in degenerate variational problems, i.e. problems whose full development involves a hierarchy of equations of constraint as well as various equations of motion. Any variational problem, degenerate or otherwise, may be called regular if the equations of the second variation provide a complete description of the infinitesimal relationships subsisting between any orbit and all its infinitesimal neighbour orbits. It is proved that Poincare’s conserved antisymmetric derived bilinear differential form in the orbit manifold of any regular degenerate problem admits no null vectors other than those which represent infinitesimal deviations due to indeterminacy in the evolution of the orbit. Conversely, it is shown how, given any continuous system of orbits endowed with a conserved antisymmetric closed bilinear differential form having this unique property of rank and nullity, one can construct at least one regular variational

Integration of one-dimensional equations of motion

2020 ◽  
pp. 3-5
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Potential Energy ◽  
Equations Of Motion ◽  
Field Change ◽  
One Dimensional ◽  
System A ◽  
Small Correction ◽  
The Law ◽  
The One ◽  
The Given

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

scholarly journals Acquiring the Symplectic Operator Based on Pure Mathematical Derivation then Verifying It in the Intrinsic Problem of Nanodevices

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1383
Author(s):  
Nie ◽  
Gui ◽  
Chen
Keyword(s):  
Finite Difference Time Domain ◽  
Intrinsic Properties ◽  
Hamilton Equation ◽  
Dispersion Error ◽  
One Dimensional ◽  
Particle Wave Function ◽  
The One ◽  
Mathematical Derivation ◽  
Difference Time ◽  
Total Energy Distribution

The symplectic algorithm can maintain the symplectic structure and intrinsic properties of the system, its cumulative error is small and suitable for multi-step calculation. At present, the widely accepted symplectic operators are obtained by solving the Hamilton equation based on artificial definitions and assumptions in advance. There are inevitable dispersion errors. We solve the equation by pure mathematical derivation without any artificial limitations and assumptions. The way to accurately obtain high-precision symplectic operators greatly reduces the dispersion error from the beginning. The numerical solution of the one-dimensional Schrödinger equation for describing the intrinsic problem of nanodevices is used as an application environment to compare the total energy distribution of the particle wave function in the box, thus verifying the properties of the Symplectic Operator based on Pure Mathematical Derivation by comparing with Finite-Difference Time-Domain (FDTD) and the widely accepted symplectic operator.

One-dimensional dislocations IV. Dynamics

1950 ◽  
Vol 201 (1065) ◽  
pp. 261-268 ◽  
Keyword(s):  
Wave Velocity ◽  
Equations Of Motion ◽  
Dislocation Model ◽  
Characteristic Wave ◽  
One Dimensional ◽  
Regular Sequences ◽  
All Solutions ◽  
Stable Equilibria ◽  
The One

Tho equations of motion of the one-dimensional dislocation model are studied, and all solutions representing disturbances propagated without change of form are obtained. These comprise: (i) dislocations, and regular sequences of the same, either of like or of alternating sign, travelling with velocity less than c , the characteristic wave velocity of the system; (ii) an iddislocations, and sequences of the spme, travelling with velocity greater than c; and (iii) waves of infinitesimal amplitude, belonging to two branches travelling respectively with velocities less than and greater than c . Only dislocations or sequences of dislocations of like sign, and waves of velocity less than c , correspond to stable equilibria. The dislocations exhibit ‘relativistic’ behaviour. The relevance of anti-dislocations to very fast slip in solids is considered, and rejected.

CRITICAL DYNAMICS OF THE XY-MODEL ON THE ONE-DIMENSIONAL SUPERLATTICE BY POSITION SPACE RENORMALIZATION GROUP

1996 ◽  
Vol 10 (22) ◽  
pp. 1077-1083 ◽  
Author(s):  
J.P. DE LIMA ◽  
L.L. GONÇALVES
Keyword(s):  
Renormalization Group ◽  
Equations Of Motion ◽  
Xy Model ◽  
Dynamic Scaling ◽  
Critical Dynamics ◽  
Renormalization Group Theory ◽  
Position Space ◽  
One Dimensional ◽  
Homogeneous Lattice ◽  
The One

The critical dynamics of the isotropic XY-model on the one-dimensional superlattice is considered in the framework of the position space renormalization group theory. The decimation transformation is introduced by considering the equations of motion of the operators associated to the excitations of the system, and it corresponds to an extension of the procedure introduced by Stinchcombe and dos Santos (J. Phys. A18, L597 (1985)) for the homogeneous lattice. The dispersion relation is obtained exactly and the static and dynamic scaling forms are explicitly determined. The dynamic critical exponent is also obtained and it is shown that it is identical to the one of the XY-model on the homogeneous chain.

scholarly journals The One-dimensional Coulomb-force Problem in Classical Radiation Reaction Theories

1977 ◽  
Vol 32 (7) ◽  
pp. 685-691
Author(s):  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Coulomb Potential ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Radiation Reaction ◽  
Coulomb Force ◽  
Stationary States ◽  
One Dimensional ◽  
The One ◽  
Force Problem ◽  
Classical Radiation

Abstract Numerical solutions of the recently proposed equations of motion for the classically radiating electron are obtained for the case where the particle moves in a one-dimensional Coulomb potential (both attractive and repulsive). The solutions are discussed and found to be meaningful also in that case, where the well-known Lorentz-Dirac equation fails (attractive Coulomb force). Discrete, stationary states are found in a non-singular version of the Coulomb potential. During the transition between those stationary states the particle looses energy by emission of radiation, which results in a smaller amplitude of the stationary oscillations.

An Application of Mixture Theory to Particulate Sedimentation

1980 ◽  
Vol 47 (2) ◽  
pp. 261-265 ◽  
Author(s):  
C. D. Hill ◽  
A. Bedford ◽  
D. S. Drumheller
Keyword(s):  
Particle Concentration ◽  
Equations Of Motion ◽  
Mixture Theory ◽  
Solid Particles ◽  
Two Phase ◽  
Small Perturbations ◽  
One Dimensional ◽  
Blood Sedimentation ◽  
The One ◽  
Extended Variational Principle

Equations for two-phase flow are used to analyze the one-dimensional sedimentation of solid particles in a stationary container of liquid. A derivation of the equations of motion is presented which is based upon Hamilton’s extended variational principle. The resulting equations contain diffusivity terms, which are linear in the gradient of the particle concentration. It is shown that the solution of the equations for steady sedimentation is stable under small perturbations. Finally, finite-difference solutions of the equations are compared to the data of Whelan, Huang, and Copley for blood sedimentation.

Green's function approach to the Ising model

1969 ◽  
Vol 47 (7) ◽  
pp. 769-777 ◽  
Author(s):  
K. C. Lee ◽  
Robert Barrie
Keyword(s):  
Ising Model ◽  
Green’S Function ◽  
Green's Function ◽  
Body Problem ◽  
Equations Of Motion ◽  
Function Technique ◽  
Many Body ◽  
One Dimensional ◽  
Spinless Fermion ◽  
The One

It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.

A separated-flow model for collapsible-tube oscillations

1985 ◽  
Vol 157 ◽  
pp. 375-404 ◽  
Author(s):  
Claudio Cancelli ◽  
T. J. Pedley
Keyword(s):  
Equations Of Motion ◽  
Separated Flow ◽  
Propagation Speed ◽  
Adverse Pressure Gradient ◽  
Collapsible Tube ◽  
State Collapse ◽  
One Dimensional ◽  
Longitudinal Wall ◽  
Wave Propagation Speed ◽  
The One

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

scholarly journals Violent Relaxation and Mixing in 1-D Gravitational Systems

1987 ◽  
Vol 127 ◽  
pp. 523-524
Author(s):  
Marc Luwel
Keyword(s):  
Surface Density ◽  
Gravitational Force ◽  
Equations Of Motion ◽  
Dimensional System ◽  
Double Precision ◽  
One Dimensional ◽  
Gravitational Systems ◽  
Gravitational Model ◽  
The One

The one dimensional gravitational model consists of N mass sheets with surface density mi, parallel to the (y, z)–plane and constrained to move along the x-axis under influence of their mutual gravitational force Fij = −2πGmimj sgn(xi – xj). in order to study the evolution of this one–dimensional system, the N Newtonian equations of motion are integrated numerically, using an “exact” double precision algorithm.

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