Principle Of Least Constraint Research Articles

Viscous attractor for the Galton board.

We analyze the Galton Board [or periodic "Lorentz Gas"] with a point mass scattered by elastic disks of diameter sigma, using a constant driving field g and a constant-viscosity linear drag force -p/tau, where p is the point-mass momentum. This combination leads to a nonequilibrium steady state which depends only upon the dimensionless ratio gtau(2)/sigma. The long-time-averaged trajectory leads to multifractal phase-space structures closely resembling those we found earlier using isokinetic equations of motion derived from Gauss' Principle of Least Constraint. A highly damped [small tau] creeping-flow limit describes our results for gtau(2)/sigma less than about 0.2. The lightly damped Green-Kubo linear-response limit for the model provides an accurate description of the dissipative dynamics for gtau(2)/sigma greater than about 2.0.

Jet methods in nonholonomic mechanics

Classical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle τ: M→R. The framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly. For systems with linear constraints a new interpretation is given of Appell’s equations as first-order differential equations associated with a suitable vector field. The d’Alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints. The equivalence between the equations of motion arising from this generalization, the ones set up on Gauss’ principle of least constraint and Hertz’s principle of least curvature is established.

Design Principles Applied to Structural Functions of Machine Components

The functional designer can derive help in his work from various principles, such as the minimization of unsprung mass and the principle of least constraint. This paper lists three groups of such principles which bear upon the structural aspects of design. These are structural design principles proper, general principles bearing upon the structural aspects and principles relating to joints. Some examples from mature technology are given, but the chief use of principles is in cases of evolution or innovation. Some related concepts (efficiencies of use) are given and a brief mention is made of the possible value of such principles in the development of the P-aesthetic sense by which designers recognise good solutions at an early stage.

Reconstruction of tractor semitrailer accidents using Gauss's principle of least constraint

Tractor semitrailer accidents are often caused by impact loading to some part of the vehicle and this leads to a loss of steering control and jackknifing. Impact loading may come from contact with another vehicle, or may occur on an isolated tractor semitrailer from maneuvers such as braking with subsequent brake lock-up, from suddenly steering into a turn where the cornering may lead to lateral wheel skidding, or from some combination of these factors. Without regard to the location, magnitude, or cause of impact loading, the accident reconstruction analyses herein rely on a knowledge of the tractor semitrailer's pre- and post-jackknife trajectories leading to a rest state. The tractor semitrailer is modeled as two rigid bodies hinged together and in-plane motion, to which is applied a system energy balance and Gauss's principle of least constraints. Explicit results are derived for the pre- and post-impact vehicle speeds as a function of the system parameters. Results of a case study for a tractor semitrailer initially in straight line motion show the sensitivity of the velocities to a range of trajectory surface-tyre friction conditions. Further, for constant friction conditions, the maximum post-impact linear velocities are from 1–2% less than the velocities just prior to impact; and for impending jackknife, the angular velocities of the tractor are about three times that of the trailer.

Approximate Solutions of Canonical Heat Conduction Equations

We consider three analytical methods for finding the approximate solutions of one-dimensional, transient, and nonlinear heat conduction problems based upon the canonical equations of heat transfer. The first method can be considered as a generalization or refinement of the integral method. The second is an iterative method similar to that of Targ utilized in boundary layer theory. The third method is a variational procedure introduced in the spirit of Gauss’ variational principle of least constraint.

The principle of least constraint for systems with non-restoring constraints

Two new versions of the principle of least constraint are derived from the D'Alembert-Lagrange principle for systems with ideal holonomic and non-holonomic restoring and non-restoring constraints. The first version is similar to Boltzmann's and Bolotov's modification of Gauss's principle for systems with non-restoring constraints. The difference is that here the actual motion is determined in a certain bounded set of possible motions as the one that deviates least from the motion of the system with all non-restoring constraints and any part of the restoring constraints disengaged. According to the second version of the principle, the actual motion is found by comparing certain distinguished possible motions as to their deviation from the motion of the system obtained by eliminating any part of the non-restoring and any part of the restoring constraints. Examples are given.

Application of Gauss principle of least constraint to the infiltration into unsaturated soil

This paper presents a new method for solving the problem of infiltration into unsaturated soils using the Gauss principle of least constraint. This technique gives a reasonable qualitative as well as quantitative picture of moisture condition during the unsteady infiltration from both semicircular furrows and buried pipes. The results are in good agreement with those obtained by the alternating direction implicit (ADI) difference method. The computation time is reduced to about 1/900 of that of ADI method.

Hamilton's principle of least action in nervous excitation

In contrast to the general membrane potential as found by Teorell and by Meyer and Sievers, the nerve potential represents a real potential function resulting from the singular solution of Nernst's differential equation. Planck–Goldman's potential is found to be valid in the nerve. It furnishes the minimum value of the line integral of the force and satisfies the principle of least constraint. Beneath this minimum principle with respect to the path according to Hamilton there exists a minimum principle with respect to time. It is the principle of least action. In the present case it is given by the time integral of Planck–Goldman's equation. Both principles are contained in Hamilton's principal function. Our investigations deal with the question of whether nervous excitation is controlled by the principal function or not. Because according to calculus of variations the minimum principles result from a variation of a definite integral, and regarding that the experimentally found potential difference Δϕ= 70 –(–55)= 125 mV represents the definite line integral of the force we take it as the integrand of the principal function. Then the time integration yields the definite value of the action W= 1.05h where h is Planck's constant. This means that a single quantum can produce a definite electric peak of 125 mV. Resulting from the definite value of the action this peak is independent of the strength or the quality of the stimulus. Thus, a sequence of n quanta gives rise to a sequence of m(⩽n)definite electric peaks.

Irreversibility in the Galton board via conservative classical and quantum hamiltonian and gaussian dynamics

The periodic Galton board is analyzed from twin equivalent classical viewpoints: constrained gaussian isokinetic mechanics and unconstrained conservative hamiltonian mechanics. Both problems display chaotic fractal behavior. We also investigate the corresponding quantum Galton board using a novel numerical technique incorporating Gauss' principle of least constraint in the nonequilibrium Schrödinger equation.

The stream function and Gauss' principle of least constraint: Two useful concepts for structural geology

To the extent that rock deformation can be approximated by a two-dimensional Newtonian model, a powerful stream-function simulation method is applicable. The significance of stream functions is that velocity, strain, stress and energy derived from the same stream function satisfy automatically three basic conditions of dynamics: 1. (1) the condition of continuity. 2. (2) the Navier-Stokes equations. 3. (3) conservation of energy. Hence we state with Jaeger: “If a stream function can be found which satisfies the boundary conditions of a dynamic model the complete solution follows.” All pertinent bits of dynamic information are implied in the stream function from which they can be directly derived, guaranteed—so to speak—not to violate the basic conditions of dynamics. Stream functions useful in structural geology are solutions of: ∇ 4ϵ= ∂ 4ψ ∂x 4 +2 ∂ 4ψ ∂x 2∂y 2 + ∂ 4ψ ∂y 4 =0 A double-polynomial solution of max. degree 14 is developed, in which the coefficients are related controlled by the ▽ 4 ψ = 0 constraint, and their absolute values are determined by the boundary conditions of specific models and by the condition of maximum rate of energy dissipation or maximum rate of decline of potential energy. The polynomial stream function is applied to a collapsing viscous “nappe” consisting of a thin basal layer with low viscosity on which a thicker layer with high viscosity slides due to gravitational spreading. The velocity of forward movement depends upon absolute and relative values of the following parameters: viscosity, thickness, the aspect ratio and density. The velocity of a variety of nappes with different thicknesses, aspect ratios, viscosities and densities is determined.

Constrained molecular dynamics: Simulations of liquid alkanes with a new algorithm

We present a new algorithm for molecular dynamics simulation involving holonomic constraints. Constrained equations of motion are derived using Gauss’ principle of least constraint. The algorithm uses a fast, exact solution for constraint forces and a new procedure to correct for accumulating numerical errors. We report several simulations of liquid n-butane and n-decane performed with the new algorithm. We obtain an average trans population of 60.6±1.5% in liquid butane at T=291 K and ρ=0.583 g/ml. This result essentially agrees with that from an earlier simulation by Ryckaert and Bellemans [Discuss. Faraday Soc. 66, 95 (1978)]. However, our simulations are substantially more precise; our run lengths are typically ∼20 times longer than those of Ryckaert and Bellemans. Our result also agrees with that from a recent simulation by Wielopolski and Smith (following paper). Thermodynamic and structural data from our simulations also agree well with results from the simulations discussed in the above articles.

Osmosis and reverse osmosis. Part 2.—The separation factor of reverse osmosis and its connection with isotonic osmosis

Two variational principles control isotonic osmosis and reverse osmosis: the principle of least dissipation of energy referring to a definite time integral and the principle of least constraint referring to the stationary (extremum) value of a volume integral.

Homogeneous periodic heat flow via nonequilibrium molecular dynamics

Two nonequilibrium methods for simulating homogeneous periodic heat flow are applied to 108 three-dimensional soft spheres in both the fluid and face-centered cubic solid phases. Both nonequilibrium methods use irreversible thermodynamics to express heat conductivity in terms of the work required to generate heat flow. The Evans-Gillan method, derived from Green-Kubo theory, correctly reproduces Ashurst's heat conductivities. An approach based on Gauss' principle of least constraint, in which the heat flow is constrained to a fixed value, fails this test. Heat flow is an inhomogeneous, nonlinear function of particle velocities and coordinates. Thus, Gauss' principle cannot be relied upon for treating inhomogeneous nonlinear nonholonomic constraints.

(P1, T12) Nonequilibrium molecular dynamics via Gauss' principle of least constraint

(P1, T12) Nonequilibrium molecular dynamics via Gauss' principle of least constraint

The isothermal/isobaric molecular dynamics ensemble

Recently Gauss' principle of least constraint has been combined with Doll's tensor methods to generate equations of motion for which the ideal gas temperature and the hydrostatic pressure are constants of the motion. We derive the equilibrium distribution function for these systems.

Nonequilibrium molecular dynamics via Gauss's principle of least constraint

Gauss's principle of least constraint is used to develop nonequilibrium molecular-dynamics algo­ rithms for systems SUbject to constraints. The treatment not only includes nonholonomic constraints-those involving velocities-but it also provides a basis for simulating nonequilibrium steady states. We describe two applications of this new use of Gauss's principle. The first of these examples, the isothermal molecular dynamics of a three-particle chain, can be treated analytically. The second, the steady-state diffusiort~of a Lennard-Jones liquid, near its triple point, is studied nu­ merically. The measured diffusion coefficient agrees with inaependent estimates from eqUilibrium fluctuation theory and from Hamiltonian external fields. I. INTRODUCTION

Isothermal-isobaric molecular dynamics

We combine the Doll's tensor hamiltonian for flow deformation with Gauss' principle of least constraint to develop classical equations of motion for which the temperature and hydrostatic pressure are constants of the motion. These equations of motion define a molecular dynamics algorithm for which N , T , p are the independent thermodynamic state variables.

Nonlinear transient temperature field in a hexagon

An approximate direct method for solving nonlinear transient heat conduction problem in the hexagon, based on Gauss' principle of least constraint is presented. The problem is reduced to the algebraic minimization of a quadratic form with respect to the physical components of the temporal and spatial changes of temperature field. For various values of specific heat and thermal conductivity temperature coefficients the comparison of approximate solutions with the numerical ones is performed and an agreement is found.

The Practical Use of Gauss’ Principle of Least Constraint

In this paper Gauss’ principle of least constraint is used to obtain approximate solutions of conservative and nonconservative dynamical systems. The main feature of the method is that every particular problem is reduced to an algebraic problem of minimizing a quadratic form with respect to the physical components of the acceleration vector of the system, or with respect to the components of a generalized force. The method is illustrated by solving several problems. The method is equally applicable to ordinary as well as partial differential equations. A brief criticism of the least-squares method, as a method of solving differential equations, is given.

Applications of gauss's principle of least constraint to the nonlinear heat-transfer problem

An approximate direct method for solving linear and nonlinear heat conduction problems, based on the Gauss's principle of least constraint is presented. In every particular case, the problem is reduced to the algebraic minimization of a quadratic form with respect to some complex of physical parameters. By the help of several concrete examples the efficiency and accuracy of this new method is demonstrated.