B. Vujanovic

Noether's theory in classical nonconservative mechanics

SummaryNoethers theorem and Noethers inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamiltons type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.ZusammenfassungDer Noethersche Satz und seine Umkehrung werden für mechanische Systeme mit nichtkonservativen Kräften aufgestellt. Die Existenz von Erstintegralen hängt von der Existenz von Lösungen der verallgemeinerten Noether-Bessel-Hagen-Gleichung oder, gleichbedeutend, von der von Lösungen des Killingschen Systems partieller Differentialgleichungen ab. Die Theorie fußt auf der Idee, daß Transformationen von Zeit, verallgemeinerten Koordinaten und dissipativen Kräften die Transformation der verallgemeinerten Geschwindigkeiten bestimmen; wie im Fall von Variationen in einem Variationsprinzip von Hamiltonscher Art für rein nichtkonservative Systeme [17], [18]. Unter Verwendung dieser Theorie werden einige neue Erstintegrale nichtkonservativer Probleme erhalten.

On one variational principle of Hamilton's type for nonlinear heat transfer problem

Abstract In this paper a new Lagrangian for nonlinear heat conduction problem is constructed. Using the concept of penetration depth a computational procedure for solving the nonlinear heat equation is given. Problem with nonlinear boundary conditions (surface radiation) is also discussed. Applying the method of Yang [33–35], it is shown that the solutions can be improved. Also, the method of choosing the best trial polynomial for the description of the temperature distribution is discussed. In the light of Yangs theory the solutions obtained by means of the variational principle have some degree of optimality in comparison to other approximative solutions.

Application of the optimal linearization method to the heat transfer problem

Abstract The generation of approximate solutions for nonlinear heat conduction problem using the method of optimal linearization is considered. Examples are used to investigate the merit of this method. Radiation cooling due to arbitrary power radiation from semi-infinite solid with temperature dependent material properties is discussed also.

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

Abstract An approximate direct method for solving linear and nonlinear heat conduction problems, based on the Gausss 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.

On the inverse Lagrangian problem

SummaryAn ad-hoc assumption regarding the form of the Lagrangian for a dynamic system with one degree of freedom is made. It is shown that this assumption leads to some useful conclusions. Several interesting examples are provided.

Application of the Hamilton-Jacobi method to the study of rheo-linear oscillators

SummaryIn this paper we demonstrate that the well-known Hamilton-Jacobi method can be used in study of the rheo-linear (i.e. time dependent) harmonic oscillator with a single degree of freedom. It will be shown that the quadratic conservation laws (exact invariants) together with corresponding auxiliary equations follow immediately from the complete integral of Hamilton-Jacobi partial differential equation by application of the Jacobi theorem. Attention is also paid to the study of linear conservation laws and to the motion of rheo-linear dynamical systems.

A variational principle for the two-dimensional boundary-layer flow of non-newtonian power-law fluids

SummaryThis paper is devoted to the theoretical analysis of steady and unsteady boundary layer problems for non-Newtonian power-law fluid flow using a new variational principle ofHamiltons type. The standard method of variational calculus in the form of partial integration is a basic tool for obtaining approximative solutions. The main characteristic of the variational principle developed here is that all basic rules of variational calculus are preserved. The results are found to be in good agreement with those obtained by other authors. Several examples of practical importance, such as steady flow around a flat plate, a wedge and a circular cylinder as well as impulsive motion of a flat plate and a circular cylinder are considered in detail.ZusammenfassungDiese Arbeit befaßt sich mit der theoretischen Analyse stationärer und nicht-stationärer Grenzschicht-Probleme nicht-newtonscher Flüssigkeiten vomOstwald-deWaele-Typ mit Hilfe eines neuen VariationsprinzipsHamiltonscher Art. Die Standard-Methode der Variationsrechnung in Form einer partiellen Integration ist eines der wichtigsten Hilfsmittel zur Gewinnung von Näherungslösungen. Das Hauptkennzeichen des hier entwickelten Variationsprinzips ist die Erhaltung aller Grundvorschriften der Variationsrechnung. Die gewonnenen Ergebnisse zeigen eine gute Übereinstimmung mit denjenigen, die von anderen Autoren mitgeteilt worden sind. Verschiedene Beispiele von praktischer Bedeutung wie die stationäre Strömung um eine flache Platte, einen Keil und einen Kreiszylinder sowie die ruckartige Bewegung einer flachen Platte und eines Kreiszylinders werden im einzelnen betrachtet.

On some geometrical aspects of classical nonconservative mechanics

It is shown that a scleronomous, holonomic dynamical system with nonconservative forces moves in such a way that the differential equations of motion are geodesic lines in a linear connected space Ln. The space Ln is semimetric and semisymmetric. The geodesic line on which the tangent at a point remains tangent if it is parallel displaced along the curve is simultaneously the curve of stationary length between two points in the space Ln. A necessary condition for the stationary length is derived by making use of the noncommutation rule for the differential of variation and the variation of differential. The noncommutation rule is obtained from a quadrilateral, which is called the fundamental quadrilateral of variational calculus. By using the noncommutation rule, the variational principles of Maupertius and Hamiltonian type for nonconservative mechanical systems are presented.

Polynomial conservation laws of the generalized Emden-Fowler equation

Abstract In this paper we consider the polynomial conservation laws of fourth degree with respect to x of the generalized Emden-Fowlers equation x + bt −1 x + λx α t k = 0 . It is demonstrated that the existence of conservation laws depends upon the solution of a system of partial differential equations, usually termed the generalized Killings equations. The general form of the fourth degree conservation laws of the Emden-Fowler equation is given and some concrete examples are discussed.

On two variational methods for obtaining solutions to transport problems

Abstract Two variational techniques with wide application to various heat transfer problems are described. The first is the chain system method applicable to th