Djordje S. Djukic

A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians

Abstract In this paper a procedure is given for finding first integrals of non-conservative mechanical systems in which the Lagrangian is gauge-variant under infinitesimal transformations as functions of time, position and generalized velocities. The procedure is founded on the generalized Noetherian theorem. The existence of first integrals depends on the existence of solutions of the system of partial differential equations. Two examples are analyzed in detail using this theory. The considerations are based on mechanical systems but the results obtained can be used on all problems in physics, engineering and mathematics for which one can construct an action integral in Hamiltons sense.

Conservation laws in classical mechanics for quasi-coordinates

Noethers theorem and Noethers inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunovs function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.

Noether's theorem for optimum control systems

Abstract In this paper Noethers theorem of classical mechanics and the variational calculus is developed for optimum control systems. Also, it is shown that the existence of first integrals of Pontryagins maximum principle equations depend on the existence of solutions of a system of partial differential equations—generalized Killings equations —for optimum control systems.

A new first integral corresponding to Lyapunov's function for a pendulum of variable length

ZusammenfassungMit Hilfe des Noetherschen Satzes und den verallgemeinerten Killingschen Gleichungen werden neue erste Integrale der Bewegungsdifferentialgleichungen für eine Klasse von nichtkonservativen mechanischen Systemen mit einem Freiheitsgrad, die das Pendel mit veränderlicher Länge als Sonderfall enthält, hergeleitet. Diese Integrale stellen Ljapunovsche Funktionen dar, mit denen sich die Stabilitätsbedingungen ergeben.SummaryUsing Noethers theorem and the generalized Killing equations [1], new first integrals of the differential equation of motion for a class of non-conservative mechanical systems with one degree of freedom, a special case of which is a simple pendulum of variable length, are obtained. These integrals are identified as Lyapunovs functions for non-autonomous systems. The stability conditions are established.

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.

Unsteady magnetic boundary layer flow of power‐law fluids

The Galerkin approximation technique is used to solve the flow problem of an infinite plate immersed in a non‐Newtonian power law fluid in the presence of a constant transverse magnetic field. The velocity outside the boundary layer depends exponentially on time.

On the rotating rod with shear and extensibility

The problem of determining the stability boundary and post-critical behavior of a heavy rotating rod is studied. Generalized constitutive equations are used so that both extensibility of the rod axis and the influence of shear stresses are taken into account. It is shown that, at the eigenvalues of the linearized equations the rod could exhibit both sub- and super-critical bifurcation patterns. An extremum variational principle for the system of equations describing the rod configuration is constructed and used for obtaining approximate solutions of the equilibrium shapes.

Adiabatic invariants for the nonconservative Kepler's problem

This paper considers adiabatic invariants for the classical Kepler problem with resisting forces. The analysis is based on the theory of integrating factors and theory of adiabatic invariants in the Krylov-Bogoliubov-Mitropolski variables. The adiabatic invariants are series with respect to a small parameter. Also, for every particular case of nonconservative forces, it is shown that, with a complete set of adiabatic invariants, an approximate solution of the problem can be obtained. Four problems are analyzed in detail where approximate solutions are compared with numerical.

Extremum variational principle for an inextensible rotating rod

Abstract An extremum variational principle for equilibrium state of an inextensible rotating rod is constructed. Its extremity is proved calculating first and second variations of the functional. The principle is used for finding an approximate solution of the problem by the Ritz technique. The approximate solution has satisfactory accuracy in comparison to the numerical solution of the same problem.

An extremum variational principle for some non-linear diffusion problems

Abstract An extremum variational principle for two non-linear boundary value problems is formulated. The first boundary value problem corresponds to the coupled diffusion reaction with high-order kinetics. The second boundary value problem describes zero-order chemical kinetics in a single catalyst pellet with Robin boundary conditions at the pellets outer surface. For both problems, approximate solutions and their error estimates for several values of the parameters are obtained.