Dj. S. Djukic

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.

Adiabatic invariants for dynamical systems with one degree of freedom

Abstract Adiabatic invariants for dynamical systems with one degree of freedom, whose equation of motion is (1), and where the existence of the corresponding Hamilton action integral is not imposed, are established. The adiabatic invariants may vary according to their structure. Using the theory a few particular problems, including non-autonomous Duffing and Van der Pol oscillators, are analysed. Finally, it is indicated how the adiabatic invariants can be used for finding approximate solutions, stability analyses and for plotting phase curves.

Error bounds via a new extremum variational principle, mean square residual and weighted mean square residual

Abstract Error bounds for a wide class of nonlinear one-dimensional boundary value problems are derived from a new extremum variational principle. A new least-squares approximate technique, based on a weighted mean square residual, is established. Also, the value of the weighted mean square residual and value of the classical mean square residual are used for error estimate. The results are illustrated by four examples.

Noether's theory for non-conservative generalised mechanical systems

Noethers theorem and Noethers inverse theorem for generalised mechanical systems described by Lagrangian functions of the second order and non-conservative forces are established. The existence of the first integral depends on the existence of solutions of the generalised Noether-Bessel-Hagen equation. The theory is based on the idea that the transformations of time and generalised coordinates together with non-conservative forces determine the transformations of velocities and accelerations. An illustrative problem is discussed.

Integrating factors and conservation laws for nonconservative dynamical systems

Abstract A general approach to the construction of conservation laws for classical nonconservative dynamical systems is presented. The conservation laws are constructed by finding corresponding integrating factors for the equations of motion. Necessary conditions for existence of the conservation laws are studied in detail. A connection between an a priori known conservation law and the corresponding integrating factors is established. The theory is applied to two particular problems.

Effect of shear on stability and nonlinear behavior of a rotating rod

SummaryThe nonlinear equations describing in-plane deformation of a rotating elastic rod, taking into account shear effect, are derived. It is shown that the critical rotation speed is determined from the linearized equation. The nonlinear equilibrium equations are solved numerically and the effect of shear on maximal deflection is studied.ÜbersichtDie nichtlinearen Differentialgleichungen für die ebene Verformung einer rotierenden elastischen Welle mit Schubspannungseinfluß auf die Biegelinie wird hergeleitet. Wir zeigen, daß die kritische Rotationsgeschwindigkeit aus den linearen Gleichgewichtsbedingungen folgt. Die nichtlinearen Gleichungen werden numerisch gelöst und der Schubspannungseinfluß auf die Durchbiegung wird untersucht.

An extremum variational principle for classical Hamiltonian systems

A variational principle for a mechanical systems withn-degrees of freedom, which is subject to conservative generalized forces, is formulated. Necessary conditions for its extremum are given in detail. Two possibilities for constructing the functional are established. The first one is applicable if the canonical equations have certain simple algebraic properties, the second one is applicable in the general case. 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 Hamiltonian description. Finally, the theory is used for obtaining approximate solution of nonlinear mechanical problem with two degrees of freedom.

A note on the classical brachistochrone

SummaryIt is shown that the classical Bernoullis brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.ZusammenfassungIn der Arbeit ist gezeigt, daß das klassische Bernoullische brachistochrone Problem und Brachistochronen im zentralen Schwerefeld durch das Pontryagin Maximum Prinzip gelöst werden können. Gemäss Theorie der optimalen Regelung sind diese Probleme singulär.

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.