Quasi-Lagrangian Systems of Newton Equations

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J. Math. Phy

Effect of interface resistance on thermoelectric properties in (1-x)La0.95_{0.95}Sr0.05_{0.05}Co0.95_{0.95}Mn0.05_{0.05}O3_3/(x)WC composite

In this study, the synergistic effect of the particle size of the dispersed phase and the interface thermal resistance (R$_{int}$) between the phases on the phonon thermal conductivity ($\kappa_{ph}$) of the (1-x)La$_{0.95}$Sr$_{0.05}$Co$_{0.95}$Mn$_{0.05}$O$_3$/(x)WC thermoelectric composite, is demonstrated. Further, the correlation between the R$_{int}$ and the Kapitza radius is discussed using the Bruggeman's asymmetrical model. In particular, the polycrystalline La$_{0.95}$Sr$_{0.05}$Co$_{0.95}$Mn$_{0.05}$O$_3$ sample is synthesized using a standard-solid state route. The presence of WC nanoparticle is confirmed from the electron microscopy images. Electrical conductivity ($\sigma$) increases, and the Seebeck coefficient ($\alpha$) decreases with the increase in conducting WC volume fraction in the composite. The simultaneous increase in $\sigma$ and a decrease in $\kappa_{ph}$ with the WC volume fraction results in an increased figure of merit (zT) for (1-x)La$_{0.95}$Sr$_{0.05}$Co$_{0.95}$Mn$_{0.05}$O$_3$/(x)WC composite. A maximum zT $\sim$ 0.20 is obtained for (1-x)La$_{0.95}$Sr$_{0.05}$Co$_{0.95}$Mn$_{0.05}$O$_3$/(x)WC composite for x=0.010 at 463 K. The results obtained in the present study shows promise to design thermoelectric composites with desired phonon thermal conductivity considering the elastic properties between the phases.Comment: 15 pages, 6 figures, 1 tabl

ν-invariants on manifolds with cylindrical end

AbstractWe study the ν-function of an operator A of Dirac type on a non-compact Riemannian manifold X∞, which is obtained from a compact manifold X with boundary Y by attaching the infinite cylinder X∞ = (-∞, 0] x Y ∪Y X. We assume that the metric structure is a product on the cylinder and that the operator B, the tangential part of the operator A on the cylinder, is non-singular. We show that the ν-function νA(s) shares all the properties of the ν-function of an operator of Dirac type defined on a closed manifold. In particular, νA(s) is a holomorphic function for Re(s) > -2

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Piknik Naukowy Online

Artykuł zamieszczony jest w : Życie Uczelni : biuletyn informacyjny Politechniki Łódzkiej nr 152, lipiec 2020Autor artykułu jest pracownikiem Centrum Nauczania Matematyki i Fizyki Politechniki Łódzkiej (CMF PŁ)24. Piknik Naukowy miał się odbyć na Stadionie Narodowym w Warszawie. Tak jak wszystkie imprezy masowe musiał przenieść się do wirtualnej rzeczywistości