Seminar for Geometry, education and visualization with applications

PROGRAM

CETVRTAK, 1. novembar 2007. u 17 sati
Srdjan Vukmirovic
Kalibracije (3.deo)

Nastavljamo tamo gde smo proslog puta stali. Kratko cemo spomenuti jos dva tipa kalibracija: specijalne Lagranzove kalibracije i kalibracije asocijativnih ravni u algebri oktoniona. One su bile tema nekih ranijih sastanaka Seminara. Potom nastavljamo sa vaznim teoremama u teoriji kalibracija: Teoremom u uglovima i Torusnom lemom.

CETVRTAK, 8. novembar 2007. u 17 sati
Srdjan Vukmirovic
Kalibracije (4.deo)

Prosli put smo dali deo dokaza Teoreme o uglovima. Ovaj put cemo kompletirati dokaz upotrebom Torusne leme. U nastavku cemo posmatrati generalisane Nansove kalibracije i prostore kompleksnih struktura na $R^{2n}$.

CETVRTAK, 15. novembar 2007. u 17 sati
Srdjan Vukmirovic
Kalibracije (5.deo)

Apstrakt: Ovo je poslednje predavanje mini kursa o kalibracijama. Bice ukratko predstavljeni neki rezultati teorije kalibracija koji se ne mogu naci u knjizi Harvey-a. Takodje ce biti reci o primenama kalibracija na teoriju mehura od sapunice (radovi Morgana).

Drugo predavanje:
Predavac: Kristina Obrenovic
Naziv predavanja: Primeri iskosenih povrsi na sferi S^6

Dacemo karakterizaciju iskosenih dvodimenzionih sfera koje su presek trodimenzione ravni i sfere S^6. U karakterizaciji i dokazu se sustinski koriste pojam oktoniona i asocijativnih ravni.

CETVRTAK, 22. novembar 2007. u 17 sati
Radu Miron, Faculty of Mathematics, University "SAl.I.Cuza" Iasi, Iasi, Romania
Lagrangian and Finslerian Mechanics of Nonconservative Mechanical Systems

The geometry of nonconservative mechanical systems, where the external force field depends on both position and velocity, was rigorously investigated by Klein [5] and Godbillon [4]. The dynamical system of a nonconservative mechanical systems is a second order vector field, or a semispray, and it has been uniquely determined by Godbillon [4] using the symplectic structure and the energy of the Lagrange space and the external force field. Using the external force field of the nonconservative mechanical system, Klein [5] introduces a force tensor, which is a second rank skew symmetric tensor. In this paper we extend the geometric investigation of nonconservative mechanical systems, using the associated evolution nonlinear connection. We show that the evolution nonlinear connection is uniquely determined by two compatibility conditions with the metric structure and the symplectic structure of the Lagrange space, [3]. The covariant derivative of the Lagrange metric tensor with respect to the evolution nonlinear connection is a second rank symmetric tensor, which uniquely determines the symmetric part of the connection. The difference between the symplectic structure of the Lagrange space and the almost-symplectic structure of the nonconservative mechanical system is the force tensor introduced by Klein [5], and used recently by Miron [6]. The force tensor, which is the vertical differential of the external force, uniquely determines the skew-symmetric part of the evolution nonlinear connection. The force tensor vanishes in the work of Bloch [2] and therefore the symplectic geometry of the nonconservative mechanical system coincides with the symplectic geometry of the underlying Lagrange space as it has been developed by Abraham and Marsden [1]. One can determine the equations of evolution either from a Lagrangian function by writing these equations as the Euler-Lagrange equations or by using a Legendre transformation, determining a Hamiltonian function and considering the Hamilton equations. One can use then Lagrange or Hamilton geometries for a geometric theory of the evolution problem. A geometrical approach of this problem on the phase space for the Riemannian case has been proposed by Munoz-Lecanda and Yaniz-Fernandez, [9]. This theory has been developed recently, for the case of Finsler and Lagrange spaces in [3] and [7]. For a mathematical model of the geomagnetic field, which has aperiodic reversals, Yajima and Nagahama proposed recently in [10] a mathematical model that corresponds to a nonlinear dynamical system (Rikitake system). This way the chaotic behavior of the system is expressed with the above mentioned geometric and topologic invariants. In this paper we study dynamical systems on the phase space that are defined by systems of second order differential equations that result from the theory of scleronomic, holonomic mechanical systems given by Lagrange equations when the external forces are a priori given. The main idea is to determine a semispray S, whose integral curves give the evolution curves. We shall determine the evolution semispray of a mechanical system by using the symplectic structure of the associated Lagrangian function and the external force field. The geometry of the semispray will determine the geometry of the associated dynamical system on the phase space. We will study these problems first for a Finsler space Fn = (M,F) and a Lagrange space Ln = (M,L). [7]. If the Lagrangian function is not homogeneous of second degree with respect to the velocity-coordinates, which is the case in the Riemannian and Finslerian context, the energy of the system is different from the Lagrangian function and the evolution curves (solution of the Euler-Lagrange equations) are different from the horizontal curves of the system. Therefore, we shall study the variation of both energy and Lagrangian function along the evolution curves and horizontal curves. Canonic nonlinear connection of a Lagrange manifold is the unique nonlinear connection that is metric and symplectic. Conditions by which the evolution nonlinear connection is either metric or symplectic are determined in terms of the symmetric or skew-symmetric part of a (1,1)-type tensor field associated with the external force field.

References:
[1] ABRAHAM, R., MARSDEN, J.: Foundation of Mechanics. Benjamin, New-York (1978).
[2] BLOCH, A.M.: Nonholonomic Mechanics and Control. Springer-Verlag (2003).
[3] BUCATARU, I., MIRON, R.: Nonlinear connections for nonconservative mechanmical systems. Reports on Mathematical Physics, 59 (2), 225-241 (2007).
[4] GODBILLON, C.: Geometrie Differentielle et Mecanique Analytique, Hermann, Paris (1969).
[5] KLEIN, J.: Espaces Variationnels et Mechaniques. Ann. Inst. Fourier, Grenoble, 12, 1-124 (1962).
[6] MIRON, R.: The Lagrangian mechanical systems and associated dynamical systems. Tensor (N.S.), 66 (1), 53-58 (2005).
[7] MIRON, R., FRIGIOIU, C.: Finslerian mechanical systems. Algebras Groups Geom., 22 (2), 151-167 (2005).
[8] MIRON, R., NIMINET, V.,The Lagrangian Geometrical Model and the Associated Dynamical System of a Nonholonomic Mechanical System. Facta Universitatis, vol.5 no.1, 2006
[9] MUNOZ-LECANDA, M.C., YANIZ-FERNANDEZ, F.J.: Dissipative control of mechanical systems. A geometric approach. SIAM J. Control Op-tim., 40 (5), 1505-1516 (2002).
[10] YAJIMA, T., NAGAHAMA, H.: KCC-theory and geometry of the Riki-take system. Journal of Physics A-Mathematical and Theoretical, 40 (11), 2755-2772 (2007).

Sednice seminara odrzavaju se u zgradi Srpske akademije nauka i umetnosti, Beograd, Knez Mihailova 35, na prvom spratu u sali 2.