Dynamical Systems Admitting Normal
نویسندگان
چکیده
Let S be a hypersurface in Riemannian manifold M . One of the ways for deforming S consists in shifting points, which constitute S, along trajectories of some Newtonian dynamical system. Such situation arises in describing the propagation of electromagnetic wave (light) in non-homogeneous media in the limit of geometric optics. Hypersurface S models wave front set (the set of points with constant phase), while trajectories of shift model light beams. Newtonian dynamics of points of Riemannian manifold M in local coordinates x, . . . , x in M is described by a system of n ordinary differential equations
منابع مشابه
A ug 2 00 1 DYNAMICAL SYSTEMS ADMITTING NORMAL SHIFT AND WAVE EQUATIONS
Abstract. High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton equations being special case for Newton’s equations. In simplest cas...
متن کاملSecond Problem of Globalization
Problem of global integration of geometric structures arising in the theory of dynamical systems admitting the normal shift is considered. In the case when such integration is possible the problem of globalization for shift maps is studied.
متن کاملDilations for $C^ast$-dynamical systems with abelian groups on Hilbert $C^ast$-modules
In this paper we investigate the dilations of completely positive definite representations of (C^ast)-dynamical systems with abelian groups on Hilbert (C^ast)-modules. We show that if ((mathcal{A}, G,alpha)) is a (C^ast)-dynamical system with (G) an abelian group, then every completely positive definite covariant representation ((pi,varphi,E)) of ((mathcal{A}, G,alpha)) on a Hilbert ...
متن کاملGlobal Geometric Structures Associated with Dynamical Systems Admitting Normal Shift of Hypersurfaces in Riemannian Manifolds
One of the ways of transforming hypersurfaces in Riemannian manifold is to move their points along some lines. In Bonnet construction of geodesic normal shift, these points move along geodesic lines. Normality of shift means that moving hypersurface keeps orthogonality to the trajectories of all its points. Geodesic lines correspond to the motion of free particles if the points of hypersurface ...
متن کاملNewtonian Normal Shift in Multidimensional Riemannian Geometry
Explicit description for arbitrary Newtonian dynamical system admitting the normal shift in Riemannian manifold of the dimension n > 3 is found. On the base of this result the kinematics of normal shift of hypersurfaces along trajectories of such system is studied.
متن کاملNORMAL FORM SOLUTION OF REDUCED ORDER OSCILLATING SYSTEMS
This paper describes a preliminary investigation into the use of normal form theory for modelling large non-linear dynamical systems. Limit cycle oscillations are determined for simple two-degree-of-freedom double pendulum systems. The double pendulum system is reduced into its centre manifold before computing normal forms. Normal forms are obtained using a period averaging method which is appl...
متن کامل