Dynamical systems on a Riemannian manifold that admit normal shift

1995 ◽  
Vol 103 (2) ◽  
pp. 543-549
Author(s):  
A. Yu. Boldin ◽  
V. V. Dmitrieva ◽  
S. S. Safin ◽  
R. A. Sharipov
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Normal Shift

Dynamical systems: Approximate Lagrangians and Noether symmetries

2019 ◽  
Vol 16 (10) ◽  
pp. 1950160 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Kinetic Energy ◽  
Equations Of Motion ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Second Order Equations

We determine the approximate Noether point symmetries of the variational principle characterizing second-order equations of motion of a particle in a (finite-dimensional) Riemannian manifold. In particular, the Lagrangian comprises of kinetic energy and a potential [Formula: see text], perturbed to [Formula: see text]. We establish a convenient system of approximate geometric conditions that suffices for the computation of approximate Noether symmetry vectors and moreover, simplifies the problem of the effect of higher orders of the perturbation. The general results are applied to several practical problems of interest and we find extra Noether symmetries at [Formula: see text].

scholarly journals Isometries and certain dynamical systems

1984 ◽  
Vol 30 (2) ◽  
pp. 239-246 ◽  
Author(s):  
Saber Elaydi ◽  
Hani R. Farran
Keyword(s):  
Dynamical System ◽  
Riemannian Manifold ◽  
Dynamical Systems

It is shown that there exists a metric under which a diffeomorphism f on a Riemannian manifold M becomes an isometry, provided that the dynamical system generated by f is of characteristic 0± and all its orbits are closed. Furthermore, it is shown that the foliation given by the suspension of f is parallel in this case.

Pseudo-Riemannian manifold of mixmaster dynamical systems

1990 ◽  
Vol 148 (5) ◽  
pp. 239-245 ◽  
Author(s):  
M. Szydłowski ◽  
A. Ł;apeta
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems

scholarly journals Global geometric structures associated with dynamical systems admitting normal shift of hypersurfaces in Riemannian manifolds

2002 ◽  
Vol 30 (9) ◽  
pp. 541-557
Author(s):  
Ruslan A. Sharipov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Explicit Formula ◽  
Riemannian Manifolds ◽  
Second Law ◽  
System Of Equations ◽  
Geometric Structures ◽  
Geodesic Lines ◽  
Free Particles ◽  
Normal Shift

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 are treated as physical entities obeying Newton's second law. An attempt to introduce some external forceFacting on the points of moving hypersurface in Bonnet construction leads to the theory of dynamical systems admitting a normal shift. As appears in this theory, the force fieldFof dynamical system should satisfy some system of partial differential equations. Recently, this system of equations was integrated, and explicit formula forFwas obtained. But this formula is local. The main goal of this paper is to reveal global geometric structures associated with local expressions forFgiven by explicit formula.

Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

1992 ◽  
Vol 12 (1) ◽  
pp. 123-151 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Topological Properties ◽  
Smooth Measure ◽  
Ergodic Properties ◽  
Special Invariant ◽  
Hyperbolic Attractors ◽  
Hyperbolic Behavior

AbstractWe introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

scholarly journals Multidimensional dynamical systems that admit normal shift

1994 ◽  
Vol 100 (2) ◽  
pp. 997-1000
Author(s):  
A. Yu. Boldin ◽  
R. A. Sharipov
Keyword(s):  
Dynamical Systems ◽  
Normal Shift
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