scholarly journals A characterization of 3D steady Euler flows using commuting zero-flux homologies

2020 ◽  
pp. 1-16
Author(s):  
DANIEL PERALTA-SALAS ◽  
ANA RECHTMAN ◽  
FRANCISCO TORRES DE LIZAUR
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Euler Flows ◽  
Homological Characterization ◽  
Steady Solutions ◽  
Volume Preserving

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

scholarly journals C-Flows on a Lie Group for Euler Equations

1970 ◽  
Vol 40 ◽  
pp. 67-84
Author(s):  
Yoshihei Hasegawa
Keyword(s):  
Euler Equations ◽  
Lie Group ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Left Invariant Riemannian Metric ◽  
Invariant Vector Fields

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

Structure and stability of solutions of the Euler equations: a lagrangian approach

1990 ◽  
Vol 333 (1631) ◽  
pp. 321-342 ◽  
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Saddle Points ◽  
Relaxation Processes ◽  
Two Dimensional ◽  
Scalar And Vector Fields ◽  
Euler Flows ◽  
Relaxation Procedure ◽  
Unbounded Fluid

This paper reviews methods that are essentially lagrangian in character for determination of solutions of the Euler equations having prescribed topological characteristics. These methods depend in the first instance on the existence of lagrangian invariants for convected scalar and vector fields. Among these, the helicity invariant for a convected or ‘frozen-in’ vector field has particular significance. These invariants, and the associated topological interpretation are discussed in §§1 and 2. In §3 the method of magnetic relaxation to magnetostatic equilibria of prescribed topology is briefly described. This provides a powerful method for determining steady Euler flows through the well-known exact analogy between Euler flows and magnetostatic equilibria. Stability considerations relating to magnetostatic equilibria obtained in this way and to the analogous Euler flows are reviewed in §4. In §5 the related relaxation procedure is discussed; for two-dimensional and axisymmetric situations this technique provides stable solutions of the Euler equations for which the vorticity field has prescribed topology. The concept of flow signature is described in §6: this is the relevant topological characteristic for two-dimensional or axisymmetric situations, which is conserved during frozen-field relaxation processes. In §§7 and 8, the formation of tangential discontinuities as a normal part of the relaxation process when saddle points of the frozen-field are present is discussed. Section 9 considers briefly the application of these ideas to the theory of vortons, i.e. rotational disturbances that propagate without change of structure in an unbounded fluid. The paper concludes with a brief discussion, with comment on the possible development of the results in the context of turbulence.

scholarly journals Steady Euler flows and Beltrami fields in high dimensions

2020 ◽  
pp. 1-24
Author(s):  
ROBERT CARDONA
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
High Dimensions ◽  
Beltrami Fields ◽  
Open Books ◽  
Euler Flows ◽  
Reeb Field ◽  
Beltrami Field ◽  
Volume Preserving

Abstract Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

Conformal vector fields on Finsler manifolds

2020 ◽  
Vol 31 (12) ◽  
pp. 2050095
Author(s):  
Qiaoling Xia
Keyword(s):  
Sectional Curvature ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Finsler Manifolds ◽  
Conformal Vector ◽  
Conformal Fields ◽  
Special Cases ◽  
Conformal Vector Fields ◽  
Equivalent Characterization

In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].

scholarly journals Existence of periodic orbits for geodesible vector fields on closed 3-manifolds

2009 ◽  
Vol 30 (6) ◽  
pp. 1817-1841 ◽  
Author(s):  
ANA RECHTMAN
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Global Section ◽  
Riemannian Metric ◽  
Torus Bundle ◽  
Reeb Vector ◽  
Ambient Manifold ◽  
Volume Preserving

AbstractIn this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume-preservingCωgeodesible vector fields, and prove the existence of periodic orbits forCωgeodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits ofC2geodesible vector fields on some closed 3-manifolds.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Talat Körpınar ◽  
Yasin Ünlütürk
Keyword(s):  
Vector Fields ◽  
Elastic Surface ◽  
Geometrical Characterization ◽  
Lorentzian Space ◽  
New Approach ◽  
Darboux Vector ◽  
Moving Particle ◽  
The Relationship ◽  
Surface Curve

AbstractIn this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

scholarly journals Controllability of Continuous Bimodal Linear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Josep Ferrer ◽  
Juan R. Pacha ◽  
Marta Peña
Keyword(s):  
Linear Systems ◽  
Higher Dimensions ◽  
Linear Dynamics ◽  
Explicit Characterization ◽  
Separating Hyperplane

We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions.

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