اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn an integrable magnetic geodesic flow on the two-torus
115
0
0.0
(
0
)
تأليف
I.A. Taimanov
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = cos (x) dx wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.
قيم البحث
اقرأ أيضاً
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show tha
t those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of s
tationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.
In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable geodesic flows on the Klein bottle.
Let $(M,g)$ be a closed Riemannian manifold and $sigma$ be a closed 2-form on $M$ representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic
flow of the pair $(g,sigma)$ can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle $T^*E$ of a suitable $S^1$-bundle $E$ over $M$ or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of $E$. We then study the relation between the stability property of energy hypersurfaces in $(T^*M,dpwedge dq+pi^*sigma)$ and of the corresponding codimension 2 coisotropic submanifolds in $(T^*E,dpwedge dq)$ arising via symplectic reduction. Finally, we reprove the main result of [9] in this setting.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be inter
ested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
