Lagrangian manifold

Figure 6. From top to bottom action surface Lagrangian manifold (LM) and extreme paths calculated [80] for the system (17) using equations (21). The parameters for the system were A = 0.264 and

Figure 5.6 The blue lines show the Lagrangian manifolds Af A ° and...

Figure 5.7 The blue lines show the Lagrangian manifolds A,, A, A,, and A associated with the eigenstates and respectively.The arrows...

Le Hong Van, and Fomenko, A T. "Lagrangian manifolds and Maslov s index in the theory of minimal surfaces. Dokl, Akad, Nauk SSSR (1987) (in print). 

The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. 

Remark 7.2. In the above argument, we assumed that the C -action on a holomorphic symplectic Kahler manifold X satisfies iploJc = tujc for t E C. This is possible only when X is non-compact. The reason is as follows. If X is compact there exists a critical manifold corresponding to the maximum of the Morse function for which the unstable manifold is open submanifold of X, but this contradicts the propostion which asserts that every unstable manifold is Lagrangian. 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

S Yj is Lagrangian, the normal bundle is identified with T (S Y). Moreover T S Y) is dense in since it is identified with the stable manifold of S Y. Hence we may say... 

This shows that As looks roughly like T Als- These are not exactly the same since may not be stable for ( , ) e As- On the other hand we have shown that C T A Since SnT, is Lagrangian, the normal bundle is identified with T (5n ). Moreover T (5n ) is dense in T since it is identified with the stable manifold of Hence we may say that t Y looks like Sn(T Y,) but not exactly. 

Within this context, ordinary differential equations are viewed as vector fields on manifolds or configuration spaces [2]. For example, Newton s equations are second-order differential equations describing smooth curves on Riemannian manifolds. Noether s theorem [4] states that a diffeomorphism,3 < ), of a Riemannian manifold, C, indices a diffeomorphism, D< >, of its tangent4 bundle,5 TC. If 4> is a symmetry of Newton s equations, then Dt(> preserves the Lagrangian o /Jc ) = jSf. As opposed to equations of motion in conventional... 

The hypothesis that the constituents of the mixture have a Lagrangian microstructure (in the sense of Capriz [3]) means that each material element of a single body reveals a microscopic geometric order at a closer look then it is there assigned a measure Vi(x) of the peculiar microstructure, read on a manifold Mi of finite dimension rnp e.g., the space of symmetric tensor in the theory of solids with large pores or the interval [0, v) of real number, with v immiscible mixture (see [5, 9]). We do not fix the rank of the tensor order parameter u%. 

The universally observed flow of time is another example of a broken symmetry. A theoretical formulation of this proposition is not known, but in principle it should parallel the theory of superconductivity. A high-symmetry state could be associated with Euclidean Minkowski space that spontaneously transforms into a curved manifold of lower symmetry. In this case the hidden symmetry emerges from a Lagrangian which is invariant under the temporal evolution group... 

In Fig. 23 a, an intersection between two-dimensional stable and unstable manifolds is displayed in a three-dimensional phase space. In order to see the intersection in a space of reduced dimensionality, only its location is indicated on the unstable manifold in Fig. 23b. Thus, we can single out the information on how they intersect, although we sacrifice the information on how these manifolds are folded as they intersect. (How they are folded can be also studied in a similar way using the Lagrangian singularity caused by folding. See the details in Ref. 12.)... 

By the geodesic flow of the Riemannian manifold M with the metric ds = Y Gijdqidqj we mean a Lagrangian system in the tangent bundle TM with the Langrange function L — Identifying TM with T M by means of the... 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

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