On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp₆(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ?³V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ?³V.     

On the trivectors of a 6-dimensional symplectic vector space. II

Let V be a 6-dimensional vector space over a field $\mathbb{F}$, let f be a nondegenerate alternating bilinear form on V and let $\mathit{Sp}(V\text{,}f)?{\mathit{Sp}}_6(\mathbb{F})$ denote the symplectic group associated with $(V\text{,}f)$. The group $\mathit{GL}(V)$ has a natural action on the third exterior power ${?}^{3}V$ of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field $\mathbb{F}$. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of $\mathbb{F}$ that are contained in a fixed algebraic closure $\overline{\mathbb{F}}$ of $\mathbb{F}$. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of $\mathit{Sp}(V\text{,}f)?\mathit{GL}(V)$ on ${?}^{3}V$.     

On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space

Translated from Itogi Nauki i Tekhniki, Seriya Sovremenaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 20, Topologia-3, 1994.     

On an extended phase space

Summary This paper presents a generalization of the dynamical concept phase space. The generalization involves a relative increase in dimensionality and employs a basic function which contains higher order derivatives. To Enrico Bompiani on his scientific Jubile.     

On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space

In this paper, we obtain accurate analytic free vibration solutions of rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected.     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

On groups acting in phase space

The problem of the motion of a material point in a central field of general type is considered. It is shown that in the infinite-dimensional group of canonical transformations which leave the Hamiltonian function invariant there are no finite-dimensional subgroups which are significantly larger than the three-dimensional group of rotations (exact formulations in Sec. 3 and Sec. 5).Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 55–61, January, 1969.I am indebted to A. A. Kirillov for helpful discussions.     

On symplectic transformations

This is an English translation¹ of the Ph.D. thesis ‘Over symplectische transformaties’ that Tonny Albert Springer, ‘born in’s-Gravenhage in 1926’, submitted as thesis for – as is stated on the original frontispiece – the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m., with promotor Prof. dr. H. D. Kloosterman.     

On the stability of KdV multi-solitons

We consider the stability of multi- or n-soliton solutions to the Korteweg-de Vries equation (KdV) posed on the real line. It is shown that in the standard variational characterization of KdV multi-solitons as critical points, the n-solitons actually realize non-isolated constrained minimizers. (The case n = 1 was already known to Benjamin; see [6].) From this fact a precise dynamic stability result for multi-solitons follows, namely, that initial data close to a given n-soliton evolves in time so as to remain close (in the Hⁿ(??) Sobolev norm) to the n-dimensional manifold of all n-solitons with appropriate wave speeds, i.e., to the set of constrained minimizers. Our techniques are also applicable to other Hamiltonian systems with several conserved quantities. In particular the inverse scattering formalism of KdV is not explicitly exploited. © 1993 John Wiley & Sons, Inc.     

The symplectic normal space of a cotangent-lifted action

For the cotangent bundle T^∗Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.     

Curvature of the space of associated metrics on a symplectic manifold

Kemerovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 132–139, January–February, 1992.     

On the complex structure of symplectic quotients

Science China Mathematics - Let K be a compact group. For a symplectic quotient Mλ of a compact Hamiltonian Kähler K-manifold, we show that the induced complex structure on Mλ is...