We complete the list of normal forms for effective 3-forms with constant coefficients with respect to the natural action of symplectomorphisms in R 6 . We show that the 3-form which corresponds to the Special Lagrangian equation is among the new members of the classification. The symplectic symmetry algebras and their Cartan prolongations for these forms are computed and a local classification theorem for the corresponding Monge-Ampère equations is proved.A classical problem of the geometric theory of differential equations is the problem of local equivalence: when do two differential equations represent the same equation modulo local change of dependent and independent coordinates (a diffeomorphism on the corresponding jet space)? We can consider this problem to be a special case of the general equivalence problem in differential geometry (see [1], chapter 7). This point of view enables us to recognize equivalent structures, objets, etc., by means of a set of so-called "scalar" differential invariants. Generally speaking, a differential invariant of order at most k is a smooth function on the jet space J k invariant under the diffeomorphisms generated by the local diffeomorphisms of the base manifold.An important particular case of this problem is the question of linearization: when is a differential equation equivalent to a linear one? The Monge-Ampère equations (MAE hereafter) provide a natural class of nonlinear second order differential equations for this problem. Sophus Lie posed in 1874 this problem of linearization for 2-dimensional MAE with respect to the (pseudo) group of contact diffeomorphism Ct and it was studied in the classical works of G.Darboux and E. Goursat as well as in many recent papers.An adequate description of the general classification problem for the MAE is achieved by computing of a complete system of differential invariants with a complete set of relations between them. Scalar differential invariants of MAE are interpreted as smooth functions on a "diffiety" quotient J ∞ (M AE)/Ct, the object of the category of differential equations corresponding to MAE (see [9]). This quotient is quite singular and admit a stratification (like a spaces of orbits, total spaces of general foliations, etc.). Locally the quotient J ∞ (M AE)/Ct has the form E ∞ of the infinite prolongation of a system of differential equations E, which is defined by the relation between differential invariants. The stratification of E ∞ is given by the singular lociwhich correspond to a reduction of the number of variables. Due to a generalized "Kerr theorem" ([11], ch. 7.2.3), this reduction is provided by a sufficiently large local symmetry group, so the most "symmetric" MAE should correspond to a very "singular strata". We will call such equations "special MAE". We will restrict ourself throughout this paper to the classification problem of these special MAE. A modern geometric approach to special MAE was proposed by V. Lychagin [10] (after an idea of E. Cartan) and was applied to this classification problem by V.
We prove that locally any hyper-Kähler metric with torsion admits an HKT potential.
We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to MongeAmpère structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction. The MongeAmpère structure for incompressible flow in two dimensions is shown to be hyper-symplectic.
The pressure in the incompressible three-dimensional Navier-Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of threeforms in six dimensions. By studying the linear algebra of the vector space of three-forms L 3 W Ã where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of L 3 W Ã to the sign of the Laplacian of the pressure-and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Dp, satisfies DpO0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Dp!0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.