Related papers: The Euler-Lagrange Cohomology and General Volume-P…
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given…
The standard Hamiltonian machinery, being applied to field theory, leads to infinite-dimensional phase spaces. It is not covariant. In this article, we present covariant finite-dimensional multimomentum Hamiltonian formalism for field…
We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the…
Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in…
Moduli spaces of compact stable $n$-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes. We give various new formulas for the…
We show that the following three systems related to various hydrodynamical approximations: the Korteweg--de Vries equation, the Camassa--Holm equation, and the Hunter--Saxton equation, have the same symmetry group and similar bihamiltonian…
In this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H in R, there is a full measure subset of…
This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the…
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small…
These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the…
In this study, we introduce Euler-Lagrange and Hamiltonian equations on (R2; g; J) being a model of para-Kaehlerian Space Forms. Finally, some geometrical and physical results on the related mechanic systems have been discussed.
In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector…
In this paper we consider the ``symplectic'' version of the Euler equations studied by Ebin \cite{ebin}. We show that these equations are globally well-posed on the Sobolev space $H^s(\mathbb{R}^{2n})$ for $n \geq 1$ and $s > 2n/2+1$. The…
We generalize an inequality of Besson-Courtois-Gallot about volume and simplicial volume of closed manifolds to the $\ell_1$-norm of all the homology classes of complete manifolds. The inequality involves the critical exponent of the…
In this paper we developed an analysis of the compressible, isentropic Euler equations in two spatial dimensions for a generalized polytropic gas law. The main focus is rotational flows in the subsonic regimes, described through the…
We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the H\'enon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family…
Properties of Hamiltonian symmetry flows on hyperbolic Euler-type Liouvillean equations E' are analyzed. Description of their Noether symmetries assigned to the integrals for these equations is obtained. The integrals provide Miura…
If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are…
We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second order partial differential equation for the corresponding time function $\tau(x)$ at…
We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean)…