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We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its "boundary depth," and establish basic results about how the boundary depths of different Hamiltonians are related. As…
In this paper, we prove that any $C^{1}$-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with…
Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic…
We present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space. A suitably defined new Hamiltonian on the extended phase space leads to equations of motion that can be…
Let L be a D-dimensional submanifold of a 2D-dimensional exact symplectic manifold (M, w) and let f be a symplectic diffeomorphism onf M. In this article, we deal with the link between the dynamics of f restricted to L and the geometry of L…
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…
We introduce a new global Lagrangian descriptor that is applied to flows with general time dependence (altimetric datasets). It succeeds in detecting simultaneously, with great accuracy, invariant manifolds, hyperbolic and non-hyperbolic…
This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M, \omega) to a group that both intersects every connected component of Symp(M) and characterizes…
We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed…
We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this…
The purpose of this paper is to present some results on the existence of homologous, nonisotopic symplectic or lagrangian surfaces embedded in a simply connected symplectic 4-dimensional manifold.
Contact Hamiltonian dynamics is a subject that has still a short history, but with relevant applications in many areas: thermodynamics, cosmology, control theory, and neurogeometry, among others. In recent years there has been a great…
In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. More precisely, we prove that if a Euclidean ball is symplectically embedded in the…
The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid.…
This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of…
We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture) that can be put into correspondence with the usual…
In the symplectic Lagrangian framework we newly embed an irreducible massive vector-tensor theory into a gauge invariant system, which has become reducible, by extending the configuration space to include an additional pair of scalar and…
We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both…
We study loops of symplectic diffeomorphisms of closed symplectic manifolds. Our main result, which is valid for a large class of symplectic manifolds, shows that the flux of a symplectic loop vanishes whenever its orbits are contractible.…
This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.