On a horizontal structure on differentiable manifolds
On a theorem of Chekanov
A proof of the Chekanov theorem is discussed from a geometric point of view. Similar results in the context of projectivized cotangent bundles are proved. Some applications are given.
On Anosov energy levels of convex Hamiltonian systems.
On control theory and its applications to certain problems for Lagrangian systems. On hyperimpulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Applications to Lagrangian systems
See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case , and satisfy conditions (11.7) when is a polynomial in , conditions (C)-i.e. (11.8) and (11.7) with -are proved to be necessary for treating satisfactorily 's hyper-impulsive motions (in which positions can suffer first order discontinuities)....
On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.
In [1] I and II various equivalence theorems are proved; e.g. an ODE with a scalar control is linear w.r.t. iff its solution with given initial conditions (chosen arbitrarily) is continuous w.r.t. in a certain sense, or iff
On Elliptic Lower Dimensional Tori in Hamiltonian Systems.
On embedding the 1:1:2 resonance space in a Poisson manifold.
On Fixed Points of Symplectic Maps.
On minimal sets in 2-manifolds.
On quadratic symplectic mappings.
On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters
In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system , referred to a system of Lagrangian co-ordinates, and to try and write some handy conditions,...
On the density of the range for some nonlinear operators
On the existence and non-existence of closed trajectories for some Hamiltonian flows.
On the existence of homoclinic solutions for almost periodic second order systems
On the geometric structure of the set of solutions of Einstein equations [Book]
On the -instability and -controllability of steady flows of an ideal incompressible fluid
In the existing stability theory of steady flows of an ideal incompressible fluid, formulated by V. Arnold, the stability is understood as a stability with respect to perturbations with small in vorticity. Nothing has been known about the stability under perturbation with small energy, without any restrictions on vorticity; it was clear that existing methods do not work for this (the most physically reasonable) class of perturbations. We prove that in fact, every nontrivial steady flow is unstable...
On the Lagrange-Souriau form in classical field theory
The Euler-Lagrange equations are given in a geometrized framework using a differential form related to the Poincare-Cartan form. This new differential form is intrinsically characterized; the present approach does not suppose a distinction between the field and the space-time variables (i.e. a fibration). In connection with this problem we give another proof describing the most general Lagrangian leading to identically vanishing Euler-Lagrange equations. This gives the possibility to have a geometric...
On the mechanics with noncontinuous hamiltonians
On the optimal control of implicit systems
On the optimal control of implicit systems
In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding manifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold, defined by Rabier and Rheinboldt for determined implicit differential equations,...