On implicit Lagrangian differential systems
Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).
Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).
Let be a constrained mechanical system locally referred to state coordinates . Let be an assigned trajectory for the coordinates and let be a scalar function of the time, to be thought as a control. In [4] one considers the control system , which is parametrized by the coordinates and is obtained from by adding the time-dependent, holonomic constraints . More generally, one can consider a vector-valued control which is directly identified with . If one denotes the momenta conjugate...
We give different notions of Liouville forms, generalized Liouville forms and vertical Liouville forms with respect to a locally trivial fibration π:E → M. These notions are linked with those of semi-basic forms and vertical forms. We study the infinitesimal automorphisms of these forms; we also investigate the relations with momentum maps.
The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved. The existence of such solutions was established some time ago. Here we report a uniqueness result that uses the Oleinik entropy condition and a cohesion condition. Both of these conditions are automatically satisfied by solutions obtained in previous existence results. Important tools in the proof of uniqueness are regularizations, generalized...
This Note is the continuation of a previous paper with the same title. Here (Part II) we show that for every choice of the sequence , 's trajectory after the instant tends in a certain natural sense, as , to a certain geodesic of , with origin at . Incidentally is independent of the choice of applied forces in a neighbourhood of arbitrarily prefixed.
In this Note (which will be followed by a second) we consider a Lagrangian system (possibly without any Lagrangian function) referred to coordinates , , with to be used as a control, and precisely to add to a frictionless constraint of the type . Let 's (frictionless) constraints be represented by the manifold generally moving in Hertz's space. We also consider an instant (to be used for certain limit discontinuity-properties), a point of , a value for 's momentum conjugate...
The notion of an implicit Hamiltonian system-an isotropic mapping H: M → (TM,ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated.
The external derivative on differential manifolds inspires graded operators on complexes of spaces , , stated by dual to a Lie algebra . Cohomological properties of these operators are studied in the case of the Lie algebra of the Lie group of Euclidean motions.
In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of endowed with the standard symplectic form . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example, discontinuous...