Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

2005 (revised version of September 2007)

The aim of this paper is twofold: first of all, we show that the $C^0$ limit of a pair of commuting Hamiltonians commute. This means on one hand that if the limit of the Hamiltonians is smooth, the Poisson bracket of their limit still vanishes, and on the other hand that we may define ``commutation'' for $C^0$ functions. The second part of the paper deals with solving ``multi-time'' Hamilton-Jacobi equations using variational solutions. This extends the work of Barles and Tourin in the viscosity case to include the case of $C^0$ Hamiltonians, and removes their convexity assumption, provided we are in the framework of ``variational solutions".

Functors and computations in Floer cohomology. Part II.(postscript)

1996 (2003 revision)

The results in this paper have been announced in a talk at the ICM 94 in Züurich. They concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I, others will be given in forthcoming papers.

Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique (Séminaire d'équations aux dérivées partielles de l'X)


Dans ce texte, on consid\`ere une m\'ethode g\'eom\'etrique pour construire des solutions d'\'equations d'Hamilton-Jacobi (cas d'\'evolution du premier ordre) $${\partial \over \partial t}u(t,x) = H(t,x,u,Du) \tag {HJ}$$ $$u(0,x)=u_0(x)$$ o\`u $Du$ d\'esigne la diff\'erentielle de $u$ par rapport \`a la variable $x$. Comme d'habitude, la variable $t$ sera appel\'ee "temps", la variable $x \in N$, "espace". On montre que cette construction, due \`a Sikorav et Chaperon, bas\'ee sur une m\'ethode de fonction g\'en\'eratrice fournit des solutions qui partagent de nombreuses propri\'et\'es des solutions de viscosit\'e de Crandall et Lions, mais peuvent \^etre diff\'erentes.

Solutions généraliséees pour l'équation d'Hamilton-Jacobi dans le cas d'évolution.


This paper studies the variational solutions of Hamilton-Jacobi equations that have been defined by Sikorav and Chaperon. A consequence of "uniqueness of generating functions" from Symplectic topology as the geometry of generating functions is the uniqueness of such varational solutions as well as the possibility to extend their defintion to the case of continuous Hamiltonians, for continuous initial conditions. We prove however, that in contrast to viscosity solutions of Lions and Crandall, the solution operator does not have the Markovian property $T_{s+t}=T_s\Circ T_t$.

Symplectic invariants and symplectic reduction: details of the proof of the camel problem

The aim of this note is to clarify the proof of the camel problem from the paper Symplectic topology as the geometry of generating functions, p.706