# Geometric description of Jacobi's theorem on complete integrals of HJ eqn.

I am not sure if this question is adapted to this site, if it is not, then I will delete it.

The Hamilton--Jacobi theory is about the connection between:

1. the solutions of an Hamilton--Jacobi equation $H\circ dS=0$ on a smooth $n$-dimensional manifold $M$, and
2. the integral manifolds of the characteristic distribution generated by $X_H$ on the coisotropic submanifold $H^{-1}(0)$ of $T^\ast M$ with its cannical symplectic form $d\lambda_M$.

The direction $2)\to 1)$ of the connection is carried out proving the theorem of local existence and unicity for solution of initial value problem specified by non-characteristic Cauchy data.
Infact the solution is the primitive of the closed differential $1$-form on $M$ whose image is developed by the flow $X_H$ through a lift of the Cauchy data.

Instead the direction $1)\to 2)$ of the connection is described by the Jacobi's theorem on complete integral. Usually it is stated as: from the knowledge of a complete integral of $1)$ we can construct the integral trajectories of $X_H$.

My question is this:

Is there a more geometric description of this Jacobi's theorem?

Somewher I read that Souriau, in his seminar Geometrie Symplectique Differentielle'' at the Strasbourg Conference in 1953, among the others thing, put this theorem in a geometric picture. But I am not access to such source.

I would also appreciate informations on this work of Souriau.

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