The geometry of generating functions for a class of Hamiltonians in the noncompact case

We consider a class of Hamiltonians H:T⋆Rn⟶RH:T⋆Rn⟶R and the related flows ϕHt:T⋆Rn⟶T⋆Rn, proving the existence and uniqueness of generating functions quadratic at infinity for its graph Λt=T∗Rn×ϕHt(T∗Rn). As a consequence, we obtain the same results for the Lagrangian submanifolds Lt≔ϕHt(L0)⊂T⋆Rn Hamiltonianly isotopic to the zero section L0≃RnL0≃Rn. This problem was also considered by Chaperon, Sikorav and Viterbo in the case of closed manifolds. The assumption on the class of Hamiltonians is an asymptotic behaviour of polynomial type on the phase space. In particular, we deal with a family of Hamiltonian systems arising from usual mechanical problems, for which we study the structure of the corresponding generating functions, showing their main analytical properties. The results presented in the paper are applied to prove the existence and uniqueness of minmax solutions for a class of Hamilton–Jacobi equations on T⋆RnT⋆Rn.