By Mariano Giaquinta, Stefan Hildebrandt

ISBN-10: 3540579613

ISBN-13: 9783540579618

This e-book by way of of the major researchers and writers within the box is the 1st a part of a treatise that covers the topic in breadth and intensity, paying distinct recognition to the historic origins of the idea. either separately and jointly those volumes have already develop into regular references.

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Extra info for Calculus of variations II

Sample text

Exemple d’application Consid´erons dans R3 le syst`eme m´ecanique d´efini par le Lagrangien L= 1 m(x˙ 2 + y˙ 2 + z˙ 2 ) − U (x, y, z). 2 Le passage en coordonn´ees sph´eriques s’´ecrit ϕ : (x, y, z) → (r, θ, ϕ) que l’on rel`eve en un diff´eomorphisme symplectique → − ϕ : (x, y, z, px , py , pz ) → (r, θ, ϕ, pr , pθ , pϕ ). Le Hamiltonien s’´ecrit H= 1 2m p2r + p2ϕ p2θ + r2 r2 sin2 θ + U (r, θ, ϕ). 1 Int´egrabilit´e 31 Consid´erons un champ de potentiel de la forme b(θ) . r2 Ici ϕ est une coordonn´ee cyclique et pϕ est une int´egrale premi`ere.

En particulier si la transformation est autonome, le syst` eme est un syst`eme Hamiltonien autonome et on a H = εH. 6 Notes et sources Les r´esultats g´en´eraux sur la g´eom´etrie symplectique et les champs Hamiltoniens proviennent de [3], [31] et [53]. La forme normale de type Jordan d’un champ de vecteurs Hamiltonien est due a` Williamson [72] (voir [22] pour une pr´esentation plus moderne). Pour une introduction g´ en´erale au calcul des variations, voir [29] et pour le principe du maximum faible voir par exemple [10], et aussi l’article [30] pour une pr´esentation heuristique du principe du maximum.

N } le spectre de A et on suppose que pour tout i, Re λi < 0. On suppose ´egalement que l’on est dans le cas non r´esonant. Alors x˙ = Ax + . . peut ˆetre lin´earis´e par un germe de diff´eomorphisme analytique. En particulier l’origine est asymptotiquement stable. Th´ eor` eme 15. Soit x˙ = Ax + . . un champ de vecteurs analytique. Si les valeurs propres de A sont de type (C, ν) alors ce champ peut ˆetre lin´earis´e par une germe de diff´eomorphisme analytique. 5 Forme normale d’un syst` eme Hamiltonien equilibre au voisinage d’un ´ La construction de formes normales formelles d´evelopp´ee par Poincar´e-Dulac s’applique aussi dans le cadre Hamiltonien.

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