On the Hamiltonian Flow Box Theorem
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Qualitative Theory of Dynamical Systems
On the Hamiltonian Flow Box Theorem Hildeberto E. Cabral
Received: 30 October 2011 / Accepted: 21 August 2012 / Published online: 26 September 2012 © Springer Basel AG 2012
Abstract The goal of this note is to give a new proof of the Hamiltonian Flow Box Theorem which is intrinsic and has a strong geometric flavor. For other proofs of this theorem see references Meyer et al. (Introduction to Hamiltonian dynamical systems and the N-body problem, 2009) and Moser and Zehnder (Notes on dynamical systems, 2005) . Keywords flow box
Hamiltonian systems · Flow box · Symplectic mappings · Simultaneous
1 Introduction Let f : U → Rm be a C r -smooth mapping, r ≥ 1, in the open set U ⊂ Rm and let φ(t; z) be the flow of the differential equation z˙ = f (z).
(1)
An equilibrium of (1) is a point z 0 ∈ U where f vanishes. If f (z 0 ) = 0 we say that z 0 is a regular point. In the neighborhood of an equilibrium the flow may be quite complicate but in the case of a regular point it is extremely simple as it is basically a parallel flow. This is the content of the following theorem, proved in courses on differential equations.
To Ken, on his 75th anniversary. H. E. Cabral (B) Departamento de Matemática, UFPE, Recife, Brazil e-mail: [email protected]; [email protected]
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H. E. Cabral
Theorem 1 (Flow Box Theorem) Let f : U → Rm be a C r -smooth mapping, r ≥ 1, in the open set U ⊂ Rm and let z 0 ∈ U be such that f (z 0 ) = 0. Then, there is a C r -diffeomorphism from a box I × onto a neighborhood Z of z 0 , given by z = F(ζ ) such that the system in ζ corresponding to z˙ = f (z) is given by ζ˙1 = 1, ζ˙2 = 0, . . . , ζ˙m = 0. In the case where Eq. (1) is Hamiltonian, that is m = 2n and f = J ∇ H , we can take the diffeomorphism to be symplectic. This is the content of the following theorem. Theorem 2 (Hamiltonian Flow Box Theorem) Let H : U → R be a function of class C r +1 , r ≥ 1, in an open set U ⊂ R2n . Let z 0 ∈ U be such that ∇ H (z 0 ) = 0. Then, there is a symplectic diffeomorphism of class C r from a box I × onto a neighborhood Z of z 0 given by z = F(ζ ) such that the new Hamiltonian is given by H(ζ ) = η1 + const, where ζ = (ξ1 , . . . , ξn , η1 , . . . , ηn ). 2 Proof of Theorem 2 Proof By a translation we can assume z 0 = 0. Let e1 and en+1 be unit vectors with directions f (0) = J ∇ H (0) and ∇ H (0), respectively. Let ω(X, Y ) = X T J Y,
J=
O I , −I O
be the symplectic form of R2n . The subspace U generated by e1 and en+1 is symplectic since ω(e1 , en+1 ) = 1, therefore the subspace U ⊥ = {v ∈ R2n ; ω(u, v) = 0, for all u ∈ U } is symplectic also. Complete {e1 , en+1 } with a symplectic basis of U ⊥ to form a symplectic basis e1 , . . . , e2n of R2n . Let ζ1 , . . . , ζ2n be the coordinates in R2n relative to this basis. We want to find symplectic coordinates ζ1 . . . , ζ2n so that the transformed Hamiltonian be given by H(ζ ) = H (z) = ζn+1 + const. For ζ ∈ R2n near the origin consider the solution z = φ(t, ζ ) of the equation z˙ = J ∇ H (z) which pa
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