Formal stability of Hamiltonian systems

✍ Scribed by James Glimm

• Publisher: John Wiley and Sons
• Year: 1964
• Tongue: English
• Weight: 796 KB
• Volume: 17
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160170408

✦ Synopsis

This is motivated by the formula (P 0 Z)-1 = P-'(l + P-'(P 0 z -P)-1 = P-' c [-P-l(P 0 2 -P)].. Proof: We first show that vq = 6 + [v,,]~~+~ + . -€9;. When p = 0 this is true with respect to the term ("q"pp of v. Let p # 0; then The first term has d, = d,(tmqnpP) -1 and d, = d,(E"qnpP). Thus d ,5 j ( d ,123

The first term is in S i and d, 2 2j + 8. The second term has its degree d, 5 Iml + In1 + 1 -j IpIj = d,(("q"pP) -( j -1) 2 2 j + 8 and d, = d,(EmqnpP) + j . Thus d , 5 j ( d , + j -3 ( j + 2)) < j ( d , -2 ( j + 2)) I This proves that vq E .F;, that vq -5 starts with terms at least of degree d, = 2 j + 8, and furthermore that the determination of a term of degree d, , d, of vq depends only on the terms of u of degree dAd; with d; 5 d, , d ; 5 dt + j -1.

2 j + 7, let c$~ = 0. If we substitute $1 + * * -+ &+, for 4 in (1.6), then (1.6) is an identity up to terms of order j .