ABSTRACT
D A symplectic structure on M consists in giving a nondegenerate closed 2-form at each point of M called symplecticform w 0 A symplectic manifold (M,w) is a manifold M provided with a symplectic form w on M
1.1 DARBOUX THEOREM AND SYMPLECTIC MATRIX
1.1.1 Moser lemma
L Given symplectic forms WI (supposed differentiable for every t E [0,1]), for every x E M there are a neighborhood U of x and a family of local transformations
fIJ,:U -+U such that
fIJ~ = id and fIJ;w, == WO° Proof. The problem is to know if there are vector fields XI on U such that
d -flJIX =XI (fIJIx)dt
with fIJ;WI == Wo°
By refering to (6-6) with here w,(t,x), that is by introducing the notion of the Lie derivative Lu linked to the flow, we have:
~fIJ;WI == fIJ;(Luwl )== rp;(OIW, +Lx w,)dt ' == fIJ;(o,w, +dix WI)',
but the form °IWI is closed since do,wl ==oldw, =0;
therefore, from the Poincare lemma, there is a neighborhood U of x on M such that the form 0IW, is exact and we can write:
°IW, =dp, and thus
d. .d( ° )-flJIW, = fIJ, p, + 'x W, .dt '
To look for '1'; such that
amounts to search for Xi such that rp;d(p, + ix li>,) =0,
or such that ix,li>, = -P"
In a local chart, we write: li>, =1!li>AB(t,x) dxA/\ dxB (A,B =1, ...,2n) P, =PA(t,X) dxA
. XAdxB=> 1Xli>, =li>AB . So, the problem amounts to solve (for XA(t,x» the following system
WAB(t,x)XA(t,x) + PB(t,X) = 0, that has a unique solution because the forms li>, are nondegenerate. The field X, is thus determined and consequently also the forms li>, such that rp;li>, =wo, the lemma is so proved.
