Darboux theorem

Radiation fields are especially interesting since they are usually represent photon states. Moreover, it is known that, because of the Darboux theorem [59— 61], the Faraday form of any electromagnetic field F and its dual can be written, locally, as... 

So, for each point x and M one can point out an open neighbourhood with symplectic coordinates. Covering M with such neighbourhoods, we obtain a symplectic atlas. We omit the proof of the Darboux theorem. 

PROOF By virtue of the Darboux theorem (see Theorem 1.2.1) it suffices to prove this assertion only for the case where M is the simplest symplectic manifold with the canonical structure (jj = Y dpi A dqi. Let the form (jj be preserved by a one-parameter group gt, that is, = 0, where 7(f) are integral trajectories... 

In the case of a two-dimensional symplectic manifold, the condition of the locally Hamiltonian character of the field admits another vivid geometrical interpretation. Let gij be a Riemannian metric on and let u) = y/det gij)dx A dy be the form of the Riemannian area. By virtue of the Darboux theorem, one can always choose local coordinates p and q such that the form oj be written in the canonical form dp A dq. Here p and q are certain functions of x and y (and vice versa). Let t be a locally Hamiltonian field t = (i (a , y),Q(x,y)), where P and Q are coordinates of the field in the local system of coordinates p and q. Let us interpret the field v as a velocity field of the flow of liquid of constant density (equal to unity) on the surface M. Let us investigate the variation of the mass of the liquid bounded by an infinitesimal rectangle on the surface when it is shifted along integral trajectories of the field v. It is clear that the mass of this liquid is equal to the area of the rectangle. Therefore, the mass of the liquid contained in a bounded (sufficiently small) domain on is equal to the area of the domain. 

Proof Note, first, that cjl is the so-called precanonical structure on L (the non-degeracy condition is lifted, but the rank is constant, and the closedness of the form d(u)L) = (doi)L = 0 is retained). In his paper [200], Losco made use of the fact that the Darboux theorem remains valid for precanonical structures. But we shall take a different approach, the one suggested by Tatarinov. Note that if a vector field Y (on L) is such that Y z) G l cn... 

Theorem 1.2.1 (Darboux). Let be a symplectic structure on M. Then, for any point xE M, there always exists an open neighbourhood with local regular coordinates Pi )Pni9i>-such that the form w in them is written in the... 

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