Locally Hamiltonian vector field

Hamiltonian and Locally Hamiltonian Vector Fields and the Poisson Bracket We will denote by V[M) an infinite-dimensional linear space of all smooth vector fields on a manifold M. If is a symplectic manifold, then in V M) there exists a... 

Thus, in the case of a two-dimensional symplectic manifold, the locally Hamiltonian vector fields are exactly the flows of incompressible liquid, that is, the vector fields with zero divergence. In other words, the condition for the local Hamiltonian properties of the field v in the two-dimensional case is equivalent to the condition div(v) = 0. 

We have given an example of a locally Hamiltonian vector field on R 0 which is not a Hamiltonian field. The symplectic manifold was not compact here. But such kind of examples may be constructed on compact closed (i.e., without a boundary) manifolds as well. Take, for instance, an ordinary flat torus supplied with a Euclidean metric gij = Si, Then the 2-form of the area on this torus will be written in Cartesian coordinates x, y as follows dx A dy, i.e., this 2-form is canonical. On the torus, consider a vector field v = (1,0) given by a uniform liquid motion along its parallels (Fig. 7). 

Thus, we have proved not only item 4, but item 5 as well. Indeed, as has become clear, the commutator of two locally Hamiltonian vector fields Ui and V2 is a globally Hamiltonian field. Therefore, Hamiltonian fields form the ideal in the space of locally Hamiltonian fields. This completes the proof of the theorem. 

Much of our understanding of critical phenomena is based on the Landau-Ginzburg model of a ferromagnet. This model concentrates on the local magnetization, represented by an m-component vector field Sa (r), a = 1,..., m. often called a classical spin field . The interaction of the spin field is described by the Landau-Ginzburg Hamiltonian... 

Definition 1.2.9 A smooth vector field u on a symplectic manifold M is called locally Hamiltonian if for any point x G Af, there exists such an open neighbourhood U[x) of the point x and such a smooth function Fu defined on this neighbourhood that V — sgrad Fez, that is, the field v is Hamiltonian in the neighbourhood U with the local Hamiltonian Fu ... 

FVom proposition 1.2.5, it follows that the Poisson bracket /, is a locally constant function, and if the manifold Af is connected, then the function is constant on the entire manifold Af. Thus, we have proved that two Hamiltonian field t and w commute (like vector fields) if and only if the Poisson bracket of their Hamiltonians, /) ) is locally constant (or constant, in case the manifold is connected). 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

The broad line spectra of nuclei with spin I = 1/2 in the solid state are mainly a consequence of the dominant contribution of the dipolar Hamiltonian HD (Eq. (4)), which gives rise to a local field B)oc. Its magnitude varies as a function of the angle 0.. between the intemuclear vector r.j and the applied magnetic field B0. Depending on the nature of spin system, two general types of interactions can be distinguished ... 

It is important to note that these treatments, both at the DFPT and SOS level, pertain to hUed bands that is, for insulators or cold semiconductors the treatment of partially filled bands requires the characterization of scattering effects as well. Moreover, since incident rather than macroscopic electric fields (or vector potentials) generally enter into the perturbation Hamiltonian [Eqs. (4) and (6)], the various linear and nonlinear susceptibihties have to be corrected if there are no imphcit or explicit local field corrections. 

O A is a shielding constant tensor of the nucleus A. Due to this shielding, nucleus A feels a local field Hioc = (l < a)H — H — a aH instead of the external field H applied (due to the tensor character of o a, the vectors Hioc and H may differ by their length and direction). The formula assumes that the shielding is proportional to the external magnetic field intensity that causes the shielding. Thus, the first term in the Hamiltonian ii may also be written asa YaHJ Ja-... 

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