Flows
Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the symplectic 2-form and hence the symplectic-volumeform, Liouville's theorem in Hamiltonian mechanics follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.
Since {H,H} = XH(H) = 0, the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy.
If the first Betti number of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and symplectic isotopy of symplectomorphisms coincide.
We can show that the equations for a geodesic may be formulated as a Hamiltonian flow.
Read more about this topic: Symplectomorphism
Famous quotes containing the word flows:
“Between flattery and admiration there often flows a river of contempt.”
—Minna Antrim (1861–?)
“Freudianism is much more nearly a religion than a science, inasmuch as the relation between analyst and patient has a great deal in common with that between priest and communicant at confessional, and such ideas as the Oedipus complex, the superego, the libido, and the id exert an effect upon the converted which is almost identical with what flows to the devout Christian from godhead, trinity, grace, and immortality.”
—Robert Nisbet (b. 1913)
“The point of the dragonfly’s terrible lip, the giant water bug, birdsong, or the beautiful dazzle and flash of sunlighted minnows, is not that it all fits together like clockwork--for it doesn’t ... but that it all flows so freely wild, like the creek, that it all surges in such a free, finged tangle. Freedom is the world’s water and weather, the world’s nourishment freely given, its soil and sap: and the creator loves pizzazz.”
—Annie Dillard (b. 1945)