The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoint orbits have a natural symplectic structure, the global geometry of the degenerate (constrained) Poisson manifold can be very complex. Som...

In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain lower bounds for number critical periods these systems. For potential degree n, provide bound n?2 periods, acquire n2/2+n?5/2 when n is odd n2/2?2 even periods. To best our knowledge, are new, moreover latter one twice existing results up to dominant term.

It is proved in this paper that the maximum number of limit cycles of system { dx dt = y, dy dt = kx− (k + 1)x2 + x3 + (α + βx + γx2)y is equal to two in the finite plane, where k > 11+ √ 33 4 , 0 < | | 1, |α| + |β| + |γ| = 0. This is partial answer to the seventh question in [2], posed by Arnold.

and Applied Analysis 3 Here, we assume that the equivariant symmetry S acts antisymplectically and S2 I. Now, we also consider the symmetric property of periodic solutions. This property was not studied for Hamiltonian vector fields without the other structure previously. 2. Main Results Theorem 2.1. Consider an equilibrium 0 of a C∞ equivariant Hamiltonian vector field f , with the equivariant...

We present a nonperturbative recipe for directly computing the $S$-matrix in strongly-coupled QFTs. The method makes use of spectral data obtained Hamiltonian framework and can be applied to wide range theories, including potentially QCD. demonstrate utility this prescription specific example 2+1d $O(N)$ model at large $N$, using energy eigenstates computed with truncation reproduce full $2 \to...