The infinite-dimensional mechanics of fluids and plasmas can be formulated as “noncanonical” Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that foliate the phase space, imposing topological constraints on the dynamics. Here we proffer a physical interpretation of Casimir elements as adiabatic invariant...

We give a simple geometrical picture of the basic structures of the covariant Sp(2) symmetric quantization formalism – triplectic quantization – recently suggested by Batalin, Marnelius and Semikhatov. In particular, we show that the appearance of an even Poisson bracket is not a particular property of triplectic quantization. Rather, any solution of the classical master equation generates on a...

It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate “momenta”, which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace ...

The Lagrangian and Hamiltonian structures for an ideal gaugecharged fluid are determined. Using a Kaluza–Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin–Noether theorem is obtained. The Hamiltonian formulation determines a non-canonical...

Let G be a simple complex algebraic group equipped with a factorizable Poisson Lie structure. Let U~(g) be the corresponding quantum group. We study U~(g)-equivariant quantization C~[G] of the affine coordinate ring C[G] along the Semenov-Tian-Shansky Poisson Lie bracket. For a simply connected group G we prove an analog of the KostantRichardson theorem stating that C~[G] is a free module over ...