Casimir invariants and the Jacobi identity in Dirac’s theory of constrained Hamiltonian systems
نویسندگان
چکیده
Abstract. We consider constrained Hamiltonian systems in the framework of Dirac’s theory. We show that the Jacobi identity results from imposing that the constraints are Casimir invariants, regardless of the fact that the matrix of Poisson brackets between constraints is invertible or not. We point out that the proof we provide ensures the validity of the Jacobi identity everywhere in phase space, and not just on the surface defined by the constraints. Two examples are considered: A finite dimensional system with an odd number of constraints, and the Vlasov-Poisson reduction from VlasovMaxwell equations.
منابع مشابه
Hamiltonian closures for fluid models with four moments by dimensional analysis
Fluid reductions of the Vlasov–Ampère equations that preserve the Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian closures using the first four moments of the Vlasov distribution are obtained, and all closures provided by a dimensional analysis procedure for satisfying the Jacobi identity are found. Two Hamiltonian models emerge, for which the explicit closures a...
متن کاملJa n 19 96 Non - canonical Quantization of a Quadratic Constrained System ∗
We propose an alternative to Dirac quantization for a quadratic constrained system. We show that this solves the Jacobi identity violation problem occuring in the Dirac quantization case and yields a well defined Fock space. By requiring the uniqueness of the ground state, we show that for non-constrained systems, this approach gives the same results as Dirac quantization. After the formulation...
متن کاملHamiltonian derivation of a gyrofluid model for collisionless magnetic reconnection
We consider a simple electromagnetic gyrokinetic model for collisionless plasmas and show that it possesses a Hamiltonian structure. Subsequently, from this model we derive a two-moment gyrofluid model by means of a procedure which guarantees that the resulting gyrofluid model is also Hamiltonian. The first step in the derivation consists of imposing a generic fluid closure in the Poisson brack...
متن کاملA hierarchy of noncanonical Hamiltonian systems
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...
متن کاملA general theory for gauge-free lifting
A theory for lifting equations of motion for charged particle dynamics, subject to given electromagnetic like forces, up to a gauge-free system of coupled Hamiltonian Vlasov-Maxwell like equations is given. The theory provides very general expressions for the polarization and magnetization vector fields in terms of the particle dynamics description of matter. Thus, as is common in plasma physic...
متن کامل