Related papers: Proof of Jacobi identity in generalized quantum dy…
In view of the recent interest in a short proof of the Jacobi identity for the Poisson-brackets, we provide an alternative simple proof for the same. Our derivation is based on the validity of the Leibnitz rule in the context of dynamical…
We investigate the conditions under which the Jacobi identity holds for a class of recently introduced anti-symmetric brackets for the hybrid plasma models with kinetic ions and massless electrons. In particular, we establish the precise…
Making use of the theory of infinitesimal canonical transformations, a concise proof is given of Jacobi's identity for Poisson brackets.
Jacobi brackets (a generalization of standard Poisson brackets in which Leibniz's rule is replaced by a weaker condition) are extended to brackets involving an arbitrary (even) number of functions. This new structure includes, as a…
A carefully motivated symmetric variant of the Poisson bracket in ordinary (not Grassmann) phase space variables is shown to satisfy identities which are in algebraic correspondence with the anticommutation postulates for quantized Fermion…
We introduce a symmetric Poisson bracket that allows us to describe anticommuting fields on a classical level in the same way as commuting fields, without the use of Grassmann variables. By means of a simple example, we show how the Dirac…
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…
By definition the identities $[x_1, x_2] + [x_2, x_1] = 0$ and $[x_1, x_2, x_3] + [x_2, x_3, x_1] + [x_3, x_1, x_2] = 0$ hold in any Lie algebra. It is easy to check that the identity $[x_1, x_2, x_3, x_4] + [x_2, x_1, x_4, x_3] + [x_3,…
A new family of $n$-dimensional solutions of the Jacobi identities is characterized. Such a family is very general, thus unifying in a common framework many different well-known Poisson systems seemingly unrelated. This unification is not…
We consider a hierarchy of Poisson structures defined on rational functions on the Riemann sphere. This hierarchy is originated in the theory of the integrable Camassa-Holm equation associated with the Krein's string spectral problem.…
Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds --…
We discuss dimensional reduction for Hamiltonian systems which possess nonconstant Poisson brackets between pairs of coordinates and between pairs of momenta. The associated Jacobi identities imply that the dimensionally reduced brackets…
We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and…
We consider a general formalism for treating a Hamiltonian (canonical) field theory with a spatial boundary. In this formalism essentially all functionals are differentiable from the very beginning and hence no improvement terms are needed.…
Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are mustered: we involve the invariance under the action of a general Lie group of the balance of substructural…
We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible type parameters $\alpha,\beta$ in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the…
We obtain a finite form of Jacobi's identity and present a combinatorial proof based on the structure of synchronized partitions.
Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation…
On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is…
A new Poisson bracket for Hamiltonian forms on the full multisymplectic phase space is defined. At least for forms of degree n-1, where n is the dimension of space-time, Jacobi's identity is fulfilled.