Symmetries and reduction Part I — Poisson and symplectic picture

@article{Marmo2020SymmetriesAR,
  title={Symmetries and reduction Part I — Poisson and symplectic picture},
  author={Giuseppe Marmo and Luca Schiavone and Alessandro Zampini},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism. 
4 Citations
Symmetries and Reduction Part II - Lagrangian and Hamilton-Jacobi Picture
Following the analysis we have presented in a previous paper (that we refer to as [I]), we describe a Noether theorem related to symmetries, with the associated reduction procedures, for classical
Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds
As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a
Abstract dynamical systems: Remarks on symmetries and reduction
  • G. Marmo, A. Zampini
  • Mathematics, Physics
    International Journal of Geometric Methods in Modern Physics
  • 2021
In this paper, we review how an algebraic formulation for the dynamics of a physical system allows to describe a reduction procedure for both classical and quantum evolutions.
A geometric approach to the generalized Noether theorem
We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented by Zhang P-M et al (2020 Eur. Phys. J. Plus 135 223). Our version of the generalized

References

SHOWING 1-10 OF 58 REFERENCES
Generalized Reduction Procedure: Symplectic and Poisson Formalism
We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on
Normal Forms, symmetry, and linearization of dynamical systems
We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter dependent family - via the Poincar\'e Normal Form approach. We
Noether's theorem for singular Lagrangians
The correspondence between constants of motion and symmetries of a singular Langrangian system is studied. It is shown to be a one-to-one correspondence after an appropriate definition of both
Generalized Hamiltonian Dynamics
  • P. Dirac
  • Physics
    Canadian Journal of Mathematics
  • 1950
The author’s procedure for passing from the Lagrangian to the Hamiltonian when the momenta are not independent functions of the velocities is put into a simpler and more practical form, the main
Reduction of Poisson manifolds
Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as
The inverse problem in the Hamiltonian formalism: integrability of linear Hamiltonian fields
The authors study the inverse problem for Hamiltonian dynamics, with restriction to the case of linear homogeneous vector fields. An algebraic necessary and sufficient condition is derived. They
Dynamical Aspects of Lie--Poisson Structures
Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these
Symmetries and constants of the motion for singular Lagrangian systems
A classification of infinitesimal symmetries of singular autonomous and nonautonomous Lagrangian systems is obtained. The relationship between infinitesimal symmetries and constants of the motion is
A general setting for reduction of dynamical systems
The reduction of dynamical systems is discussed in terms of projecting vector fields with respect to foliations of the manifold on which the dynamics take place. Examples of established reduction
...
...