Related papers: On Symplectic Reduction in Classical Mechanics
We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. Correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular…
Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a…
A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a…
Modern learning systems act on internal representations of data, yet how these representations encode underlying physical or statistical structure is often left implicit. In physics, conservation laws of Hamiltonian systems such as…
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and…
Using a basic idea of Sullivan's rational homotopy theory, one can see a Lie groupoid as the fundamental groupoid of its Lie algebroid. This paper studies analogues of Lie algebroids with non-trivial higher homotopy. Using various homotopy…
We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical…
We develop a unified geometric framework for dissipative mechanical systems based on uniform $q$-contact manifolds, which provide an extended phase space equipped with multiple contact $1$-forms. Within this setting, we construct both…
The concept of symmetries in physics is briefly reviewed. In the first part of these lecture notes, some of the basic mathematical tools needed for the understanding of symmetries in nature are presented, namely group theory, Lie groups and…
In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…
Spherical symmetry for f(R)-gravity is discussed by searching for Noether symmetries. The method consists in selecting conserved quantities in form of currents that reduce dynamics of f(R)-models compatible with symmetries. In this way we…
Noether's theorem is reviewed with a particular focus on an intermediate step between global and local gauge and coordinate transformations, namely linear transformations. We rederive the well known result that global symmetry leads to…
A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new…
Among theoretical issues in General Relativity the problem of constructing its Hamiltonian formulation is still of interest. The most of attempts to quantize Gravity are based upon Dirac generalization of Hamiltonian dynamics for system…
Transformer models contain substantial internal redundancy arising from coordinate-dependent representations and continuous symmetries, in model space and in head space, respectively. While recent approaches address this by explicitly…
In this paper we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of [1,3]. We illustrate the theory with classical examples describing the dynamics of…
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a…
A lattice Maxwell system is developed with gauge-symmetry, symplectic structure and discrete space-time symmetry. Noether's theorem for Lie group symmetries is generalized to discrete symmetries for the lattice Maxwell system. As a result,…
This article introduces two reduction schemes for Hamiltonian systems on an exact symplectic manifold admitting Lie group symmetries. It is demonstrated that these reduction procedures are equivalent by employing a modified…
Any symmetry reduces a second-order differential equation to a first-order equation: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion…