Noether theorem applied to variational problems on hyperbolic surfaces.
problem Variational problems on hyperbolic surfaces.
method Noether's theorem on symmetry and conservation laws.
result Application to geometric problems on hyperbolic surfaces.
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48 results for “Noether symmetries”
The Noether theorem is extended to stochastic control problems using contact symmetries.
problem Stochastic optimal control problems.
method Exploiting jet bundles and contact geometry, the authors prove the existence of conserved quantities.
result Optimal control problems admit infinitely many conserved quantities in the form of local martingales.
Noether's framework reveals symmetry-breaking in neural networks.
problem Understanding the role of symmetry breaking in neural networks.
method Developed a theoretical framework using Lagrangian mechanics.
result Identified 'kinetic symmetry breaking' and its effect on learning dynamics.
We prove Noether's direct and inverse second theorems for Lagrangian systems on fiber bundles in the case of gauge symmetries depending on derivatives of dynamic variables of an arbitrary order. The appropriate notions of reducible gauge symmetries and Noether's identities are formulated, and their equivalence by means…
We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transpar…
Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian -symplectic formalismmath-ph
In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the -symplectic formalism, using the Frölicher-Nijenhuis formalism on the space of -velocities of the configuration manifold. For the case , it is well known that Cartan sy…
The paper analyzes symmetries of Vaidya-Bonner geodesics.
problem Investigating invariance properties of Vaidya-Bonner geodesics.
method Classification of Lie point symmetries and Noether symmetries, determination of optimal system of subalgebras.
result Determination of optimal system of subalgebras for Vaidya-Bonner geodesics.
Introduces a variational framework for indefinite Lagrangians with specific symmetries.
problem Handling indefinite Lagrangians with complex symmetries.
method Develops a variational setting for an indefinite Lagrangian with a specific Noether charge.
result Validates the existence of a variational setting for a broad class of Lagrangians.
Noether's theorem clarifies how symmetries in neural networks influence learning.
problem Understanding how symmetries in neural networks affect learning.
method Systematic study of symmetry interactions with learning algorithms using Noether's theorem.
result Symmetries impose restrictions on the optimization path, leading to conserved quantities.
New theorem links symmetries to first integrals in plasma physics.
problem Understanding the relationship between symmetries and first integrals in divergence-free fields.
method Developed a Noether-type Theorem reformulation for three-dimensional divergence-free vector fields.
result Converse of the Noether-type Theorem holds on the toroidal region, proving the existence of flux coordinates.
This work connects symmetries and conserved quantities in machine learning.
problem Improving machine learning models by learning conserved quantities.
method Using Noether's theorem, learn symmetries and conserved quantities directly from data.
result Correctly identifies conserved quantities and improves model performance.
We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to th…
Paper extends Noether's Theorem to nonholonomic systems, proving conserved momentum.
problem Link between symmetries and first integrals broken in nonholonomic systems.
method Constructive method proving conserved momentum under certain conditions.
result Conserved momentum map in nonholonomic systems, extending Noether's Theorem.
In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are …
Develops a reduction theory for covariant field theories with gauge symmetries.
problem Handling gauge symmetries in covariant field theories.
method Utilizes generalized principal connections and fiberwise action of Lie groups.
result Relates vertical reduced equations to the Noether theorem.
Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain condi…
In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the -dimensional optima…
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
An infinite sequence of commuting nonpolynomial contact symmetries of the two-dimensional minimal surface equation is constructed. Local and nonlocal conservation laws for -dimensional minimal area surface equation are obtained by using the Noether identity.
We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with independent and dependent variables ( systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation …
In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid …
Euler derived elastica equation using modern mathematical concepts.
problem Euler's original derivation of elastica equation has not been properly interpreted.
method Euler used Noether's theorem and the Goldstein-Petrich scheme.
result Euler's equation is the static modified KdV equation.
In this paper we will present Lagrangian and Hamiltonian -symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will o…
This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noe…
In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conse…
A framework for reducing PDEs by symmetry, preserving key structures.
problem Reducing PDEs while preserving geometric structures and symmetries.
method Systematic calculation of reduced forms for various geometric structures.
result Noether's theorem is inherited in reduced systems, preserving conservation laws.
A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in…
New Lagrangian approach for optimal control of second-order systems.
problem Optimal control of second-order differential equations derived from force-controlled Lagrangian systems.
method Proposes a new hyperregular control Lagrangian and control Hamiltonian, providing necessary optimality conditions.
result Defines an extended Tulczyjew's triple with controls and studies the relationship between Noether symmetries.
We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a diff…
Defines new canonical lifts for field theories, analyzing Klein-Gordon, Polyakov string, and Einstein-Cartan gravity.
problem Analyzing natural Noether symmetries and conserved quantities in field theories.
method Defining canonical lifts to study field theories and applying Noether's theorem.
result New geometrical interpretation of Virasoro constraint in string theory.
The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.
problem Understanding scaling symmetries and their impact on central configurations in symplectic geometry.
method Introducing conformally symplectic maps, conformally Hamiltonian systems, and generalized momentum maps.
result Relative equilibria of scaling symmetries are solutions to specific equations involving the conformal momentum map and primitive one-form.
Noether's Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler…
We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been studied using the Frenet-Serret frame. The Rotation Minimising frame has many a…
The study explores Hesse manifolds and their symmetries in multifield cosmological models.
problem Understanding symmetries in multifield cosmological models.
method Analyzes Hesse functions and their properties on Riemannian manifolds.
result Complete Hesse manifolds are characterized by their index and are hyperbolic.
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also con…
Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of…
A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-tri…
The paper explores how symmetries and noise in SGD influence parameter dynamics.
problem Understanding the dynamics of parameter updates in SGD with symmetries.
method Proved the existence of noise equilibria and showed their role in balancing gradient noise.
result Gradient noise creates a systematic motion of parameters to a unique fixed point, called noise equilibria.
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both th…
Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networksphysics.data-an
Understanding complex systems with their reduced model is one of the central roles in scientific activities. Although physics has greatly been developed with the physical insights of physicists, it is sometimes challenging to build a reduced model of such complex systems on the basis of insights alone. We propose a nov…
A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of are of superconformal simple type, and that the numerical invariants of 4-man…
Given a generic Lagrangian system, its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condi…
We formulate the variational problem for AdS gravity with Dirichlet boundary conditions and demonstrate that the covariant counterterms are necessary to make the variational problem well-posed. The holographic charges associated with asymptotic symmetries are then rederived via Noether's theorem and `covariant phase sp…
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and -algebras.
result Extension of conserved currents to spacetime vector fields.
Following work by I. Anderson, in this note we present a formulation of Noether's Second Theorem that is valid on any natural bundle.
This paper investigates Lie Quandles and Leibniz Racks, extending Noether's first theorem.
problem Classifying and understanding Lie Quandles and Leibniz Racks.
method Investigates linear/nonlinear correspondences and classifies generalizations.
result Describes a nonlinear analogue of Noether's first theorem.
We present constructions of simply connected symplectic 4-manifolds which have (up to sign) one basic class and which fill up the geographical region between the half-Noether and Noether lines.