General Lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered. Its non-trivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second Noether theorem. In contrast with gauge symmetries, non-trivial Noether and higher-stage Noether identities of Lagrangian theory can be intrinsically defined by constructing the exact Koszul-Tate complex. The inverse second Noether theorem that we prove associates to this complex the cochain sequence with the ascent operator whose components define non-trivial gauge and higher-stage gauge symmetries. These gauge symmetries are said to be algebraically closed if the ascent operator can be extended to a nilpotent operator. The necessary conditions for this extension are stated. The characteristic examples of Yang-Mills supergauge theory, topological Chern-Simons theory, gauge gravitation theory and topological BF theory are presented.
We prove twist positivity and positivity of the pair correlation function for combined spatial and internal symmetries of free bosonic Lagrangians. We work in a general setting, extending the results obtained in Twist Positivity [1].
Given a finite order Lagrangian L on a fibre bundle, its global generalized symmetries depending on higher order derivatives of dynamic variables are considered. The first variational formula is obtained. It leads both to the corresponding Lagarngian conservation laws and the characteristic equation for generalized symmetries of L.
By generalizing the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first order jet bundles $J^1π$ of a fiber bundle $π:E\to {\mathbb R}^k$ where ${\mathbb R}^k$ is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether Theorem is proved.
This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.
We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with $m$ commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the $(m+1)$-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with $m$ commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpar…
The investigation of approximate symmetries of reparametrization invariant Lagrangians of n + 1 degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the…
By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether's theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, we provide a systematic method for constructing Lagrangian multiforms for which the closure property and the multiform Euler-Lagrange (EL) equations both hold. We present three examples, including the first known example of a Lagrangian 3-form: a multiform for the Kadomtsev-Petviashvili equation. We also present a new proof of the multiform EL equations for a Lagrangian k-form for arbi…
After a brief survey of the definition and the properties of Lambda-symmetries in the general context of dynamical systems, the notion of "Lambda-constant of motion'' for Hamiltonian equations is introduced. If the Hamiltonian problem is derived from a Lambda-invariant Lagrangian, it is shown how the Lagrangian Lambda-invariance can be transferred into the Hamiltonian context and shown that the Hamiltonian equations turn out to be Lambda-symmetric. Finally, the "partial'' (Lagrangian) reduction of the Euler-Lagrange equations is compared with the reduction obtained for the corresponding Hamilt…
In the late 80s - early 90s J. Moser and A. P. Veselov considered Lagrangian discrete systems on Lie groups with additional symmetry conditions imposed on Lagrangians. They observed that such systems are often integrable time-discretizations of integrable Euler equations on these Lie groups. In recent papers we studied Lagrangian discrete systems with additional symmetry requirements on certain infinite-dimensional Lie groups. We will discuss some interesting properties of these systems.
The one-dimensional shallow water equations in Eulerian and Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian (potential) coordinates and symmetries and conservation laws in mass Lagrangian variables. For equations in Lagrangian coordinates with a flat bottom an invariant difference scheme is constructed which possesses all the difference analogues of the conservation laws: mass, momentum, energy, the law of center of mass motion. Some exact invariant solutions are constructed for the invariant scheme, while the scheme ad…
Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approx…
We analyse the complex-valued Klein-Gordon Equation from an integrability perspective by the implementation of the Lie Theory of Continuous Groups, where this equation is governed by power-law nonlinearity. We write the equations in terms of real dependent variables and examine both the two-dimensional and three-dimensional models. In the case of the two-dimensional model the analysis is extended by the determination of Noether point symmetries and conserved vectors via the construction of a Lagrangian.
We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schrödinger equation is the key that takes classical mechanics into quantu…
Generalized symmetries and supersymmetries depending on derivatives of dynamic variables are treated in a most general setting. Studding cohomology of the variational bicomplex, we state the first variational formula and conservation laws for Lagrangian systems on fiber bundles and graded manifolds under generalized symmetries and supersymmetries of any order. Cohomology of nilpotent generalized supersymmetries are obtained.
This letter investigates the Lie point symmetries and conserved quantities of the Lagrangian systems on time scales, which unify the Lie symmetries of the two cases for the continuous and the discrete Lagrangian systems. By defining the infinitesimal transformations' generators and using the invariance of differential equations under infinitesimal transformations, the determining equations of the Lie symmetries on time scales are established. Then the structure equations and the form of conserved quantities with delta derivatives are obtained. The letter also gives brief discussion on the Lie …
Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches…
We present a new method based on Lie symmetries and Jacobi last multipliers which allows one to find many non-standard Lagrangians for dissipative dynamical systems. In particular, it is demonstrated that for every non-standard Lagrangian one can generate a new non-standard Lagrangian associated to a new equation of motion. We point out that the knowledge of Lie symmetries for a given dynamical system generates Jacobi last multipliers which can be used to obtain new non-standard Lagrangians for dissipative dynamical systems in a simple and straightforward way. We exemplify the new method by ap…
In this paper we will present Lagrangian and Hamiltonian $k$-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 obtain new kinds of conservation laws for $k$-symplectic Hamiltonian systems and $k$-symplectic Lagrangian systems.
We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green-Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries,…
We prove a theorem concerning the Noether symmetries for the area minimizing Lagrangian under the constraint of a constant volume in an n-dimensional Riemannian space. We illustrate the application of the theorem by a number of examples.
It is shown that an arbitrary singular Lagrangian theory (with first and second class constraints up to $N$-th stage in the Hamiltonian formulation) can be reformulated as a theory with at most third-stage constraints. The corresponding Lagrangian $\tilde L$ can be obtained by pure algebraic methods, its manifest form in terms of quantities of the initial formulation is found. Local symmetries of $\tilde L$ are obtained in closed form. All the first class constraints of the initial Lagrangian turn out to be gauge symmetry generators for $\tilde L$.
We analyze the relation of the notion of pluri-Lagrangian systems, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether.
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 dynamics within the Lagrangian and the Hamilton-Jacobi formalism.
New symmetry transformations for the n-dimensional Toda lattice are presented. Their existence allows for the construction of several first order Lagrangian structures associated to them. The multi-Hamiltonian structures are derived from Lagrangians in detail. The set of symmetries generates a Lie algebra.
A Lagrangian formalism for variational second-order delay ordinary differential equations (DODEs) is developed. The Noether operator identity for a DODE is established, which relates the invariance of a Lagrangian function with the appropriate variational equations and the conserved quantities. The identity is used to formulate Noether-type theorems that give the first integrals for DODE with symmetries. Relations between the invariance of the variational second-order DODEs and the invariance of the Lagrangian functions are also analyzed. Several examples illustrate the theoretical results.
We investigate the relation between pluri-Lagrangian hierarchies of $2$-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys. 24:sup1, 121--145 (2017)] for ordinary differential equations. We consider hierarchies of $2$-dimensional Lagrangian PDEs (many of which have a natural $(1+1)$-dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian …
Motivated by BRST theory, we study generalized symmetries and supersymmetries depending on derivatives of dynamic variables in a most general setting. We state the first variational formula and conservation laws for higher order Lagrangian systems on fiber bundles and graded manifolds under generalized symmetries and supersymmetries of any order. Cohomology of nilpotent generalized supersymmetries are considered.
Making use of the Lagrange anchor construction introduced earlier to quantize non-Lagrangian field theories, we extend the Noether theorem beyond the class of variational dynamics.
One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain…
We show that if a Lagrangian is invariant under a transformation (with the invariance defined in the standard manner), then the equations of motion obtained from it maintain their form under the transformation. We also show that the converse is not true, giving examples of equations of motion that are form-invariant under a transformation, but these equations can be derived from a Lagrangian that is not invariant under such transformation. The conclusions are valid for discrete or continuous systems.
We propose a procedure which allows one to construct local symmetry generators of general quadratic Lagrangian theory. Manifest recurrence relations for generators in terms of so-called structure matrices of the Dirac formalism are obtained. The procedure fulfilled in terms of initial variables of the theory, and do not implies either separation of constraints on first and second class subsets or any other choice of basis for constraints.
Lagrangian systems with nonholonomic constraints may be considered as singular differential equations defined by some constraints and some multipliers. The geometry, solutions, symmetries and constants of motion of such equations are studied within the framework of linearly singular differential equations. Some examples are given; in particular the well-known singular lagrangian of the relativistic particle, which with the nonholonomic constraint v^2=c^2 yields a regular system.
We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying their equivalence. We define several kinds of symmetries for contact dynamical systems, as well as the notion of dissipation laws, prove a dissipation theorem and give a way to construct conserved quantities. Some well-known examples of dissipative systems are discussed.
This paper analyses the Noether symmetries and the corresponding conservation laws for Chern-Simons Lagrangians in dimension $d=3$. In particular, we find an expression for the superpotential of Chern-Simons gravity. As a by-product the general discussion of superpotentials for 3rd order natural and quasi-natural theories is also given.
The language of Lagrangian submanifolds is used to extend a geometric characterization of the inverse problem of the calculus of variations on tangent bundles to regular Lie algebroids. Since not all closed sections are locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are necessary but not sufficient, so they give a weaker definition of the inverse problem. As an application the Helmholtz conditions on Atiyah algebroids are obtained so that the relationship between the inverse problem and the reduced inverse problem by symmetries can be described. Some examples and c…
We study the reduction of non-autonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system $Q\times\real\to\real$. These kind of symmetries generalize the idea of ``time-invariance'' (which corresponds to taking the trivial connection in the above trivial bundle).
In their simplest form, metric-like Lagrangians for higher-spin massless fields display constrained gauge symmetries, unless auxiliary fields are introduced or locality is foregone. Specifically, in its standard incarnation, gauge invariance of Maxwell-like Lagrangians relies on parameters with vanishing divergence. We propose an alternative form of the corresponding local symmetry involving unconstrained parameters of mixed-symmetry type, described by rectangular two-row Young diagrams and entering high-derivative gauge transformations. The resulting gauge algebra appears to be reducible and …
In this paper we address the problem of identifying contracting systems among dynamical systems appearing in mechanics. First, we introduce a sufficient condition to identify contracting systems in a general Riemannian manifold. Then, we apply this technique to establish that a particular type of dissipative forced mechanical system is contracting, while stating immediate consequences of this fact for the stability of these systems. Finally, we use the previous results to study the stability of particular types of Contact and Nonholonomic systems.
Noether theorem establishes an interesting connection between symmetries of the action integral and conservation laws of a dynamical system. The aim of the present work is to classify the damped harmonic oscillator problem with respect to Noether symmetries and to construct corresponding conservation laws for all over-damped, under damped and critical damped cases. For each case we obtain maximum five linearly independent group generators which provide related five conserved quantities. Remarkably, after obtaining complete set of invariant quantities we obtain analytical solutions for each cas…
We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagr…
We analyze a structure of the singular Lagrangian $L$ with first and second class constraints of an arbitrary stage. We show that there exist an equivalent Lagrangian (called the extended Lagrangian $\tilde L$) that generates all the original constraints on second stage of the Dirac-Bergmann procedure. The extended Lagrangian is obtained in closed form through the initial one. The formalism implies an extension of the original configuration space by auxiliary variables. Some of them are identified with gauge fields supplying local symmetries of $\tilde L$. As an application of the formalism, w…
We give a geometrical interpretation of the notion of $μ$-prolongations of vector fields and of the related concept of $μ$-symmetry for partial differential equations (extending to PDEs the notion of $λ$-symmetry for ODEs). We give in particular a result concerning the relationship between $μ$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $μ$-symmetries and nonlocal standard symmetries.
For a theory with first and second class constraints, we propose a procedure for conversion of second class constraints based on deformation the structure of local symmetries of the Lagrangian formulation. It does not require extension or reduction of configuration space of the theory. We give examples in which the initial formulation implies a non linear realization of some global symmetries, therefore is not convenient. The conversion reveals hidden symmetry presented in the theory. The extra gauge freedom of conversed version is used to search for a parameterization which linearizes the equ…
Combining Symmetries and Helmholtz's Conditions to Construct Lagrangians
Center for Research and Advanced Studies（高级研究中心）；National Polytechnic Institute（国立理工学院）；Higher School of Mechanical and Electrical Engineering, National Polytechnic Institute（国立理工学院高等机械电气工程学院）
Constants of motion, Lagrangians and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained b…
In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the $k$-symplectic formalism, using the Frölicher-Nijenhuis formalism on the space of $k^1$-velocities of the configuration manifold. For the case $k=1$, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's Theorem and its converse. For the case $k>1$, we provide a new proof for Noether's Theorem, which shows that, in the $k$-symplectic formalism, each Cartan symmetry induces a conserva…
Continuing the analysis in a unified scheme for treating generalized superselection sectors based upon the notion of selection criteria for states of relevance in quantum physics, we extend the Doplicher-Roberts superselection theory for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the basic structural features of the theory with spontaneously broken symmetry, are clarified in a clear-cut way, such as the degenerate vacua parametrized by …
This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\T^EE^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
We prove that, given a time-independent Lagrangian defined in the first tangent bundle of configuration space, every infinitesimal Noether symmetry that is defined in the $n$-tangent bundle and is not vanishing on-shell, can be written as a canonical symmetry in an enlarged phase space, up to constraints that vanish on-shell. The proof is performed by the implementation of a change of variables from the the $n$-tangent bundle of the Lagrangian theory to an extension of the Hamiltonian formalism which is particularly suited for the case when the Lagrangian is singular. This result proves the as…
We present Noether's second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether's second theorem for Lagrangian systems on fiber bundles over X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved.
Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always po…
We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lema\^ıtre-Robertson-Walker spacetime is a well-known result in the literature.
Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.
The investigation of contact symmetries of re--parametrization invariant Lagrangians of finite degrees of freedom and quadratic in the velocities is presented. The main concern of the paper is those symmetry generators which depend linearly in the velocities. A natural extension of the symmetry generator along the lapse function $N(t)$, with the appropriate extension of the dependence in $\dot{N}(t)$ of the gauge function, is assumed; this action yields new results. The central finding is that the integrals of motion are either linear or quadratic in velocities and are generated, respectively …
