A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. Our main result establishes that both the Euler-Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the first prolongation of the original LagrangianAfter discussing certain features of the formulation, we introduce the so-called inherited constants of the motion and relate them to the Noether constants of the extended system.
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…
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 variational Lie derivative of classes of forms in the Krupka's variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application we determine the condition for a Noether--Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.
A relation between variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that for a system of differential equations in Eulerian variables corresponding Lagrangian description is related to introducing nonlocal variables. The connection between these descriptions is obtained in terms of differential coverings. The relation between variational principles of a system of equations and its symplectic structures is discussed. It is shown that if a system of equations in Lagrangian variables can be derived from a variational pr…
We present a variational formulation for the Navier-Stokes-Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in \cite{GBYo2016b} as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representatio…
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.
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and, in consequence, with a very good energy behavior. Additionally, the symmetries of the discrete Lagrangian imply that mome…
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.
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.
In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite dimensional case of discrete systems as well as for the infinite dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton's principle, we show, with the help of thermodynamic systems with gradually increasing level complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical fric…
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 …
Based on the works of Gordon ([4]) and Zhang-Zhou([8])) on the variational minimizing properties for Keplerian orbits and Lagrangian solutions of Newtonian 2-body and 3-body problems, we use the constrained variational principle of Ambrosetti-Coti Zelati ([1]) to compute the Lagrangian actions on Keplerian and Lagrangian elliptical solutions with fixed energies, we also find an interesting relationship between period and energy for Lagrangian elliptical solutions with Newtonian potentials.
Second gradient theories have to be used to capture how local micro heterogeneities macroscopically affect the behavior of a continuum. In this paper a configurational space for a solid matrix filled by an unknown amount of fluid is introduced. The Euler-Lagrange equations valid for second gradient poromechanics, generalizing those due to Biot, are deduced by means of a Lagrangian variational formulation. Starting from a generalized Clausius-Duhem inequality, valid in the framework of second gradient theories, the existence of a macroscopic solid skeleton Lagrangian deformation energy, dependi…
An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the actio…
A boundary value problem is commonly associated with constraints imposed on a system at its boundary. We advance here an alternative point of view treating the system as interacting "boundary" and "interior" subsystems. This view is implemented through a Lagrangian framework that allows to account for (i) a variety of forces including dissipative acting at the boundary; (ii) a multitude of features of interactions between the boundary and the interior fields when the boundary fields may differ from the boundary limit of the interior fields; (iii) detailed pictures of the energy distribution an…
By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa-Holm system (1). We show that the two-component Camassa-Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.
We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find that a covariant strongly conserved current is canonically associated with the deformed Lagrangian obtained by contracting Euler--Lagrange equations of the original Lagrangian with (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms ly…
Lagrangian formalism on graded manifolds is phrased in terms of the Grassmann-graded variational bicomplex, generalizing the familiar variational bicomplex for even Lagrangian systems on fiber bundles.
The non-standard Lagrangians (NSLs) for dissipative-like dynamical systems were introduced in an ad hoc fashion rather than being derived from the solution of the inverse problem of variational calculus. We begin with the first integral of the equation of motion and solve the associated inverse problem to obtain some of the existing results for NSLs. In addition, we provide a number of alternative Lagrangian representations. The case studies envisaged by us include (i) the usual modified Emden-type equation, (ii) Emden-type equation with dissipative term quadratic in velocity and (iii) Lokta-V…
It is shown that the Zakharov-Mihailov (ZM) Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure. We show that, as a consequence of this multiform structure, we can formulate a variational principle for the Lax pair itself, a problem that to our knowledge was never previously considered. As an example, we present an integrable $N\times N$ matrix system that contains the AKNS hierarchy, and we exhibit the Lagrangian multiform structure of the scalar AKNS hierarchy by presenting the components corresponding t…
We justify the Hamilton least action principle for the Maxwell-Lorentz equations with Abraham's rotating extended electron. The main novelty in the proof is application of the variational Poincare equations on the Lie group SO(3). The variational equations allow to derive the corresponding conservation laws from general Noether theory of invariants.
Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite element methods. Multisymplectic variational integrators are a class of geometric numerical integrators for Lagrangian and Hamiltonian field theories, and they yield methods that preserve the multisymplectic structure and momentum-conservation properties of the continuous system. In this paper, we investigate the synthesis of these two approaches, by constructin…
We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- comprising both the original coordinates and the compatible ``virtual displacements'' joining two solutions of the original system. The variational principle is Hamilton's with the new Lagrangian. We use this formulation to obtain constants of motion in the Jacobi equations of any L…
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework.
We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms we are able to obtain a Lorentz invariant set of equ…
Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vectors. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete ta…
In this paper, we introduce the notion of port-Lagrangian systems in nonequilibrium thermodynamics, which is constructed by generalizing the notion of port-Lagrangian systems for nonholonomic mechanics proposed in Yoshimura and Marsden [2006c], where the notion of interconnections is described in terms of Dirac structures. The notion of port-Lagrangian systems in nonequilibrium thermodynamics is deduced from the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,2017b]. It is a type of Lagrange-d'Alembert principle associated to a specific cla…
We present a variational approach which shows that the wave functions belonging to quantum systems in different potential landscapes, are pairwise linked to each other through a generalized continuity equation. This equation contains a source term proportional to the potential difference. In case the potential landscapes are related by a linear symmetry transformation in a finite domain of the embedding space, the derived continuity equation leads to generalized currents which are divergence free within this spatial domain. In a single spatial dimension these generalized currents are invariant…
We introduce an generalized action functional describing the equations of motion and the variational equations for any Lagrangian system. Using this novel scheme we are able to generalize Noether's theorem in such a way that to any $n$-parameter continuous symmetry group of the Lagrangian there exist 1) the usual $n$ constants of motion and 2) $n$ extra constants valid in the variational equations. The new constants are related to the infinitesimal generators of the symmetry transformation by relations similar to the ones that stem from the `nonextended' Noether theorem.
We consider the assimilation of Lagrangian data into a primitive equations circulation model of the ocean at basin scale. The Lagrangian data are positions of floats drifting at fixed depth. We aim at reconstructing the four-dimensional space-time circulation of the ocean. This problem is solved using the four-dimensional variational technique and the adjoint method. In this problem the control vector is chosen as being the initial state of the dynamical system. The observed variables, namely the positions of the floats, are expressed as a function of the control vector via a nonlinear observa…
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.
In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincaré--Cartan and Euler--Lagrange forms, momenta and momenta of generating forms, a new geometric object arising in variational sequences. Finally, we provide a simple but important example of special Lagrangian, namely the Hilbert--Einstein Lagrangian.
We study certain two-dimensional variational systems, namely pluri-Lagrangian systems on the root lattice $Q(A_{N})$. Here, we follow the scheme which was already used to define two-dimensional pluri-Lagrangian systems on the lattice $\mathbb{Z}^{N}$ and three-dimensional pluri-Lagrangian systems on the lattice $\mathbb{Z}^{N}$ as well as on $Q(A_{N})$. We will show that the two-dimensional pluri-Lagragian systems on $Q(A_{N})$ are more general than the ones on $\mathbb{Z}^{N}$, in the sense that they can encode several different pluri-Lagrangian systems on $\mathbb{Z}^{N}$. This also means th…
We propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler-Lagrange equations of a variational principle. The method is based on the invariantization of standard, non-invariant discrete Lagrangian functionals using equivariant moving frames. The invariant variational schemes are given by the Euler-Lagrange equations of the corresponding invariantized discrete Lagrangian functionals. We showcase this general method by constructing invariant variational schemes of ordinary differential equations that prese…
The Carathéodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrisic geometric operations applied to the well-known Poincaré--Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carathéodory equivalent of the Hilbert Lagrangian in general relativity is discussed.
In 1966, Arnold [1] showed that the Lagrangian flow of ideal incompressible fluids (described by Euler equations) coincide with the geodesic flow on the manifold of volume preserving diffeomorphisms of the fluid domain. Arnold's proof and the subsequent work on this topic rely heavily on the properties of Lie groups and Lie algebras which remain unfamiliar to most fluid dynamicists. In this note, we provide a simple derivation of Arnold's result which only uses the classical methods of calculus of variations. In particular, we show that the Lagrangian flow maps generated by the solutions of th…
In this paper we provide a variational derivation of the Euler-Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the previous construction to the formal derivation of the variational error for numerical integrators of forced Euler-Poincaré equations and the application of this theory to the derivation of geometric integrators for forced systems.
In this paper a new approach is proposed to quantize mechanical systems whose equations of motion can not be put into Hamiltonian form. This approach is based on a new type of variational principle, which is adopted to a describe a relation: a damping particle may shares a common phase curve with a free particle, whose Lagrangian in the new variational principle can be considered as a Lagrangian density in phase space. According to Feynman's theory, the least action principle is adopted to modify the Feynman's path integral formula, where Lagrangian is replaced by Lagrangian density. In the ca…
A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property holds for both continuous dynamical systems as well as for discrete ones defined in discrete space-time. Possibly the simplest example in the discrete case is that of a linear quadrilateral lattice equation, which can be viewed as a linearised version of the well-known lattice potential Korteweg-de Vries (KdV) equation. In spite of the linearity, the MDC pr…
We study higher--order variational derivatives of a generic second--order Lagrangian ${\cal L}={\cal L}(x,ϕ,\partialϕ,\partial^2ϕ)$ and in this context we discuss the Jacobi equation ensuing from the second variation of the action. We exhibit the different integrations by parts which may be performed to obtain the Jacobi equation and we show that there is a particular integration by parts which is invariant. We introduce two new Lagrangians, ${\cal L}_1$ and ${\cal L}_2$, associated to the first and second--order deformations of the original Lagrangian ${\cal L}_0$ respectively; they are in fa…
In this paper we have chosen to work with two different approaches to solving the inverse problem of the calculus of variation. The first approach is based on an integral representation of the Lagrangian function that uses the first integral of the equation of motion while the second one relies on a generalization of the well known Noether's theorem and constructs the Lagrangian directly from the equation of motion. As an application of the integral representation of the Lagrangian function we first provide some useful remarks for the Lagrangian of the modified Emden-type equation and then obt…
In the present work, we formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.
Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained.
Conformal geodesics are solutions to a system of third order of equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third--order conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth order ODEs arising from this Lagrangian, and show that some of its integral curves are spirals.
Geometric continuum models for fluid lipid membranes are considered using classical field theory, within a covariant variational approach. The approach is cast as a higher-derivative Lagrangian formulation of continuum classical field theory, and it can be seen as a covariant version of the field theoretical variational approach that uses the height representation. This novel Lagrangian formulation is presented first for a generic reparametrization invariant geometric model, deriving its equilibrium condition equation, or shape equation, and its linear and angular stress tensors, using the cla…
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental "pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the "pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl.…
