The Lagrange--Poincaré equations for the mechanical system describing the motion of a scalar particle on a Riemannian manifold with a given free and isometric action of a compact Lie group is obtained. In an arising principle fibre bundle, the total space of which serves as a configuration space of the considered mechanical system, the local description of the reduced motion is done in terms of dependent coordinates. In obtaining of the equations we use the variational principle developed by Poincaré for the mechanical systems with a symmetry.
A special case of the Lagrange-Poincaré equations for the gauge field interacting with a scalar field is obtained. For description of the dynamics on the configuration space, the adapted coordinates are used. After neglecting the group variables the obtained equations describe the evolution on the gauge orbit space of the principal fiber bundle which is related to the system under the consideration.
Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincaré and Lie-Poisson reduction is also studied in detail.
We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the higher-order Lagrange-Poincaré equations of classical mechanical systems with higher-order constraints are obtained from classical Lagrangian reduction. Higher-order Lagrange-Poincaré operator is introduced to characterize higher-order Lagrange-Poincaré equations. Interesting applications are derived as, for instance, the optimal control of an underactuated E…
This note is a continuation of our earlier articles arXiv:1612.08897 and arXiv:1709.09030, where using the dependent coordinates the local Lagrange-Poincaré equations were obtained for a mechanical system with symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and vector space), on which a free proper and isometric action of a compact semisimple Lie group is given. Assuming the existence of the parametric representations for local sections in the principal bundle, we make the transiti…
The Lagrange--Poincaré equations for a mechanical system which describes the interaction of two scalar particles that move on a special Riemannian manifold, consisting of the product of two manifolds, the total space of a principal fiber bundle and the vector space, are obtained. The derivation of equations is performed by using the variational principle developed by Poincaré for the mechanical systems with a symmetry. The obtained equations are written in terms of the dependent variables which, as in gauge theories, are implicitly determined by means of equations representing the local sectio…
We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory.
The Lagrange-Poincare equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin-Noether theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, metamorphosis image dynamics, and molecular strands illustrate various aspects of the theory.
Using the dependent coordinates, the local Lagrange-Poincaré equations and equations for the relative equilibria are obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space) on which a free proper and isometric action of a compact semi-simple Lie group is given. As in gauge theories, dependent coordinates are implicitly determined by means of equations representing the local sections of the principal fiber bundle.
In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of Lagrange-Poincare reduction. We model the combined fluid-structure system as a geodesic curve on the total space of a principal bundle on which a diffeomorphism group acts. After reduction by the diffeomorphism group we obtain the fluid-structure interactions where the fluid evolves by the inviscid fluid equations. Along the way, we describe various geometric structures appearing in fluid-structure interactions: principal connections, Lie groupoids, Lie algebroids, etc. We…
A Poincaré covariant Lagrange anchor is found for the non-Lagrangian relativistic wave equations of Bargmann and Wigner describing free massless fields of spin $s>1/2$ in four-dimensional Minkowski space. By making use of this Lagrange anchor, we assign a symmetry to each conservation law and perform the path-integral quantization of the theory.
A Poincare invariant Lagrange anchor is found for the non-Lagrangian relativistic wave equations of Bargmann and Wigner describing free massless fields of spin s > 1/2 in four-dimensional Minkowski space. By making use of this Lagrange anchor, we assign a symmetry to each conservation law.
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with distinguished curves in a Lagrangian submanifold of a symplectic manifold. For this purpose, we will combine the concept of a Tulczyjew triple with Marsden-Weinstein symplectic reduction.
We study the following functional equation that has arisen in the context of mechanical systems invariant under the Poincare algebra: \sum\limits_{i=1}^{n+1}\dfrac{\partial}{\partial x_{i}}\prod\limits_{j\neq i}f(x_{i}-x_{j}) =0,\qquad n \geq 2. New techniques are developed and the general solution within a certain class of functions is given. New solutions are found.
We study the groups of local BRST cohomology associated to the general systems of ordinary differential equations, not necessarily Lagrangian or Hamiltonian. Starting with the involutive normal form of the equations, we explicitly compute certain cohomology groups having clear physical meaning. These include the groups of global symmetries, conservation laws and Lagrange structures. It is shown that the space of integrable Lagrange structures is naturally isomorphic to the space of weak Poisson brackets. The last fact allows one to establish a direct link between the path-integral quantization…
The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include as particular cases the classic…
Based on the d'Alembert-Lagrange-Poincaré variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincaré-Hamilton equations, and study a version of corresponding Poincaré-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincaré variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain lin…
A solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is the target of this paper. We present the mechanical systems called by use, mechanical (?; ?)-systems, Lagrange mechanical (?; ?)-systems or Finsler mechanical (?; ?)-systems and we develop their geometries. We obtain the canonical (?; ?)-semi(spray) associated to a mechanical (?; ?)-system. The Lagrange mechanical (?; ?)-systems are the spaces necessary to develop a Lagrangian formalism. We obtain the (?; ?)-semispray associated to a regular Lagrangian L and external force Fe and we derive the equa…
\emph{Mechanical systems} called by use, \emph{mechanical}$\left(ρ,η\right) $\emph{-systems, Lagrange mechanical}$\left(ρ,η\right) $\emph{-systems} or \emph{Finsler mechanical}$\left(ρ,η\right) $\emph{-systems} are presented. The canonical $\left(ρ,η\right) $\emph{-}semi(spray) associated to a mechanical $\left(ρ,η\right) $-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical $(ρ,η)$% -systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the $(ρ,η)$-semispray associated to a regular Lagrangian …
The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincaré and Lagrange Poincaré type equations is studied. Reduction and reconstruction results for such systems are established.
In this study, we present a new analogue of Euler-Lagrange and Hamilton equations on an almost Kähler model of a Finsler manifold. Also, we give some corollories about the related mechanical systems and equations.
We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kaehler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange- and Hamilton-Fedosov manifolds.
We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincaré-Cartan 1-form of a …
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\R$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The cor…
A mechanical covariant equation is introduced which retains all the effectingness of the Lagrange equation while being able to describe in a unified way other phenomena including friction, non-holonomic constraints and energy radiation (Lorentz-Abraham-Dirac force equation). A quantization rule adapted to the dissipative degrees of freedom is proposed which does not pass through the variational formulation.
We review the construction and applications of exactly Poincaré invariant quantum mechanical models of few-degree of freedom systems. We discuss the construction of dynamical representations of the Poincaré group on few-particle Hilbert spaces, the relation to quantum field theory, the formulation of cluster properties, and practical considerations related to the construction of realistic interactions and the solution of the dynamical equations. Selected applications illustrate the utility of this approach.
We formulate Euler-Poincaré and Lagrange-Poincaré equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. The geometric construction applies to order parameter spaces consisting of either unsigned unit vectors (directors) or symmetric matrices (alignment tensors). On the Hamiltonian side, we provide the corresponding Poisson brackets in both Lie-Poisson and Hamilton-Poincaré formulations. The explicit form of the helicity invariant for uniaxial nem…
Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-…
We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructio…
We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.
This paper develops the theory of discrete Dirac reduction of discrete Lagrange-Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange-Pontryagin principle, and show that they both lead to the same discrete Lagrange-Poincaré-Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last,…
We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincaré canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the…
In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{é}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{é}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.
In this paper we study a Hamiltonization procedure for mechanical systems with velocity-depending (nonholonomic) constraints. We first rewrite the nonholonomic equations of motion as Euler-Lagrange equations, with a Lagrangian that follows from rephrasing the issue in terms of the inverse problem of Lagrangian mechanics. Second, the Legendre transformation transforms the Lagrangian in the sought-for Hamiltonian. As an application, we compare some variational integrators for the new Lagrangians with some known nonholonomic integrators.
Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the complexity of the problems. However, these reduction techniques, used without care for general problems (waves, field theory), may lead to uncomfortable infinite dimensional spaces. As an alternative, the \emph{covariant} formulation allows to consider a finite dimensional configuration space by increasing the number of independent variables. But the geometric eleme…
In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combination of quadrics having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and the hyperbolic space by means of central projection. With the same approach, we also extend these results to…
We describe a recipe to generate "nonlocal" constants of motion for ODE Lagrangian systems. As a sample application, we recall a nonlocal constant of motion for dissipative mechanical systems, from which we can deduce global existence and estimates of solutions under fairly general assumptions. Then we review a generalization to Euler-Lagrange ODEs of order higher than two, leading to first integrals for the Pais-Uhlenbeck oscillator and other systems. Future developments may include adaptations of the theory to Euler-Lagrange PDEs.
Being based on V. Konoplev's axiomatic approach to continuum mechanics, the paper broadens its frontiers in order to bring together continuum mechanics with classical mechanics in a new theory of mechanical systems. There are derived motion equations of `abstract' mechanical systems specified for mass-points, multibody systems and continua: Newton-Euler equations, Lagrange equations of II kind and Navier-Stokes ones. Quasi-linear constitutive equations are introduced in conformity with V. Konoplev's definition of stress and strain (rate) matrices.
The purpose of this paper is to show that, at least for Lagrangians of mechanical type, nonholonomic Euler-Lagrange equations for a nonholonomic linear constraint D may be viewed as non-constrained Euler-Lagrange equations but on a new (generally not Lie) algebroid structure on D. The proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of Neoether Theorem producing actual first integrals in case of symmetries.
We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler-Lagrange equations and the physical invariants of the motion associated with the Poincaré symmetry of Minkowski space, the mass and the spin of the particle, are expressed in a simple way in terms of these curvatures. The simplest non-trivial model of this form, with the lagrangian depending on the first FS (or geodesic) curvature, is i…
Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adap…
The aim of this paper is to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them, here we consider the generalization of the Hamiltonian principle for nonholonomic systems with nonzero transpositional relations. By applying this variational principle which takes into the account transpositional relations different from the classical ones we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. The…
We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a nonlinear connection.
This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincaré-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory …
We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincaré-Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincaré-Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular the Kepler problem. Finally, we prove a variant of the theorem on complete (non-commutative) integrability in terms of Noether symmetries of time-dependent Hamiltonian systems.
We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a k-connection to provide a reconstruction method. We show …
