Morse families and constrained mechanical systems. Generalized hyperelastic materials (original) (raw)
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Nonholonomic constraints: A new viewpoint
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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. understand as the lagrangian submanifold Γ of the symplectic manifold (TT * M, d T ω M ) equipped with tangent lift d T ω M of the canonical symplectic form ω M of T * M . Here, M represents the configuration manifold of the system and Γ is obtained from the lagrangian submanifold dL(M ) ⊂ T * TM induced by the Lagrangian L : TM → R via the canonical isomorphism ε M : T * TM → TT * M . In other words, the phase dynamics, as well as the Euler-Lagrange equations, are obtained in a simple way by means of the Tulczyjew differential Λ L = ε M • dL : M → TT * M . It is important to observe that both TT * M and T * TM are double vector bundles over T * M and TM (see and references therein). The resulting submanifold Γ of TT * M is a particular case of modelling dynamical systems as implicit differential equations defined by differential inclusions (see ).
A new look at classical mechanics of constrained systems
Annales De L Institut Henri Poincare-physique Theorique, 1997
Nous presentons dans ce papier une formulation de la Mecanique Analytique qui permet de traiter des systemes non-holonomes. Cette etude preliminaire concerne la geometrie de l'espace des etats cinetiques du systeme. Cela nous a conduit a revoir la definition de Chetaev du travail virtuel, qui joue ici un role remarquable dans la realisation d'un modele mecanique compatible avec le principe de determinisme. L'ensemble de ce travail nous a permis de l'appliquer aux systemes non-holonomes ideaux, de comparer les formulations de D'Alembert et de Gauss et d'ecrire de facon explicite les equations de la dynamique.
Nonideal Constraints and Lagrangian Dynamics
Journal of Aerospace Engineering, 2000
This paper deals with mechanical systems subjected to a general class of non-ideal equality constraints. It provides the explicit equations of motion for such systems when subjected to such nonideal, holonomic and/or nonholonomic, constraints. It bases Lagrangian dynamics on a new and more general principle, of which D'Alembert's principle then becomes a special case applicable only when the constraints become ideal. By expanding its perview, it allows Lagrangian dynamics to be directly applicable to many situations of practical importance where non-ideal constraints arise, such as when there is sliding Coulomb friction.
Lagrangian systems with higher order constraints
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A class of mechanical systems subject to higher order constraints ͑i.e., constraints involving higher order derivatives of the position of the system͒ are studied. We call them higher order constrained systems ͑HOCSs͒. They include simplified models of elastic rolling bodies, and also the so-called generalized nonholonomic systems ͑GNHSs͒, whose constraints only involve the velocities of the system ͑i.e., first order derivatives in the position of the system͒. One of the features of this kind of systems is that D'Alembert's principle ͑or its nonlinear higher order generalization, the Chetaev's principle͒ is not necessarily satisfied. We present here, as another interesting example of HOCS, systems subjected to friction forces, showing that those forces can be encoded in a second order kinematic constraint. The main aim of the paper is to show that every HOCS is equivalent to a GNHS with linear constraints, in a canonical way. That is to say, systems with higher order constraints can be described in terms of one with linear constraints in velocities. We illustrate this fact with a system with friction and with Rocard's model ͓Dynamique Générale des Vibrations ͑1949͒, Chap. XV, p. 246 and L'instabilité en Mécanique; Automobiles, Avions, Ponts Suspendus ͑1954͔͒ of a pneumatic tire. As a by-product, we introduce some applications on higher order tangent bundles, which we expect to be useful for the study of intrinsic aspects of the geometry of such bundles.
On the variational mechanics with non-linear constraints
Journal de Mathématiques Pures et Appliquées, 2004
This paper concerns a geometric formulation of the so-called variational mechanics for mechanical systems with non-linear constraints. Given a smooth Lagrangian L on the tangent bundle of the configuration space M of the constrained mechanical system, its variational trajectories are defined, through a generalization of Hamilton's principle of stationary action, as extremals of the smooth Lagrangian functional γ → L(γ) defined on a convenient Banach manifold of curves compatible with the constraint manifold C ⊂ TM. In the particular case of a Lagrangian given by the positive definite quadratic form induced by a metric tensor on M, this amounts to a generalization of sub-Riemannian geometry. Among the main results, it is proven that, under a regularity condition on the Lagrangian L, the normal extremals of the Lagrangian functional are given by the projections on M of a Hamiltonian vector field defined on the generalized mixed bundle W.
Mechanical systems with nonlinear constraints
International Journal of Theoretical Physics, 1997
A geometrical formalism for nonlinear nonholonomic Lagrangian systems is developed. The solution of the constrained problem is discussed by using almost product structures along the constraint submanifold. Constrained systems with ideal constraints are also considered, and Chetaev conditions are given in geometrical terms. A Noether theorem is also proved.
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Numerical Algebra, Control and Optimization, 2013
This paper develops a new, simple, general, and explicit form of the equations of motion for general constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system whose mass matrix is positive definite and which is subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. A simple, unified fundamental equation that gives in closed-form both the acceleration of the constrained mechanical system and the constraint force is obtained. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics.
The general explicit equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of constrained motion where the forces of constraint may be ideal and/or non-ideal. The general problem of obtaining the equations of motion of a constrained discrete mechanical system is one of the central issues in analytical dynamics. While it was formulated at least as far back as Lagrange, the determination of the explicit equations of motion, even within the restricted compass of lagrangian dynamics, has been a major hurdle. The Lagrange multiplier method relies on problem specific approaches to the determination of the mul-tipliers which are often difficult to obtain for systems with a large number of degrees of freedom and many non-integrable constraints. Formulations offered by Gibbs, Volterra, Appell, Boltzmann, and Poincare require a felicitous choice of problem specific quasi-coordinates and suffer from similar problems in dealing with systems with large numbers of degrees of freedom and many 0096-3003/$-see front matter