Journal of Geometric Mechanics
March 2015 , Volume 7 , Issue 1
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The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems, including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus, we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrarily high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.
We prove that a fixed configuration of $N-1$ masses in the plane can be extended to a central configuration of $N$ masses by adding a specified additional mass only in finitely many ways. This holds for a family of potential functions including the Newtonian gravitational case and the classical planar point vortex model.
In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$. Moreover, we show that any $D-$preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$. Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints.
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.
We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.
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