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Journal of Geometric Mechanics

September 2019 , Volume 11 , Issue 3

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Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies
Víctor Manuel Jiménez Morales, Manuel De León and Marcelo Epstein
2019, 11(3): 301-324 doi: 10.3934/jgm.2019017 +[Abstract](253) +[HTML](129) +[PDF](462.58KB)
Abstract:

A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of \begin{document}$ G- $\end{document}structures is discussed in detail. An illustrative example is presented as an application of the theory.

A symmetry-adapted numerical scheme for SDEs
Francesco C. De Vecchi, Andrea Romano and Stefania Ugolini
2019, 11(3): 325-359 doi: 10.3934/jgm.2019018 +[Abstract](224) +[HTML](110) +[PDF](1008.07KB)
Abstract:

We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE puts in evidence notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established.

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity
Jordi Gaset and Narciso Román-Roy
2019, 11(3): 361-396 doi: 10.3934/jgm.2019019 +[Abstract](259) +[HTML](143) +[PDF](572.45KB)
Abstract:

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

Self-organization on Riemannian manifolds
Razvan C. Fetecau and Beril Zhang
2019, 11(3): 397-426 doi: 10.3934/jgm.2019020 +[Abstract](247) +[HTML](107) +[PDF](2252.82KB)
Abstract:

We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials and an application of the model to aggregation on the rotation group \begin{document}$ SO(3) $\end{document} are also presented.

Relative periodic solutions of the $ n $-vortex problem on the sphere
Carlos García-Azpeitia
2019, 11(3): 427-438 doi: 10.3934/jgm.2019021 +[Abstract](179) +[HTML](108) +[PDF](303.46KB)
Abstract:

This paper gives an analysis of the movement of \begin{document}$ n\ $\end{document}vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.

Improving E. Cartan considerations on the invariance of nonholonomic mechanics
Waldyr M. Oliva and Gláucio Terra
2019, 11(3): 439-446 doi: 10.3934/jgm.2019022 +[Abstract](256) +[HTML](138) +[PDF](311.21KB)
Abstract:

This paper concerns an intrinsic formulation of nonholonomic mechanics. Our point of departure is the paper [6], by Koiller et al., revisiting E. Cartan's address at the International Congress of Mathematics held in 1928 at Bologna, Italy ([3]). Two notions of equivalence for nonholonomic mechanical systems \begin{document}$ ( {\mathsf{{M}}}, {{\mathsf{{g}}}}, {\mathscr{D}}) $\end{document} are introduced and studied. According to [6], the notions of equivalence considered in this paper coincide. A counterexample is presented here showing that this coincidence is not always true.

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