ISSN:

1941-4889

eISSN:

1941-4897

## Journal of Geometric Mechanics

September 2019 , Volume 11 , Issue 3

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**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

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**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.

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**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.

*+*[Abstract](247)

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**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

*+*[Abstract](179)

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**Abstract:**

This paper gives an analysis of the movement of

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**Abstract:**

This paper concerns an intrinsic formulation of *nonholonomic mechanics*. Our point of departure is the paper [

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