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Journal of Geometric Mechanics

June 2020 , Volume 12 , Issue 2

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Invariant structures on Lie groups
Javier Pérez Álvarez
2020, 12(2): 141-148 doi: 10.3934/jgm.2020007 +[Abstract](405) +[HTML](238) +[PDF](270.72KB)

We approach with geometrical tools the contactization and symplectization of filiform structures and define Hamiltonian structures and momentum mappings on Lie groups.

Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures
Hassan Najafi Alishah
2020, 12(2): 149-164 doi: 10.3934/jgm.2020008 +[Abstract](405) +[HTML](212) +[PDF](368.78KB)

We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a time re-parametrization, is provided using Dirac\begin{document}$ \backslash $\end{document}big-isotropic structures. Using the equivalence between replicator and Lotka-Volterra (LV) equations, the set of conservative LV equations is enlarged. Our approach generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system. In the case of predator-prey model, our method does allow interaction between different predators and between different preys.

Nonholonomic and constrained variational mechanics
Andrew D. Lewis
2020, 12(2): 165-308 doi: 10.3934/jgm.2020013 +[Abstract](282) +[HTML](103) +[PDF](1063.2KB)

Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.

The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.

Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle.

A note on Hybrid Routh reduction for time-dependent Lagrangian systems
Leonardo J. Colombo, María Emma Eyrea Irazú and Eduardo García-Toraño Andrés
2020, 12(2): 309-321 doi: 10.3934/jgm.2020014 +[Abstract](153) +[HTML](64) +[PDF](401.58KB)

This note discusses Routh reduction for hybrid time-dependent mechanical systems. We give general conditions on whether it is possible to reduce by symmetries a hybrid time-dependent Lagrangian system extending and unifying previous results for continuous-time systems. We illustrate the applicability of the method using the example of a billiard with moving walls.

2019  Impact Factor: 0.649



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