# American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

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## Journal of Geometric Mechanics

March 2020 , Volume 12 , Issue 1

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2020, 12(1): ⅰ-ⅱ doi: 10.3934/jgm.2020001 +[Abstract](88) +[HTML](52) +[PDF](235.95KB)
Abstract:
2020, 12(1): 1-23 doi: 10.3934/jgm.2020002 +[Abstract](98) +[HTML](36) +[PDF](460.27KB)
Abstract:

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of \begin{document}$k$\end{document}-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of \begin{document}$k$\end{document}-precosymplectic structure, which is a generalization of the \begin{document}$k$\end{document}-cosymplectic structure. Next \begin{document}$k$\end{document}-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

2020, 12(1): 25-52 doi: 10.3934/jgm.2020003 +[Abstract](64) +[HTML](33) +[PDF](989.99KB)
Abstract:

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.

2020, 12(1): 53-84 doi: 10.3934/jgm.2020004 +[Abstract](72) +[HTML](33) +[PDF](1597.76KB)
Abstract:

We compare a classical non-holonomic system—a sphere rolling against the inner surface of a vertical cylinder under gravity—with certain discrete dynamical systems called no-slip billiards in similar configurations. A feature of the former is that its height function is bounded and oscillates harmonically up and down. We investigate whether similar bounded behavior is observed in the no-slip billiard counterpart. For circular cylinders in dimension \begin{document}$3$\end{document}, no-slip billiards indeed have bounded orbits, and very closely approximate rolling motion, for a class of initial conditions we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to overall downward acceleration. Concerning different cross-sections, we show that no-slip billiards between two parallel hyperplanes in arbitrary dimensions are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept relying on a factorization of the motion into transversal and longitudinal components) has period two, the motion, under no forces, is generically not bounded. This commonly occurs in planar no-slip billiards.

2020, 12(1): 85-106 doi: 10.3934/jgm.2020005 +[Abstract](66) +[HTML](36) +[PDF](571.97KB)
Abstract:

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory.

2020, 12(1): 107-140 doi: 10.3934/jgm.2020006 +[Abstract](91) +[HTML](39) +[PDF](703.04KB)
Abstract:

In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface \begin{document}$\Sigma$\end{document}. When \begin{document}$\Sigma$\end{document} has intrinsic curvature of finite Lipschitz norm, we prove the existence of global strong solutions to the Cauchy problem of the Boussinesq equations with full or partial dissipations. The issues of uniqueness and singular limits (vanishing viscosity/vanishing thermal diffusivity) are also addressed. In addition, we establish a breakdown criterion for the strong solutions for the case of zero viscosity and zero thermal diffusivity. These appear to be among the first results for Boussinesq systems on Riemannian manifolds.

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