ISSN:

1941-4889

eISSN:

1941-4897

## Journal of Geometric Mechanics

March 2020 , Volume 12 , Issue 1

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**Abstract:**

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of

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**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.

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**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 *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.

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**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.

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**Abstract:**

In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface

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