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Journal of Geometric Mechanics

2012 , Volume 4 , Issue 1

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Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds
Cédric M. Campos , Elisa Guzmán and  Juan Carlos Marrero
2012, 4(1): 1-26 doi: 10.3934/jgm.2012.4.1 +[Abstract](26) +[PDF](526.1KB)
A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold.
Homogeneity and projective equivalence of differential equation fields
Mike Crampin and  David Saunders
2012, 4(1): 27-47 doi: 10.3934/jgm.2012.4.27 +[Abstract](21) +[PDF](413.8KB)
We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths.
Variational reduction of Lagrangian systems with general constraints
Sergio Grillo and  Marcela Zuccalli
2012, 4(1): 49-88 doi: 10.3934/jgm.2012.4.49 +[Abstract](34) +[PDF](635.4KB)
In this paper we present an alternative procedure for reducing, in the Lagrangian formalism, the equations of motion of first order constrained mechanical systems with symmetry. The procedure involves two principal connections: one of them is used to define the reduced degrees of freedom and the other one to decompose variations into horizontal and vertical components. On the one hand, we show that this new procedure is particularly useful when the configuration space is a trivial principal bundle over the symmetry group, which is the case of many interesting examples. On the other hand, based on that procedure, we extend in a natural way the variational reduction methods to the Lagrangian systems with higher order constraints. Examples are discussed in order to illustrate the involved theorethical constructions.
Stable closed equilibria for anisotropic surface energies: Surfaces with edges
Bennett Palmer
2012, 4(1): 89-97 doi: 10.3934/jgm.2012.4.89 +[Abstract](23) +[PDF](606.1KB)
We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn-Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling, the Wulff shape.
Lagrangian dynamics of submanifolds. Relativistic mechanics
Gennadi Sardanashvily
2012, 4(1): 99-110 doi: 10.3934/jgm.2012.4.99 +[Abstract](29) +[PDF](354.2KB)
Geometric formulation of Lagrangian relativistic mechanics in the terms of jets of one-dimensional submanifolds is generalized to Lagrangian theory of submanifolds of arbitrary dimension.

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