4,018 research outputs found
Discrete moving frames on lattice varieties and lattice based multispace
In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalization of the jet bundle that also generalizes Olver’s one dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame
Hamiltonian evolutions of twisted gons in \RP^n
In this paper we describe a well-chosen discrete moving frame and their
associated invariants along projective polygons in \RP^n, and we use them to
write explicit general expressions for invariant evolutions of projective
-gons. We then use a reduction process inspired by a discrete
Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the
space of projective invariants, and we establish a close relationship between
the projective -gon evolutions and the Hamiltonian evolutions on the
invariants of the flow. We prove that {any} Hamiltonian evolution is induced on
invariants by an evolution of -gons - what we call a projective realization
- and we give the direct connection. Finally, in the planar case we provide
completely integrable evolutions (the Boussinesq lattice related to the lattice
-algebra), their projective realizations and their Hamiltonian pencil. We
generalize both structures to -dimensions and we prove that they are
Poisson. We define explicitly the -dimensional generalization of the planar
evolution (the discretization of the -algebra) and prove that it is
completely integrable, providing also its projective realization
Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top
We develop the theory of discrete time Lagrangian mechanics on Lie groups,
originated in the work of Veselov and Moser, and the theory of Lagrangian
reduction in the discrete time setting. The results thus obtained are applied
to the investigation of an integrable time discretization of a famous
integrable system of classical mechanics, -- the Lagrange top. We recall the
derivation of the Euler--Poinsot equations of motion both in the frame moving
with the body and in the rest frame (the latter ones being less widely known).
We find a discrete time Lagrange function turning into the known continuous
time Lagrangian in the continuous limit, and elaborate both descriptions of the
resulting discrete time system, namely in the body frame and in the rest frame.
This system naturally inherits Poisson properties of the continuous time
system, the integrals of motion being deformed. The discrete time Lax
representations are also found. Kirchhoff's kinetic analogy between elastic
curves and motions of the Lagrange top is also generalised to the discrete
context.Comment: LaTeX 2e, 44 pages, 1 figur
Nonlinear d'Alembert formula for discrete pseudospherical surfaces
On the basis of loop group decompositions (Birkhoff decompositions), we give
a discrete version of the nonlinear d'Alembert formula, a method of separation
of variables of difference equations, for discrete constant negative Gauss
curvature (pseudospherical) surfaces in Euclidean three space. We also compute
two examples by this formula in detail.Comment: 21 pages, fixed typos (v3
Integrable discretizations of some cases of the rigid body dynamics
A heavy top with a fixed point and a rigid body in an ideal fluid are
important examples of Hamiltonian systems on a dual to the semidirect product
Lie algebra . We give a Lagrangian derivation of
the corresponding equations of motion, and introduce discrete time analogs of
two integrable cases of these systems: the Lagrange top and the Clebsch case,
respectively. The construction of discretizations is based on the discrete time
Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian
reduction. The resulting explicit maps on are Poisson with respect to
the Lie--Poisson bracket, and are also completely integrable. Lax
representations of these maps are also found.Comment: arXiv version is already officia
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