140,411 research outputs found
Discrete Hamilton-Jacobi Theory
We develop a discrete analogue of Hamilton-Jacobi theory in the framework of
discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation
is discrete only in time. We describe a discrete analogue of Jacobi's solution
and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The
theory applied to discrete linear Hamiltonian systems yields the discrete
Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We
also apply the theory to discrete optimal control problems, and recover some
well-known results, such as the Bellman equation (discrete-time HJB equation)
of dynamic programming and its relation to the costate variable in the
Pontryagin maximum principle. This relationship between the discrete
Hamilton-Jacobi equation and Bellman equation is exploited to derive a
generalized form of the Bellman equation that has controls at internal stages.Comment: 26 pages, 2 figure
Statistical equilibrium of tetrahedra from maximum entropy principle
Discrete formulations of (quantum) gravity in four spacetime dimensions build
space out of tetrahedra. We investigate a statistical mechanical system of
tetrahedra from a many-body point of view based on non-local, combinatorial
gluing constraints that are modelled as multi-particle interactions. We focus
on Gibbs equilibrium states, constructed using Jaynes' principle of constrained
maximisation of entropy, which has been shown recently to play an important
role in characterising equilibrium in background independent systems. We apply
this principle first to classical systems of many tetrahedra using different
examples of geometrically motivated constraints. Then for a system of quantum
tetrahedra, we show that the quantum statistical partition function of a Gibbs
state with respect to some constraint operator can be reinterpreted as a
partition function for a quantum field theory of tetrahedra, taking the form of
a group field theory.Comment: v3 published version; v2 18 pages, 4 figures, improved text in
sections IIIC & IVB, minor changes elsewher
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