Geometry and Hamiltonian mechanics on discrete spaces

Abstract

Numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions. To this end, one either converts a 'smooth' model into a discrete (in space and time) model, or models systems directly at a discrete level. The goal of this paper is to provide a discrete analogue of differential geometry, and to define on these discrete models a formal discrete Hamiltonian structure-in doing so we try to bring together various fundamental concepts from numerical analysis, differential geometry, algebraic geometry, simplicial homology and classical Hamiltonian mechanics. For example, the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus. The theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures. Also an example of an energy-conserving algorithm on a simple harmonic oscillator is presented, and its effect on the Poisson structure is discussed.

Discrete port-Hamiltonian systems

“…[10,11]. The first requirement is to choose an appropriate discrete analogue for the reals R. We can use discrete lattices (which have a ring structure), or the space of floating point numbers F which have a quasi-ring (cf.…”

“…The first requirement is to choose an appropriate discrete analogue for the reals R. We can use discrete lattices (which have a ring structure), or the space of floating point numbers F which have a quasi-ring (cf. [10,11]) structure. Since computers uses floatingpoint numbers, and since our main focus is numerical simulation, F will be our choice.…”

“…a link) connecting each pair of points (p, q), where the pair of points are defining a discrete vector, the value f (q) − f (p). Hence, in the natural basis, we would obtain as a representation: The concept of discrete manifolds has been introduced in [10,11]. Discrete manifolds are those that locally look like F n , on these we can define the discrete analogues of charts.…”

“…There exist many special integration techniques that do preserve the energy balance, but these techniques dramatically alter the geometric structure of the Dirac framework. We have discussed these aspects in [10,11] on discrete Hamiltonian systems where we showed how the Poisson structure can get dramatically modified under structure preserving algorithms. Similar analysis also holds for Dirac structures, this will be the subject of future work.…”

Discrete port-Hamiltonian systems: mixed interconnections

“…Either we can discretize continuous systems (there exist a wide variety of techniques for doing so), or we can directly model at the discrete level itself. In previous work [6], [7], [8], [9], [10] we developed a methodology to model physical systems directly at the discrete (in space and time) level and we showed that the discrete models which we obtain as a result of our modeling process exactly coincide with discretized models!, thus offering an alternative approach towards the simulation of port-Hamiltonian systems.…”