Equations of motion for a classical 3d discrete model, whose auxialiary system is a linear system, are investigated. The Lagrangian form of the equations of motion is derived. The Lagrangian variables are a triplet of “tau functions”. The equations of motion for the Triplet of Tau functions are Three Trilinear equations. Simple solitons for the trilinear equations are given. Both the dispersion...

We construct, for a second-order homogeneous Lagrangian in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the ‘fundamental Lepage equivalent’ associated with first-order Lagrangians defined on jets of sections of a fibred manifold. We show that this form may be defined on a fourth-ord...

We provide methods of computing multivalued solutions to the Euler-Poisson system and test them in the context of a klystron amplifier. An Eulerian formulation capable of computing multivalued solutions is derived from a kinetic description of the Euler-Poisson system and a moment closure. The system of the moment equations may be closed due to the special structure of the solution in phase spa...

In this paper, we study Zeno behavior in Lagrangian hybrid systems, which are mechanical systems with unilateral constraints that are undergoing impacts. Zeno solutions involve an infinite number of impacts occurring in a finite amount of time (the Zeno time). In such systems, one is typically not able to explicitly compute the Zeno time and Zeno limit point, and even not to detect a Zeno solut...

This article describes the design methodology for a network of robotic Lagrangian floating sensors designed to perform real-time monitoring of water flow, environmental parameters, and bathymetry of shallow water environments (bays, estuarine, and riverine environments). Unlike previous Lagrangian sensors which passively monitor water velocity, the sensors described in this article can actively...

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental “pure non-Lagrangian” balance system. In the case when ...

An exact expression for the gravitational field strength in a self–gravitating dust continuum is derived within the Lagrangian picture of continuum mechanics. From the Euler–Newton system a transport equation for the gravitational field strength is formulated and then integrated along trajectories of continuum elements. It is shown that the so–obtained integral solves one of the Lagrangian equa...