The purpose of this article is to illustrate the role of connections and symmetries in the Wheeled Inverted Pendulum (WIP) mechanism an underactuated system with rolling constraints popularized commercially as the Segway, and thereby arrive at a set of simpler dynamical equations that could serve as the starting point for more complex feedback control designs. The first part of the article view...

Using the subgroup structure of the generalized Poincaré group P (1, 4), the symmetry reduction of the five-dimensional wave and Dirac equations and Euler–Lagrange– Born–Infeld, multidimensional Monge–Ampere, eikonal equations to differential equations with a smaller number of independent variables is done. Some classes of exact solutions of the investigated equations are constructed.

The aim of the paper is to announce some recent results concerning Hamiltonian theory. The case of second order Euler–Lagrange form non-affine in the second derivatives is studied. Its related second order Hamiltonian systems and geometrical correspondence between solutions of Hamilton and Euler–Lagrange equations are found.

The paper suggests a generalization of the classic Euler-Lagrange equation for circuits compounded of arbitrary elements from Chua’s periodic table. Newly defined potential functions for general (,) elements are used for the construction of generalized Lagrangians and generalized dissipative functions. Also procedures of drawing the Euler-Lagrange equations are demonstrated.

This paper is intended as a minimal introduction to the application of Lagrange equations to the task of finding the equations of motion of a system of rigid bodies. A much more thorough and rigorous treatment is given in the text “Fundamentals of Applied Dynamics” by Prof. James H. Williams, Jr., published in 1996 by John Wiley and Sons. In particular the treatment given here to the evaluation...

are called stationary Stokes equations, where u : Ω→ R denotes the velocity of the uid, p : Ω→ R denotes the pressure and f : Ω → R is the density of forces acting on the uid (e.g. gravitational force). The Stokes equations govern a ow of a steady, viscous, incompresible uid. We note that (1) is called the momentum equation and (2) is called the incompressibility equation. We supplement the sys...

are called stationary Stokes equations, where u : Ω→ R denotes the velocity of the uid, p : Ω→ R denotes the pressure and f : Ω → R is the density of forces acting on the uid (e.g. gravitational force). The Stokes equations govern a ow of a steady, viscous, incompresible uid. We note that (1) is called the momentum equation and (2) is called the incompressibility equation. We supplement the sys...