Equilibrium point

Summary

In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde{\mathbf{x}}\in \mathbb{R}^n is an equilibrium point for the differential equation :\frac{d\mathbf{x}}{dt} = \mathbf{f}(t,\mathbf{x}) if \mathbf{f}(t,\tilde{\mathbf{x}})=\mathbf{0} for all t. Similarly, the point \tilde{\mathbf{x}}\in \mathbb{R}^n is an equilibrium point (or fixed point) for the difference equation :\mathbf{x}_{k+1} = \mathbf{f}(k,\mathbf{x}_k) if \mathbf{f}(k,\tilde{\mathbf{x}})= \tilde{\mathbf{x}} for k=0,1,2,\ldots. Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the

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https://en.wikipedia.org/wiki/Equilibrium_point

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Stability of Equilibria for the so(4) Free Rigid Body

Tudor Ratiu, Murat Turhan

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in so(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.

2012

This paper deals with adhesive frictionless normal contact between one elastic flat solid and one stiff solid with rough surface. After computation of the equilibrium solution of the energy minimization principle and respecting the contact constraints, we aim at studying the stability of this equilibrium solution. This study of stability implies solving an eigenvalue problem with inequality constraints. To achieve this goal, we propose a proximal algorithm which enables qualifying the solution as stable or unstable and that gives the instability modes. This method has a low computational cost since no linear system inversion is required and is also suitable for parallel implementation. Illustrations are given for the Hertzian contact and for rough contact.

SPRINGER2018

Integrable G-strands on semisimple Lie groups

François Gay-Balmaz, Tudor Ratiu

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and Hamiltonians for these systems, and analyzes the linear stability of their equilibrium solutions in the examples of so(3) and sl(2, R).

Iop Publishing Ltd2014