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The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...
A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...
Stability diagram classifying Poincaré maps of linear autonomous system ′ =, as stable or unstable according to their features. Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points.
The standard example is the action of C * on the plane C 2 defined as (,) = (,).The weight in the x-direction is 1 and the weight in the y-direction is -1.Thus by the Hilbert–Mumford criterion, a non-zero point on the x-axis admits 1 as its only weight, and a non-zero point on the y-axis admits -1 as its only weight, so they are both unstable; a general point in the plane admits both 1 and ...
Stability diagram classifying Poincaré maps of linear autonomous system ′ =, as stable or unstable according to their features. Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points. 2-dimensional case refers to Phase plane.
The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away. Diagram of a ball placed in a stable equilibrium. Second derivative > 0
When the curve goes down to zero the point is unstable, and will flow down to zero along the action of . When the flow stays between zero and infinity, the point is in an unstable equilibrium (semi-stable). This analogy with mechanical equilibrium motivates the terminology of stability and instability.
The latter is always stable due to a theorem of Edward Teller which states that atoms can never bind in Thomas–Fermi model. [ 14 ] [ 15 ] [ 16 ] The Lieb–Thirring inequality was used to bound the quantum kinetic energy of the electrons in terms of the Thomas–Fermi kinetic energy ∫ R 3 ρ ( x ) 5 3 d 3 x {\displaystyle \int _{\mathbb {R ...