Questions tagged [stability]

Stability theory addresses the stability of potentials, solutions of differential equations, and of trajectories of dynamical systems under small perturbations of initial conditions.

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Defining a 'small disturbance which dampens in time' while identifying stable points in a nonlinear system

I'm reading the book "Nonlinear dynamics and Chaos" by S Strogatz. In section 2.2, titled "Fixed points and stability", he defines equilibrium points as solutions where ...all ...
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The classical electrodynamic atom

What methods have been used to rigorously prove that classical electrodynamics does not admit a robustly stable atom? The conclusion is often stated and I am aware of the standard responses such as ...
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Hyperbolic harmonic oscillator

The classical harmonic oscillator can be associated to the differential equation: $$y''+\omega^2y=0$$ and solutions $$y=A\cos(\omega t)+B\sin(\omega t)$$ or $$y=A\cos(\omega t+\delta)$$ The harmonic ...
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Physical significance of orbital stability

I saw the orbital stability in Wiki, I just understand it from mathematics angle. But in physical, what is its mean? Since I saw many paper talk about the stability of Schrödinger equation, I think ...
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Why physically do things in general tend to move toward a lower potential value in a potential field? [duplicate]

There are many answers on the site discussing motion of electrons in an electric potential field, See Why is voltage described as potential energy per charge? but also mass tends to move toward a ...
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Frequencies associated with boson/fermion operators

For a Hamiltonian like, $$\hat{H}=\sum_{k}\hbar\omega_{k}b_{k}^{\dagger}b_{k}$$ What does it mean to say that the frequencies $\omega_{k}$ must be positive if $b_{k}$, $b_{k}^{\dagger}$ are boson ...
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Why can't a pyramid stay balanced on a vertex? [duplicate]

A pyramid, such as a tetrahedron/3-simplex, or any other isohedron, falls from some height and lands on a vertex. It will eventually end up with a face to the ground. Why can't it stay balanced on a ...
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Usage of Floquet's Method

I'm treating with a nonlinear system of ODE, in which one of my fixed points is non-hyperbolic, that is, its eigenvalues has ($\Re(\lambda_{1,2}) = 0$). Therefore, I cannot say anything about its ...
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Why does standard model lose predictivity if it has unstable vacuum?

In String Theory In A Nutshell by Elias Kiritsis, Standard Model is unstable as we increase the energy (hierarchy problem of mass scales) and the theory loses predictivity as one starts moving far ...
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Difference between unstable fixed point and chaotic point

I am reading the Scholarpedia article on Lyapunov exponents: Given a dynamical system $$ \dot{\vec{x}}=\vec{F}(\vec{x}) $$ and a fixed point $\vec{x}_0$ such that $\vec{F}(\vec{x}_0)=\vec{0}$, the ...
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Physical meaning of relativistic saturation of an instability?

In the derivation of the Rayleigh-Taylor instability when the fluid is in the extreme relativistic limit ($\rho_0 c^2 \ll p$) and there is a large effective gravity ($ g \gg kc^2$), where $\rho_0$, $p$...
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Rayleigh-Taylor instability with negative Atwood number?

I was reading a paper entitled "The Rayleigh—Taylor instability in astrophysical fluids" by Allen & Hughes (1984) that indicates the instability can occur for $ \rho_{01} < \rho_{02} $ which ...
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Limits on phase speed for a growing instability?

When analyzing the Rayleigh-Taylor instability relativistically, a growing instability only occurs for $ \frac {\omega^2} {k^2 c^2} \ll 1 $. Why must the phase speed of the instability be sufficiently ...
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Sufficient conditions for Rayleigh-Taylor instability

I was reading the paper entitled "The Rayleigh—Taylor instability in astrophysical fluids" by Allen & Hughes (1984), and they discuss relativistically hot plasmas in the context of weak magnetic ...
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How to understand $L_4$ and $L_5$ Lagrange points gravity balance? [duplicate]

It's relative easy to understand gravity balance of Lagrange points $L_1$, $L_2$ and $L_3$. But I am having a hard time to understand how a body would be "kind of" balanced out on Lagrange points $L_4$...
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Nuclear stability [duplicate]

Why does increasing the number of neutrons in a nucleus make it more unstable? I know that adding more protons increases electrostatic repulsion, therefore the nucleus is more unstable, but as ...
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Why certain rotations are unstable? (Euler Equations)

We have the Euler equations for a rotating body as follows $$I_1\dot\omega_1+\omega_2\omega_3(I_3-I_2)=0\\ I_2\dot\omega_2+\omega_1\omega_3(I_1-I_3)=0\\ I_3\dot\omega_3+\omega_2\omega_1(I_2-I_1)=0$$ ...
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Dilemma of classical physics: stationary particles that can't be in stable equilibrium

Statement from Electricity and Magnetism (Edward Purcell): $$U = −0.8738Ne^2 /4π\epsilon_0 a$$ The negative sign shows that work would have to be done to take the crystal apart into ions. In ...
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Are all thermodynamics potentials minimized at thermodynamics equilibrium?

It is said that all thermodynamics potentials are equivalent. Some are more useful than others to describe some systems, based on which state variables are kept constant and which are allowed to vary. ...
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Why is Helium 4 so stable?

I've been looking at stuff to do with binding energies and was wondering why Helium 4 is so stable. The fact everything up to carbon is less stable seems a bit odd. Is there a reason for this or ...
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Thermal stability of solar cells

Concerning the construction of solar cells which property is referred to when researchers speak about thermal stability of solar cells?
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356 views

Human Lean Equation

For a medical experiment I am doing, I need an equation to find the angle at which someone will lean before falling. I am not mathematically inclined in terms of advanced stuff, I am more so of a ...
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Can a satellite park between the Earth and Sun?

If you put a satellite at Geostat altitude travelling the wrong way, will it remain between the Earth and Sun?
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Why does a leaning bike not fall over?

This question has been bothering me for a while now. Everywhere I look, everyone talks about 'fictitious forces' and how they apparently explain the bike being in equilibrium. However, if we just look ...
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How does a balancing toy work?

I know that the balancing toys have their center of mass under the axis on which they are balancing. That's why they stay still. But when we give a little tap on it, it re-balances itself. But how ...
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Classification of fixed points in 4D phase space

The usual classification of fixed points as used in linear stability analysis is based on planar systems (un-/stable node, un-/stable spiral point, saddle). I need to extend this classification to a ...
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339 views

How does Laplace's equation $\nabla^2U = 0$ indicate saddle points?

When I learned about saddle points I had this expression 'rt-(s^2)', where r=Dx, t=Dy, s=Dxy=Dyx. And the intuition behind why it is so was also clear. In an electric/magnetic field, in Earnshaw' ...
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How can we determine stable and unstable equilibrium points from a potential energy versus displacement graph?

In this question, how can we comment on the position of stable/unstable equilibrium if we don't know the total energy of the system? We know force is zero at points 'a', 'b', 'c' and 'd' but there ...
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Can planets orbiting a binary star trace out an orbit in the form of an 8 (or $\infty$-sign)? [duplicate]

In this article there is a lot explained about planets orbiting a binary star. It is said that a planet orbits the binary star around the CM of both stars. I couldn't find anything about planetary ...
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Stability of Photon Orbits

In the context of general relativity the photonsphere occurs in $r = 3M$ (schwarzschild), and it is a saddle (unstable) fixed point on the phase space. Is it possible for a saddle point to be an ...
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Stability, unstability (and metastability) in liquid-gas phase transition: unstable in regards to what?

I have a question about the stability, unstability (and extra questin about metastability, between the spinodal lines if you have time), when we are having a liquid gas phase transition. Here, a ...
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Will an isolated Helium atom decay if all electrons are removed?

I’m not about a plasma with the possibility of reassembling of the lost electrons. How long a He-nucleus (aka. an alpha particle) will be stable without electrons? Stimulation for my question was ...
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A question about the tennis racket theorem with degenerate eigenvalues $I_1, I_2 , I_3$

If a rigid body has a symmetry such that two of the principal moments of inertia are equals, i.e. $$I_1=I_2> I_3 \qquad{\rm or}\qquad I_1>I_2=I_3.$$ Are the rotations around the principal axes ...
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Explain why a stationary object on an inclined plane may start moving after a random time? [duplicate]

An object (such as a mug of coffee, expensive phone, laptop) is placed on an inclined plane and remains stationary due to friction. After an interval (observed as random) it will begin to move (and ...
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Wheel rolling leaned against a vertical wall

I'm new with rigid problems about. I'm trying to solve this: A massive circular disk of mass m, radius R, and negligible thickness is leaned against a vertical wall, slanting by 45 . In the ...
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Stability of a matrix with complex coefficients

while solving a physical problem of an optical beam propagating through a periodic media, I have obtained the following system of coupled differential equations \begin{gather} \frac{d}{dz}\begin{...
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What is the state of the equilibrium for a second derivative equal to zero?

Considering a potential energy of $U$, and a displacement of $x$, the force is given by $F=-\frac{\partial U}{\partial x}$. Since equilibrium is defined as the point at which $F=0$, we can express ...
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Situation of Stable, Neutral and Unstable Equilibrium

Recently, I was reading about stability of equilibrium. I came across the definitions for different types of equilibrium. Neutral Equilibrium: The kind of equilibrium of a body so placed that when ...
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Why do (short) strands of hair keep a horizontal position while falling?

When I drop a short (5 to 10 cm) strand of hair, it tends to end up falling slowly down at a horizontal position. I would expect a uniform cylinder to fall through air neutrally (at a arbitrary ...
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“Dynamically Stable” vs. “Elastically Stable”?

I am studying a group of materials using density functional theory. I am able to calculate elastic constants, phonon densities of states, and formation energies. The results imply that the materials ...
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thermodynamics and stability

Suppose this three processes, at same T and P, each of them on thermodynamical equilibrium: atoms -->molecule 1 atoms --> molecule 2 atomsdifferent --> molecule 3 Where atoms are infinitely ...
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What is the physical interpretation of Rayleigh's inflection point theorem?

Let $\boldsymbol{u} = U(z)\,\mathbf{e}_x$ be the velocity profile of an inviscid parallel flow. Rayleigh's inflection point theorem states that this flow may be linearly unstable to perturbations only ...
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Is this system in stable or unstable equilibrium? [closed]

6 positive, identical charges $+q$ are placed in a hexagon. A negative charge whose value I want to determine is placed at the center to keep the system in equilibrium. Which means $dU/dr = 0$. But ...
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Regularization: What is so special about the Coulomb/Newtonian and harmonic potential?

I wanted to know if the procedure for regularization of the Coulomb potential outlined in Celletti (2003): Basics of regularization theory could be generalized to arbitrary polynomial potentials. So ...
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Periodicity of the Kelvin-Helmholtz Instability

Why is the Kelvin-Helmholtz Instability so regular across space? If any perturbations anywhere in the boundary can lead to instability, why doesn't the pattern appear randomly across the boundary? ...
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Dzhanibekov effect in quantum systems

Dzhanibekov- or Tennis racket effect is what happens when an object with three diferent moments of inertia doesnt spin around the axis with highest or lowest moment of inertia. The object starts to ...
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Conservative central force and stable orbits

I saw a question a few days ago which referred to Bertrand's theorem. So, I now know that stable, closed orbits only occur when the potential function is $\frac{-k}{r}$ or $\tfrac{1}{2}kr^2$. If ...
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Smallest relative velocity driving a two-stream instability

The physical picture is a two-stream system of cold electrons and ions (i.e. $T_i=T_e=0$) with realtive velocities $V_i$ and $V_e$. The dispersion relation obtained is $$D(\omega)=1-\frac{\omega_{pi}^...
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Are electroweak particles stable?

Photons that [are associated with] the electromagnetic force are stable; while the W and Z bosons that [are associated with] the weak force are short lived. I guess that the high temperate electroweak ...
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Tipping point on fence panels? [closed]

I deal in temporary fence panels - and my concern is the tipping point of our product out in the field. Panels are 6' tall x 12' long (63#) Stands are 23" long x 6" wide We use sand bags (30#-40#) ...

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