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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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Is indistinguishability required for the stability of matter?

Classically, it is well-known that a charge-neutral system of electrons and nuclei is thermodynamically unstable. In simplistic terms, nothing in classical mechanics prevents electrons from binding ...
Endulum's user avatar
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Expanding growth rate (linear stability analysis) [migrated]

I have done a linear stability analysis for a system of coupled PDEs. The growth rate of perturbations, $f(\lambda)$, satisfies an equation $f(\lambda)=0$. Now I want to find the leading order terms ...
questionerno8's user avatar
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Why does chaos preclude exact solutions?

It is sometimes said that the n-body problem (using the initial positions and velocities of n point masses to calculate their future paths) has no general closed-form solution because the system is ...
Maurice Mizrahi's user avatar
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A problem to understand the stability analysis in the Cahn-Hillard equation

Let us suppose the general diffusion equation (Cahn-Hillard equation): $$\frac{\partial c}{\partial t} = M \nabla^2 \mu, \tag{1}$$ where $c (\underline{r},t)$ is the concentration of a given species ...
math-int's user avatar
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1 answer
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How can I formalize better this proof that angular momentum is conserved for a small impulse?

The book I am studying is discussing Lagrange stability of circular orbits, which assumes fixed angular momentum $L$, hence in an introductory paragraph explains why, when studying stability of a ...
ebenezer's user avatar
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Are there purely 1D instabilities in fluid dynamics?

If I have a purely 1D fluid flow governed by the 1D Navier-Stokes equations (let's assume compressible flow for more generality), are there any instabilities that can happen? It seems like you'd need ...
confusion's user avatar
1 vote
1 answer
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Solving partial differential equations using MacCormack scheme and to quantify in what situations this scheme is stable using von Neumann stability

I am trying to simulate Alfven waves and for that, I need to solve partial differential equations using the MacCormack scheme. The predictor steps are: \begin{align} u^p_j&=u_j^n-c\left(b_{j+1}^n-...
subrojitroy's user avatar
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Why when spinning over non-principal axis, it will change the axis of rotation?

Let's look at a disk which is rotating around non-principal axis. I know the explanation when looking in a rotating frame, the centrifugal force on the edges of a the disk create a torque that wants ...
Dor's user avatar
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Doubt regarding proof of Earnshaw's Theorem using Gauss's theorem

While proving Earnshaw's theorem using Gauss's theorem, we consider a small sphere surrounding our test charge, and apply Gauss law on this sphere, stating that field from all external charges must ...
Eisenstein's user avatar
1 vote
1 answer
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How stable are orbits of light objects around twin large bodies?

I know the two-body problem has a stable solution and the three-body problem does not. In the case that there are two comparable large bodies (twin planets) in a stable mutual orbit, what happens to a ...
spraff's user avatar
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Kelvin-Helmholtz instability for a continuous fluid

I have a question about Kelvin-Helmholtz instability (KHI) for a continuous fluid. I am new to hydrodynamics and I am currently working on a project about KHI. For the past few days, I am looking for ...
Jochem4T's user avatar
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Is there proof for: "Elements heavier than iron will decay to iron by processes such as fission and alpha emission"?

Freeman J. Dyson in his "Time without end: Physics and biology in an open universe", Lecture 2: Physics, part G: All matter decays to iron, claimed that on a long enough time scale "...
Sourabh Choudhary's user avatar
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How to find the stability of time dependent Lyapunov equation?

After linearization of the nonlinear equations, I want to find the covariance matrix $v$ through the numerical solution of time dependent Lyapunov equation, $$dv/dt=a*v + v*a'+ d,$$ where $a$ is my ...
Ani's user avatar
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Rotating disc with viscous damping with large initial angle (always 90 degrees) (unbalance/instability)

I have an application for a rotating disc with inertia that is put on a "knife edge" balancing fixture. The disc is then released in order to find the "heavy spot", once identified ...
Mikro1234's user avatar
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2 answers
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Coupled oscillators and stability of equilibrium points

My question is about parts (e) and (f). I have found the matrix to equation of motion to be $\frac{d}{dt}\begin{bmatrix} x_1 \\ x_2 \\ p_1 \\ p_2\end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & ...
Dave Conkers's user avatar
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On the proof of the Bertrand theorem

I was following the proof of the Bertrand theorem on Wikipedia, which is based on Goldstein "Classical mechanics" (2nd edition). The explanation was clear upto Eq (3). But then it assumes ...
watahoo's user avatar
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Applying Kato-Rellich to the hydrogen atom model to prove stability of first kind [closed]

Trying to Understand the lower bound on the Schrodinger Operator of the Hydrogen atom. Using the kato-rellich theorem. My education has been in physics and i am slowly adding to my mathematics toolset....
Gedankenhooman's user avatar
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1 answer
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How much does quantum uncertainty contribute to the uncertainty of earthquakes?

More abstractly, the topic is: amplification of quantum uncertainty within dynamically unstable systems. I'd like to have a calculable toy model, e.g. maybe a quantum version of the famous "...
Mitchell Porter's user avatar
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Motion around stable circular orbit

Hello I am to solve whether it is possible for body of mass $m$ to move around stable circular orbit in potentials: ${V_{1} = \large\frac{-|\kappa|}{r^5}}$ and ${V_{2} = \large\frac{-|\kappa|}{r^{\...
Optimammal's user avatar
23 votes
1 answer
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Why is helium-4 the only nuclide with a negative nucleon binding energy?

He-4 is very unusual as it’s the only nuclide that does not accept another nucleon. In other words, even if you force a proton or a neutron into He-4, it will be kicked out immediately. If you ...
哲煜黄's user avatar
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Most stable isobar for even-$A$ nuclei

In the Liquid Drop Model of the nucleus, the most stable isobar is the one whose atomic number $Z_{A}$ is the one corresponding to the minimum mass, and can be found from the mass parabola or, by ...
Momo's user avatar
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Is small perturbation in axial direction directly analogous to radial direction for cylindrical coordinate?

In cylindrical coordinate, the stability for a cylindrical liquid column/ligament can be analysed using perturbation theory by applying small perturbation in radial direction as follow; $$\rho(z,t)=\...
jamill1283's user avatar
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Stability of the structure made of magnets

From 4 magnets and 3 steel screwdrivers we can create a fairly stable structure, see the picture on the left. This structure is made on a rotating table, so it can be rotated around a vertical axis. ...
Alex Trounev's user avatar
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Where is the most stable orbit in the Earth/Moon system?

Cis Lunar orbit, trans Lunar orbit, lunar orbit, Earth/Moon L4/5? What altitude, eccentricity/inclination? Moon resonance? Stable means not crash into Earth/Moon or escape. It's a chaotic n>2 body ...
darsie's user avatar
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Marangoni effect and surface tension gradients in binary liquid

I have a question regarding the understanding of the Marangoni effect. A simple visualization is the deposition of a drop of soap (some surfactant-laden solution) into pure water. Because the soap ...
M. Hennes's user avatar
1 vote
0 answers
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Why can't massive nuclei combine together to release energy

I am basically confused as why can't larger nuclei undergo fushion and release energy. One reason I know is because of too much protons than neutrons which generates stronger electrostatic repulsive ...
EBoiG _XF65's user avatar
2 votes
1 answer
131 views

Is there a rigorous proof regarding the non-linear stability of the $L_4$ and $L_5$ Lagrange points?

I have found that many proofs regarding the stability of the $L_4$ and $L_5$ Lagrange points are based on linear approximations of the equations of motion near these points. However, from a dynamical ...
ChungLee's user avatar
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1 answer
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Why is the surface of a static liquid always perpendicular to the direction of net force acting on the liquid as a whole? [duplicate]

In a, let's say rectangular container, the water surface always aligns itself perpendicular to the direction of net force acting on it. Why exactly does it happen? (For example when this container is ...
Ars's user avatar
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How the $N/Z$ ratio affects the stability of isotopes and their method of radioactive decay?

Although there is a graph that tells us the number of isotopes and which ones are stable or abundant in nature, like the one below, I have come across the $N/Z$ ratio, which is the number of neutrons ...
Newton's cat's user avatar
1 vote
2 answers
135 views

Weight distribution and support polygon

I was working on a problem of supporting an object with sticks and wondering about some use cases that would fail. My approach is to place the n-sticks (for example 4) under an object with mass m ...
Ken Adams's user avatar
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0 answers
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Why can static solutions be found only in problems with Dirichlet boundary conditions in non-relativistic point particle mechanics?

Qmechanic in this answer says that only Dirichlet b.c.s are consistent with static solutions of point particle mechanics. Why is this so? E.g. The standard classical harmonic oscillator problem with b....
Sanjana's user avatar
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0 answers
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Zel'dovich pancake derivation

I am looking for a derivation of the Zel'dovich pancake. Does anyone have a reference to the derivation or a link to the original paper? Y. Zel'dovich, Gravitational instability: An Approximate theory ...
4 votes
1 answer
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Has anyone tested the Dzhanibekov Effect to see if the frequency of the 180 degree flip changes based on the objects speed or mass?

Rotation around axis 2, the intermediate axis of an object that is not perfectly symmetrical will flip 180 degrees while continuing to spin in the same direction. I'm curious if there is a way to ...
Amy Myers's user avatar
1 vote
0 answers
71 views

What is the state of matter-energy inside Black Holes? [closed]

Do we know absolutely nothing about what might be the state of matter-energy inside the black hole? Are there existing theories or research that explore the possibility of a metastable vacuum inside ...
VVM's user avatar
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0 answers
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How do I find the binodal in a ternary plot?

I have three components, $A,B,C$, with 3 exchange parameters: $\chi _{AB}, \chi _{BC}, \chi _{AC}$. I want to create a FH ternary diagram to see how such a mixture behaves and how phase separation ...
megamence's user avatar
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What is the condition for the critical point in a ternary system?

In the 2-component system, the condition for stability is $$D = \frac{\partial ^2f}{\partial \phi ^2}>0$$ and the condition for the critical point is $$\frac{\partial ^2f}{\partial \phi ^2}=0,\frac{...
bad_chemist's user avatar
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0 answers
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What is the relationship between global and local stability (thermodynamics)?

To be concrete, let us consider the energy $U$ as the relevant fundamental relation. To be even more specific, let's take a simple, single-component system: $U = U(S,V,N)$. One can then show that ...
EE18's user avatar
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2 votes
1 answer
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Decay of metastable state in classical statistical mechanics

Suppose a classical system at temperature $T$ with one variable $m$ and a free energy $F(m)$ having a metastable and a stable minimum. Suppose the system is in the metastable equilibrium at $t=0$. My ...
emilio grandinetti's user avatar
1 vote
2 answers
129 views

On the stability of an ideal gas

I am just learning about stability, but am struggling to convince myself that an ideal gas is stable with respect to perturbations in particle number. The fundamental equation of an (single-component) ...
EE18's user avatar
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1 vote
0 answers
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Proof of Rayleigh's stability criterion for a rotating inviscid fluid using linear perturbation equations

This question is concerned with how to prove Rayleigh's stability criterion for a rotating inviscid fluid. I can follow the details of the proof up to the final line, but I cannot see immediately how ...
tokyojoe's user avatar
2 votes
0 answers
97 views

What happens with natural frequency on a rope which length is changing?

I was watching violin concerto, and I thought of two scenarios of sliding on the string and taking finger off the string as shown in the image: Define $v = \sqrt{\frac{T}{\mu}} = f\lambda$, $T, \mu, ...
Cro's user avatar
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1 answer
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Relation between the 2 interpretations of the Courant–Friedrichs–Lewy (CFL) condition

In computational fluid dynamics, one tries to ensure that the Courant–Friedrichs–Lewy (CFL) number of the discretization scheme is less than 1. This is done to ensure that the errors made at a certain ...
James's user avatar
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3 votes
1 answer
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What exactly is KAM stability and how can I determine if an orbit is KAM stable or not?

I have been working on the three-body problem lately and came across KAM stability. I read that KAM stability generally means that the solution is stable at different initial conditions (that of ...
Belal Bahaa's user avatar
1 vote
1 answer
111 views

Doubt: Arnold's "Mathematical Methods of Classical Mechanics", pg. 18

I am having trouble understanding the following diagram from pg. 18 of Arnold's book: I am unable to see why local maxima of the potential energy correspond to unstable equilibria (and, reciprocally, ...
algebroo's user avatar
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4 votes
0 answers
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Ghost detection at the level of equations of motion

My question is about how to detect ghostly degrees of freedom at the level of equations of motion. It is not clear for me how does this work. Let me explain with an example: Consider the following ...
Gravitino's user avatar
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3 votes
2 answers
115 views

Where does this (hydrogen molecule energy) graph come from?

I was thinking about the good old question of 'Why do molecules have lower energy than the atoms?' And in a video (around 6:15), this good old energy graph is shown, which is stated as the 'answer' to ...
Rohit Shekhawat's user avatar
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0 answers
65 views

Are very stable super-heavy elements theoretically possible?

I was recently reading about superheavy elements. According to that article all superheavy elements currently known have only been synthesized in laboratory experiments and have a very short half-life,...
LorenzoDonati4Ukraine-OnStrike's user avatar
3 votes
1 answer
421 views

How can two black holes merge without violating No Hair? [duplicate]

Before two black holes merge, their individual event horizons must be perfectly spherical due to the No Hair theorem. If they weren't they'd be betraying information about the inside. After merging, ...
Paul's user avatar
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3 votes
0 answers
76 views

Mechanics behind Pringle Ring

Everyone knows about the famous Pringle Ring Challenge Does anyone know how I could develop a theoretical model to predict its stability?
Dekka's user avatar
  • 31
3 votes
1 answer
151 views

Metastability curve, two different definitions

I have now come across two different definitions for the spinodal curve which, together with the coexistence line, encloses two metastable phase regions. The first definition is from Tong on page 139. ...
Jahn Dorian's user avatar

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