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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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 ...
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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, ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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,...
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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, ...
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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?
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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. ...
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Intermediate axis theorem - why can't we have exponential decay? [duplicate]
I was reading about the intermediate axis theorem and its mathematical proof. Typically one starts with the torque-free Euler's equations
$$
\begin{align}
0&=I_1\dot\omega_1 + (I_3-I_2)\omega_3\...
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Is amorphous ice thermodynamically stable?
Amorphous ice is often produced by cooling liquid water below its glass transition temperature so fast that it does not have time to form ice crystals. The fact that this can only occur if the ...
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Numerical model of 1D equations for compressible flow
The equations for 1D compressible flow are
$$\frac{\partial u(x,t)}{\partial t}-A\frac{\partial v(x,t)}{\partial x}=0,$$
$$\frac{\partial v(x,t)}{\partial t}-B\frac{\partial u(x,t)}{\partial x}=0,$$
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How does the increased binding energy per nucleon help in stability?
I have some read answers on this forum, some books like Halliday Resnick and my course books. I have understood that binding energy is the energy we need to supply to break nucleus into individual ...
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Rain drop trails on a car window prefer certain paths. Why?
Why do the raindrops always flow down in the middle?
That's a question given to me by my seven year old son.
Some background:
We are sitting in a bus with two windscreen wiper blades.
I hope that ...
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Does every element have a critical mass?
We have all heard of critical mass for radioactive materials like Uranium, Plutonium etc. but does every element theoretically have a critical mass? If not, what determines if such a phenomena is ...
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How can the mass of an unstable composite particle become complex?
To show where the resonances in cross sections come from, one usually considers the exact propagator in the interacting theory, which for a scalar is
$$iG(p^2)=\frac{i}{p^2-m_R^2+\Sigma(p^2)+i\epsilon}...
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Arrow of time as the result of a feedback loop that stabilizes an inheret unstable reversible time [closed]
I am searching to find some literature on idea i am having, regarding the "arrow of time".
The main idea is an expansion of the main theme of an Asimov sci-fi novel, where he postulates ...
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What is meant by "stable" vs. "unstable" in terms of particles?
I am reading more carefully about subatomic particles, and am on the Muon Wiki page, where it says:
The muon is an unstable subatomic particle with a mean lifetime of 2.2 μs, much longer than many ...
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Renormalization constant in unstable particle propagator
On Peskin & Schroeder's QFT, section 7.3, the book discusses the unstable particle.
Before (7.57), the book gives the formula of a scalar particle propagator
For unstable particle, the book ...
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Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
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If there wasn't a weak nuclear force, what would be the heaviest stable theoretically stable isotope with equal protons and neutrons?
If there was no weak nuclear force, what would be the heaviest stable theoretically stable isotope with equal protons and Neutrons? For our universe the heaviest such isotope is calcium 40, but most ...
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Can a bicycle whose front wheel cannot turn left/right stay upright?
Most sensible explanations of why a bicycle stays upright involve the turning left/right of the front wheel.
If a bicycle's front wheel cannot turn left/right, can it stay upright while being pedalled?...
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Convection in stratified atmosphere
In the stability analysis of stratified atmosphere in the presence of gravity, we consider a blob of atmosphere and perturb it (move it up by a small height $\Delta z$). Initially, the blob had the ...
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It is possible to combine two heavy elements into one?
Is it possible to combine say two uranium atoms into very heavy new element? Periodic table ends around 115 today, but can it go forever technically?
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Mass per nucleon graph
I am a high school student and am struggling with understanding the significance of the mass per nucleon graph.
What does this graph show with regard to nuclear reactions? (y-axis: M/A, x-axis:A) I ...
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Why is a star unstable if it's adiabatic exponents are less than 4/3?
In "Introduction to Stellar Structure" by Walter J. Maciel at page 76 it is said that for a partially ionized non degenerate hydrogen gas, the star is unstable if the adiabatic exponents are ...
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Isotopes half-lives distribution - is it log-uniform?
Decay of single isotope follows exponential law.
BUT decay of nuke fallout / spent nuclear fuel will follow hyperbolic law. It is easy to explain if we assume that half-lives are log-uniformly ...
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Timescale for unstable stratification to resolve itself
The ocean is typically vertically stratified, that is the deeper you go the higher the density should be. This can be because of increased salinity or decreased temperature.
If that is not the case ...
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Why do magic number nuclei have higher first excitation energies?
I have read online that one property of stable 'magic number' nuclei is that they have higher first excitation energies. Why is this?
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Stability of Euler-Cromer method
Euler method doesn't perform well in the context of oscillatory problems like the harmonic oscillator; the amplitude of the oscillation gets bigger with time, which clearly contradicts theory as no ...
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Is there a "most unstable rigid body shape" when spun about its intermediate axis?
Is there a shape of a solid, that provides the quickest and most violent deviation of its axis of rotation, when spun around its intermediate axis? What would be this shape and does it depend on a ...
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Could there be undiscovered long-lived isomers of unstable elements? [duplicate]
There are 80 stable elements in the periodic table. These elements have at least one stable isotope. Other elements don’t have any identified stable isotopes. The existence of stable isotopes can be ...
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Time for pendulum to leave unstable equilibrium is logarithmic
I would like to show that the time it takes for a pendulum to leave the unstable equilibrium at the top is logarithmic; i.e. if the starting position of the mass is $\theta_0 = \delta$ for some small $...
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Why isn't lead-207 radioactive?
I've recently learnt from here, in an atom, the stability of an atom is described in two contexts, one according to the ratio of neutron and proton of the atom. The ratio will always be between 1 and ...
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Literature reference: example of stable and unstable manifolds in Henon-Heiles system
There is a quite classical description of chaotic systems based on the behaviour of stable and unstable manifolds around a stationary point of the Poincaré section. It is presented, for example, [here,...
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Most stable shape if Newtonian gravity was proportional to $r^\alpha$
Consider lots of mass in isolated 3D space, close to each other. Consider that only the gravitational force (Newtonian) exists. Also consider that there is no rotational motion.
It is evident that a ...
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Given a system, how to decide whether a closed orbit is homoclinic, not periodic, solely based on its phase portrait?
Background and definitions:
A system is conservative if it has at least one conserved quantity.
In a phase portrait of a nonlinear conservative system, trajectories that start and end at the same ...
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Neutron star vs. Islands of stability
"Neutron stars" are said to be almost entirely composed of neutrons. The islands of stability principle recognizes an upper limit to the number of neutrons that can be introduced to an atom. ...
