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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Stabillity of hitting pool ball aligned in a straight line
While playing pool, my friend showed me this YouTube short video (link). It shows that we can always hit the last ball.
With curiosity, I wonder if it has a stability to compensate for small ...
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Stability with the Variational Principle
In quantum mechanics you can use the variational principle to find an approximate bound to the energy of some state. My lecturer said that with this method the stability of an $H_2$ atom and a $H^+$ ...
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Lyapunov is wrong - got unstable on a stable system [closed]
I'm angry with the Lyapunov stability criteria. Consider this system:
Here, $u$ is the input and $x_1$, $x_2$ are my state variables. Now, solve for the transference of the system, defining my output ...
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Why does bonding stabilize molecules?
We are always told that bonding stabilizes molecular structures as energy is given off during bonding and a system with less energy is stable.
My question is why is energy given off during bonding , ...
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What makes classical Rutherford model unstable?
Every reference to the classical Rutherford model of atom claims that it is unstable since the electron is radiating energy and so it should collapse into the nucleus.
But I had a confusion with the ...
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Why is it that for light elements, roughly up to iron ($\rm Fe$), splitting nuclei actually costs energy, rather than energy being released?
I have read this question:
For light elements, roughly up to iron ($\mathrm{Fe}$), splitting atoms actually costs energy, rather than energy being released.
How much energy is released from the ...
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Self-confinement of $N$-body gravitational systems
Consider $N=2$ point particles (each having unit mass) interacting via Newtonian gravity in the usual 3-dimensional space. There is a simple criterion to assess whether the system is bounded or not: ...
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Why do elements heavier than calcium require more neutrons to remain stable? [duplicate]
In studying the periodic table, I noticed that calcium (with atomic number 20) is the heaviest element with stable isotopes with a 1:1 ratio of protons to neutrons. For elements heavier than calcium, ...
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How stable is the first harmonic on a vibrating string?
Occasionally, when I play my fiddle, a "forced" harmonic sounds one octave above the fingered note. If I want the note to continue, I can finesse the bow such that the note becomes more ...
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Stability of extended rigid bodies around Lagrange-points $L_1$, $L_2$ and $L_3$
$L_1$, $L_2$ and $L_3$ are known unstable for point-like bodies, cf. e.g. this Phys.SE post. They assume for point-like objects, which might not cover all practical scenarios. For example, there are ...
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Validity of Bertrand's theorem for a self-interacting system
We know that in classical mechanics, a particle of mass $m$ orbiting in a given central-force potential $V$ will satisfy the following eom:
$$\frac{d(m\dot{r})}{dt}-mr\dot{\theta}^2+\frac{\partial V}{\...
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Why a global phase transition in a supercooled liquid but a local transition in a superheated liquid?
The classic example of a supercooled liquid is supercooled water. The metastable state is easily perturbed and the whole sample transitions to water ice.
The classic example of a superheated liquid is ...
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Can light become a satellite of a black hole?
In general, stars can have satellites orbiting around them.
Then, can a photon become a satellite of a black hole?
Once a photon enters the Schwarzschild radius, it cannot escape the black hole.
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How was this equation for the stability of a finite difference model Wave Equation derived? Von Neumann analysis?
Wave Equation
In this paper, the following equation is given for a wave equation:
Stability Equation
They then go on to state the stability of such a wave equation in sample based solutions (in terms ...
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Fully confined incompressible fluid patterns under viscosity
Assume there is some energy in a closed 2d domain $\Omega$ depending on the velocity $v$ and a constant $K$:
$$ E=\int d\Omega\, K (\nabla v : \nabla v) $$
so that any spatial variation of $v$ is ...
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What is the proof of Earnshaw's theorem in this context?
I am a mathematician and do not know much physics so I would appreciate your explanations very much. In my understanding, Earnshaw's theorem says that there are two stable stationary configurations, ...
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Non-linear Eigenvalue problem with an ODE to solve for the Linear Stability Theory of a Boundary Layer
I am working on the Linear Stability Theory (LST) for analysing the stability of a boundary layer in fluid mechanics. I have conducted CFD simulations and extracted the base flow data. To form the LST ...
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Breakdown of the Fermi gas model for the nucleus
One of the predictions of the Fermi gas model for the nucleus is that the most energetically favorable situation for a nucleus is to have $N=Z$, an equal number of protons and neutrons. Observing the ...
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Are elements above 137 possible?
I have heard the argument that elements with atomic numbers above 137 are not possible but I am unsure if it's true or why.
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Can perfectly stable orbits exist in GR?
Defining "stable orbit" between two bodies as one where, in the absence of other bodies or non-gravitational forces, the distance stays between some value pair $r_{min}>0$ and $r_{max}$. ...
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Why does the imaginary time Euler-Lagrange equation imply the potential goes to zero at infinite imaginary time?
On reading up about the bounce solution for false vacuum decay in S. Coleman's The Fate of the False Vacuum I it was stated that the equation
$$0 = \frac{1}{2} \frac{\partial q}{\partial \tau} . \frac{...
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Why does fission of large nuclei always result in energy released?
When large nuclei undergo fission, the binding energy per nucleon of products is greater than the binding energy of the original nuclei. This only happens (with certainty) when the products are Iron-...
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Why are orbitals are stable even though they have wierd shapes?
I'm curious to know about why are they stable, let's talk about $p$-orbital , $p$-orbital is dumbbell shaped shouldn't electrons just fall into the nucleus because we need a centrifugal force to ...
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About exchange energy
Electrons of the same spin in degenerate orbitals undergo exchange and make the atom more stable. Why do they release energy during exchange? We can calculate the number of possibilities in which the ...
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Would an Hypothetical Ringworld Positioned Inside the Moon's Orbit Around Earth, gravitationally affect Earth?
If a ringworld were to encircle Earth within the orbit of the Moon, would it induce any gravitational alterations on Earth?
I grasp that inside a circular structure, there might not emerge a ...
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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 ...
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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 ...
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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 ...
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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 ${\bf L}$, hence in an introductory paragraph explains why, when studying stability of ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 "...
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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 ...
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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 ...
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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 & ...
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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 ...
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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....
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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 "...
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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 ${V_{2} = \large\frac{-|\kappa|}{r^{\...
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Why is helium-4 the only nuclide with a negative nucleon binding energy?
He-4 is very unusual as it’s the only nuclide that does not accept another nucleon. In other words, even if you force a proton or a neutron into He-4, it will be kicked out immediately. If you ...
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Most stable isobar for even-$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 ...
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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)=\...
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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. ...
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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 ...
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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 ...
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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 ...
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