Questions tagged [non-linear-systems]
The term non-linear or nonlinear has several definitions but is generally used to describe a system that cannot be approximated by a superposition principle or perturbative approach.
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Is there any effect of gravity in a vertical nonlinear spring? [closed]
I know that for a linear vertical spring, the governing equation of motion written in the presence of gravity is the same as the one written in the absence of gravity. We can either undergo a ...
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Why didn't Einstein propose any metric solution to his equations? [migrated]
I've read about general relativity (GR) recently and something stroke me: Einstein came up with his equations in 1915, linking the metric of spacetime to the distribution of energy (more exactly, to ...
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Reformulate Einstein equations to make them linear
Is it possible to reformulate the Einstein equation in terms of a new variable, say $k_{\mu\nu}$ in terms of the metric $g_{\mu\nu}$, in order to make the Einstein equations linear in $k_{\mu\nu}$?
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What evidence do we have for GR in the nonlinear regime?
The classical equations for Einstein's GR (modulo the cosmological constant) read
$$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \kappa T_{\mu\nu}.$$ These equations have a complicated linearization that ...
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Interactions in nonlinear chiral theories
When discussing nonlinear realizations of $SU(3)_L \times S(3)_R$ in Chiral theories, it is usual to introduce the interactions between the baryon octet ($B$) and some meson matrix $M$ as
\begin{...
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Is nonlinearity a denser encoding of information?
At the microscopic level, an $n$-particle system in 3D can be described by the Liouville equation, which governs the evolution of the distribution function in a $6n$-dimensional phase space.
Going ...
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How does convex splitting method work?
I'm an undergraduate physics student and I'm simulating some partial differential equations using finite element method. For non-linear equations I found a method called linear convex splitting ...
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Why for motion planning of quadrators the goal is to minimize the jerk/snap?
In motion planning for quadrators the optimization goal is sometimes to minimize the (norm squared of the) jerk and more often the (norm squared of the) snap. Can someone provide an intuitive and ...
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Most general nonlinear Lorentz transformation law can be built from linear transformations?
Peskin and Schroeder give a Lorentz transformation law:
$$\Phi_a(x)\rightarrow M_{ab}(\Lambda)\Phi_b(\Lambda^{-1}x).\tag{3.8}$$
Then they say that
"the most general nonlinear [Lorentz] ...
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Concerete examples of physical systems that can be (approximately) modelled using a 2D triharmonic equation?
I have some experimental measurements of input-driven standing-wave resonances in a nonlinear, 2D medium. I think it's fair to assume that the dynamics are homogeneous and isotropic, and we can think ...
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Non-linear spring systems
I've recently been re-learning some physics, and a question came to me when looking over Hooke's law:
In the following I am always assuming that the force required for permanent deformation is ...
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What does it mean for a material's elasticity to be non-linear?
Hooke's law only applies to materials with linear elasticity, usually for small displacements.
Now, if you imagine having a material that does not deform permanently when crossing a specific limit, ...
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How causality and unitarity are ensured given a non-linear electromagnetic Lagrangian?
I am reading these notes on non-linear electrodynamics (NED). On page 8, below equation (5.1) the author states that the modified electromagnetism parameter $\gamma$ should be non-negative in order to ...
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Calculating the Lyapunov exponents spectrum from particle trajectories
I am simulating a forced, compressible 2D flow, that is turbulent and statistically steady, but not stationary.
I want to calculate the Lyapunov exponents spectrum from the trajectories of Lagrangian ...
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Efficient algorithm to calculate nonlinear conductivity spectra
I'm just thinking about the efficient algorithm to calculate the photovoltaic conductivity
$$
J(0) = \sigma^{(2)}(0, \omega, -\omega)E(\omega)E(-\omega)
$$
in time domain calculation. In the case of ...
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What's the best open source alternatives of COMSOL? [closed]
Good time of day. I need to solve system of differential non-linear equations for 3D system. It requires parallel computing on a cluster. Our lab lacks the license on Comsol. It is necessary to ...
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Solubility of integrable systems and the classical XXZ model
I've been learning about integrability in the Hamiltonian sense, and trying to wrap my mind around the analytic power afforded by integrability, both in quantum and classical systems. My goal with ...
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Is it physically relevant to restrict the solution of a nonlinear PDE to positive frequencies in the Fourier transfrom?
I would like to mention that I am a mathematician and not a physicist, so I apologize in advance if my question seems obvious.
Considering any linear PDE, it is common to understand the behavior of ...
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Problem identifying type of equation (linear/nonlinear)
I've looked at the answer to this Math.SE question, but I still can't know the answer to my question here.
The following is the equation of equilibrium: divergence of stress tensor that is the sum of ...
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Does Poisson Distribution means the system is chaotic?
The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
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References for harmonic oscillator with memory
I'm reading Neu's "Singular Perturbation in the Physical Sciences" and in problems 1.1 and 1.2 he defines systems that "have memory" as the the variant of the harmonic oscillator
$$...
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On the (non)linearity of electromagnetism
As a student you are typically told that Maxwell's equations (ME) in vacuum are linear. However, it seems that for extremely high electromagnetic fields the equations for electromagnetism turn out to ...
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Deriving Non-linear acoustic wave models, equilibrium state assumption
The standard derivation in obtaining a single wave equation involves making use of the heat equation with a Taylor expansion of the equation of state, then differentiating this equation and the ...
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Second-order polarization equations
I'm reading through a tutorial about the basics of nonlinear spectroscopy, and I recently came across an equation describing the density matrix of a system that has been acted upon by a pair of laser ...
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Could you explain how the one soliton solution for the KdV equation was mathematically derived?
I’m confused as to how the solution $u(x,t)$ was attained from the KdV equation, I understand there has to be some hyperbolic integral substitution however I’m not too sure how this was attained.
If ...
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Wave propagation speed in non-linear differential equations
Could it happen than a solitary travelling wave (soliton) had a different propagation speed when seen from the usual wave equations from that in a non-linear equation. I mean, suppose a solution $F=f(...
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Reference request for QFT $SO(3)$ non-linear sigma model
I was wondering if anyone has a reference that could help me understand quantum field theories that have a nonlinear configuration space. For example, from classical mechanics if we have a three-...
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Non-linear symmetry and symmetry at quantum level
Can anyone explain me what does the statement mean:
"the BRST symmetry is a non-linear symmetry, so the BRST is also a symmetry at the quantum level"?
What does "at the quantum level&...
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Could nonlinear quantum mechanics be found by future quantum computers?
How could working quantum computers test if nonlinear quantum mechanics or another nonlinear theory is at work at deeper levels or fundamental level?
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How to justify this small angle approximation $\dot{\theta}^2=0$?
Suppose the equation of motion for some oscillating system takes the following form:
$$\ddot{\theta}+\dot{\theta}^2\sin\theta+k^2\theta\cos\theta=0$$
Applying small angle approximation to $\theta$ ...
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Is a damped, driven simple harmonic oscillator a limit cycle?
I've been reading about limit cycles and synchronization from Pikovsky's Synchronization in order to build a background for non-linear oscillators. What I know about the limit cycles is that they're ...
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Lagrangian for a non-linear wave equation
I have the following wave equation that I am trying to understand better:
$$\frac{\partial^2 \varphi}{\partial t^2}-\frac{\partial^2 \sin{\varphi}}{\partial x^2}=0.$$
This equation describes an LC ...
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A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation? [closed]
A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - closed form solutions validation?
Introduction_____________________
I am looking for simple mechanics ...
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Find the tensions in the supporting ropes of an inclined heavy beam supported by two diferent kinds of ropes [closed]
Recently I was making some late-revisit (for self reference) to undergraduate physics topics from the book Ohanian's Physics, 2E expanded-1989, which I loved to use during my freshman undergraduate ...
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Vector operations from nonlinear continnum mechanics with large deformation for curl of a vector $p_i$=$\epsilon_{ijk} s_{k,l}u_{l,j}$
I just found there is an expression from the curl operation of a vector component $p_i=\epsilon_{ijk}s_{k,l}u_{l,j}$, where $\mathbf {u}$ is deformation and $\mathbf {s}$ is another variable dependent ...
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What is the simplest PDE/ODE/model I can use to understand how nonlinearities can lead to leakage of energy to higher harmonics in an oscillator?
I came across this problem in the study of surface waves in an oscillating cylindrical vessel of liquid.
There are various eigenmodes described using Bessel functions, and energy transfer can happen ...
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Do all dynamical systems have attractors?
Do all dynamical systems have attractors?
Is there any chance that there are two or more absolutely the same sets of states in one attractor?
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Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum
Consider a system which exhibits multiperiodicity, say with oscillations of the form $x(t) = \sum_{n=0} c_n \cos(n \omega_0 t)$, $\lim_{n \to \infty} c_n = 0$. The Fourier transform $\tilde{x}(\omega)$...
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Nonlinear PDE from Chain of Oscillators
Some years ago, I was reviewing the calculation for the dynamics of limiting case for a chain of springs with transverse oscillations and found a partial differential equation for which I haven't been ...
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Almost all Liouville torus is preserved for small oscillation problems even if we don't use second-order approximation to potential energy, right?
In small oscillation problems, we use a second-order approximation to the potential energy function (suppose the oscillation is around the point $(0,\cdots, 0)$),
$$
V(x) = V(0) + \frac{\partial^2 V(0)...
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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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Prey-predator dynamical system
I'm working with a prey-predator differential equation system and I have a problem with the competitive exclusion principle.
In its simplest form, this principle states that if there are 2 predators ...
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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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Dynamic equations for a gas discharge lamp
This question asks why a gas discharge lamp is blinking and making a noise when it is turned on. I think that the lamp (with potential on) is a bi-stable system where the current-carrying "bright&...
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Trying to prove chaotic motion from the equation of a nonlinear oscillation [closed]
So I'm given the equation of a nonlinear oscillation:
$x''+ω_0^2x=λx^3$
Assume that $x_1$ and $x_2$ are solutions to the differential equation above.
Therefore;
$x = αx_1+βx_2$
$x' = αx_1'+βx_2'$
$x'' ...
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HOW are harmonics generated in longitudinal waves on the particle level?
I have been looking through the physics.se and all over the internet for weeks now honestly, and I still don't understand how harmonics are formatted on the particle level. Yes, I know that only ...
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Turbulence, Euler equations and equipartition of energy
Recently the user CBBAM asked about the inviscid limit in turbulence and the relation between Navier-Stokes equations and Euler equations when $\nu \to 0$. There I pointed out that Onsager proposed ...
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Fast solution for a nonlinear system of 3 equations of degree 3? [closed]
I have a nonlinear system of 3 equations with 27 constants and 3 unknowns x, y, z
...
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Possible Explanation for Behaviour Non-Linear Bead Knot Experiment
I have a very simple setup to study non-linear phenomena with plastic beads, as described here:
E. Ben-Naim, Z.A. Daya, P. Vorobieff, and R.E. Ecke, “Knots and Random Walks in Vibrated Granular ...
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Don’t understand how nonlinear resistors violate Ohm’s law
Ohm’s law states that the voltage across a resistor is directly proportional to the current through. This is given by the formula v=iR. But most textbooks say that this law is violated when the v vs i ...
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