37

Gravity can, of course, become turbulent if it is coupled to a turbulent fluid. The interesting question is thus, as John Rennie points out, whether a vacuum solution can be "turbulent". As far as I'm aware this is not known. If turbulence does occur in vacuum gravity, it is remarkably hard to stir up. Even in very extreme situations like colliding black ...


31

This is a very broad subject, but as a rule of thumb, highly non-linear means that the non-linearities cannot be treated with perturbation theory, as these are not negligible as compared to the linear part of the equations (and, in general, they not only are non-negligible, but actually dominate the dynamics). As an example of a non-linear theory which can ...


30

None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be ...


23

They are derivable from classical mechanics using either the continuum or molecular points of view. Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress ...


20

To add to John Rennie's excellent answer I would like to note that the linearity of a theory depends of what you are asking of it, especially which observables you are interested in. One example is the optical response of a 2-level atom (see e.g. https://en.wikipedia.org/wiki/Jaynes–Cummings_model). While being linear in the wavefunctions involved it is non-...


19

Thanks to holography, we now know that solutions to the Einstein equation in certain $d+1$ dimensional spaces are equivalent (dual) to solutions of the Navier-Stokes equation in $d$ dimensions. This is the fluid-gravity correspondence. As a result, turbulence can be studied using the Einstein equations, see, for example, http://arxiv.org/abs/1307.7267.


18

The obvious example is hydrodynamics. The interactions in a fluid all originate from the interactions between atoms and molecules that are described by quantum mechnics, and QM is as far as we know linear. However the Navier-Stokes equations are (scarily) non-linear and produce all sorts of weird behaviour. It's an interesting question whether general ...


18

The classical principle of superposition of electromagnetic field refers to the proposition that the electromagnetic field cannot be scattered by the electromagnetic field. In other words, if two electromagnetic waves pass through a point (say you point two LASERs in a way so that their beams intersect), they really pass through each other without affecting ...


17

I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), ...


13

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


13

Yes. A phase space trajectory of a smooth system$^1$ has to be a continuous curve. For it to be called "periodic", the movement has to repeat itself, both velocity and position: i.e., it must come back to the same spot in phase space. For a deterministic system, the current condition determines uniquely its future evolution, and there can not be "...


12

The interplay of Hamiltonian and Lagrangian theory is based on the following general identities, where $L$ is the Lagrangian function of the system, $$\dot{q}^k = \frac{\partial H}{\partial p_k}\:,\qquad(1)$$ $$\frac{\partial L}{\partial q^k} = -\frac{\partial H}{\partial q^k}\:.\qquad(2)$$ Above, the RH sides are functions of $t,q,p$ whereas the LH sides ...


12

Linear equations are always an approximation of real physical processes, so it's safe to say every real system is non-linear to some extent. You still get to call many systems linear, if their behaviour can be described by linear equations with sufficient precision. Likewise, if you call a system non-linear, it usually means it is substantially non-linear, ...


11

These kinds of proportionality questions are often best answered with dimensional analysis. You want to know a form a quantity with the units of time in terms of what you have. You have a quantity $k$ with units $\frac{\text{Energy}}{\text{Distance}^3} = \frac{\text{Mass}}{\text{Distance} \times \text{Time}^2}$. You also have the mass $m$ (units of Mass) ...


11

Nonlinear optical elements are called nonlinear precisely because of the behaviour you note: because the optical response of the material does not depend linearly on the driving fields. The response may then have a quadratic or higher dependence on the driver, which is usually written in the form $$ \mathbf P =\varepsilon_0 \chi^{(1)} \mathbf E + \...


11

For ordinary or partial differential equations, there is a practical distinction between weakly nonlinear systems where the "standard" solution methods (finite element, finite difference, finite volume, etc) for the corresponding linear system still work well, and strongly nonlinear systems where they may not work at all. An example of this type of weakly ...


11

First things first, there is no proof that the universe is either. An outstanding question in philosophy is the ontological question of whether the universe is defined by mathematics, or if we created mathematics to understand the universe. Your question only makes sense in the former. With sufficient feedback, you can create remarkable approximations of ...


10

The essential idea of a Poincaré map is to boil down the way you represent a dynamical system. For this, the system has to have certain properties, namely to return to some region in its state space from time to time. This is fulfilled if the dynamics is periodic, but it also works with chaotic dynamics. To give a simple example, instead of analysing the ...


9

The basic idea is that statistical properties of complex physical systems fall into a small number of universal classes. A very known example of this phenomenon is the universal law implied by the central limit theorem where the sum of a large number of random variables belonging to a large class of distrubutions converges to the normal distribution. Please ...


9

Update Recently, there was a talk titled Turbulent gravity in asymptotically AdS spacetimes which may be of interest. In these papers, spacetimes which are anti-de Sitter asymptotically with reflecting boundary conditions are considered, and the notion of turbulence in this case is that small perturbations about these spacetimes exhibit 'turbulent behavior.'...


8

Short Intro The nonlinear term or steepening term, $\left( \mathbf{V} \cdot \nabla \right) \mathbf{V}$, determines the rate of steepening of a wave. This can be balanced/offset by loss terms like dispersion (e.g., $\propto \ \beta \ \partial_{x}^{3} v$), diffusion, viscosity (e.g., $\propto \ \nu \ \partial_{x}^{2} v$), resistivity, friction (e.g., $\...


8

If you want to generalize a potential to a class that's broader than the simple $\frac12 k_2 x^2$, it is tempting as a first step to include a small perturbation of the form $\frac13k_3x^3$. Unfortunately, this drastically changes the structure of the potential, because it becomes unbounded from below. Thus, you might get a slightly perturbed behaviour from ...


8

(1) In general, what is meant by non-linear system in classical mechanics? A linear system is described by a set of differential equations that are a linear combination of the dependent variable and its derivatives. Some examples of linear systems in classical mechanics: A damped harmonic oscillator, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k x(t) ...


7

Let's start out using notation similar to the example you linked to: $$ \ddot{y}+b\dot{y}+\sin(y)=a\cos(ct)+d $$ As in the example, we'll write this in autonomous form: $$ \mathbf{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}y\\\dot{y}\\ct\end{pmatrix} $$ the differential equation can now be written: $$ \mathbf{\dot{x}}=\begin{pmatrix}\...


7

Numerical simulations are not always meaningful, as chaos theory belongs to the large subject of dynamical systems theory. Although the definitions differ, chaos generally occurs in three contexts: Sensitive dependence on initial conditions (SDIC). The set is topologically transitive. Periodic points are dense in the set. Think of two particles having some ...


7

The BMS (Bondi Metzner, Sachs) symmetry is a semi-direct product of the Poincare group of spacetime symmetry with an abelian group of translations. It is this latter part which contains gravitational memory. This paper illustrates some of this physics. What this means is that a gravitational wave interacting with a set of test masses will not return these ...


6

The dimensional analysis in zkf's answer completely solves the exercise. Nevertheless, it is possible to give a closed formula for the period $$ T~=~ 4 ~\sqrt{\frac{m}{2k}} \int_0^a\! \frac{dx}{\sqrt{a^3-x^3}} ~\stackrel{x=au}{=}~ 4 ~\sqrt{\frac{m}{2ka}} \int_0^1\! \frac{du}{\sqrt{1-u^3}}. $$ Can you see why? Unsurprisingly, this just confirms zkf's ...


6

Three different points of views on essentially the same thing: Chaotic systems are not only sensitive to numerical errors, but also to any other small perturbations, such as dynamical noise, which may simulate real conditions. Though tiny perturbations affect the detailled, microscopic future of a system, its qualitative dynamics is unaffected. And the ...


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