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21

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 ...


18

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 ...


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 ...


12

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), ...


8

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 ...


8

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 ...


7

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 ...


6

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 + ...


5

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 ...


5

@Ron Maimon has given the canonical answer to this: the wavefunction is probabilities, and to preserve probabilities one must have a linear equation (indeed, also a norm-preserving evolution operator). I offer another viewpoint, in the style of how Einstein thought about relativity, i.e. two postulates. The postulate is that it is not possible to solve ...


5

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 it’s 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 ...


4

A linear system is one whose dynamics obeys linear differential equations, in contrast with those that are non-linear whose dynamics obeys non-linear differential equations. So if the dyanmics of the variable $x(t)$ obeys a a differential equation $$f\left(x(t),\frac{d}{dt}x(t),\dots,\frac{d^n}{dt^n}x(t),t\right)=0,$$ if $x_1(t)$ and $x_2(t)$ are differente ...


4

Since the rate of change of $ x $ is the same as the rate of change of $y $ you really only a single equation of with one variable. We write, \begin{equation} x = y + c \end{equation} where the constant $ c $ is determined by your initial conditions, \begin{equation} c = x (0) - y (0) \end{equation} (in your case it is the difference between the ...


4

1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces. It was initially intended as a useful simplification to try to ...


4

It is an interesting problem. Usually you would find the infinitesimal oscillations by setting $p_i = b + \delta p_i$, $q_i = a + \delta q_i$ and expanding the Hamiltonian to second order in the $\delta p_i$, $\delta q_i$. Here though, this doesn't work as you just get the same Hamiltonian, $$ H = \sum_{i,j} M_{ij} \sqrt{(\delta p_i - \delta p_j)^2 + (\delta ...


4

(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 ...


3

In the action formalism a linear Euler-Lagrange (EL) equation corresponds to a quadratic action, i.e. an action which is quadratic in the dynamical field variables of the theory. On the other hand, self-coupling or interaction terms in the action correspond to cubic or higher terms. Such terms leads to non-linear EL equations. See also this related Phys.SE ...


3

One of the simplest measurements you can do is to characterize the spatial mode coming out of the laser. This can be done with a simple photodiode and a scanning slit (optical chopper or razor blade mounted on a movable stage). I wrote a brief note on making such a measurement here. This isn't the most interesting of measurements if you aren't going to ...


3

A classical paper on this is Weinberg's Testing Quantum Mechanics.


3

Hints: Conclude that $y-x=c$ is a constant. Use separation of variables $-k\int \!\mathrm{d}t= \int \!\frac{\mathrm{d}x}{x(x+c)}$.


3

The other answers address how to solve this analytically, but I like numerical solutions to things so here goes: $$\frac{d}{dt} \begin{bmatrix}x \\ y \end{bmatrix} = -\begin{bmatrix}kxy\\kxy\end{bmatrix}$$ which can be solved using any number of numerical methods. For simplicity, we can take the second-order Runge-Kutta method where $i$ is the time index. ...


3

There is a normalized form, though it's properly called the dimensionless Euler equations. The way to do it is define: scale time $t_0$ scale density $\rho_0$ scale length $L_0$ and then derive the scales from these: $$ v_0 = \frac{L_0}{t_0},\quad p_0=\rho_0v_0^2 $$ NB: it is possible to use other combinations, but I find that these are often the ...


3

As requested by the OP, I gather my points in an answer. Linear systems Linear systems are systems which are linear with respect to a physical quantity. Mathematically, their evolution can be written as a (possibly differential) equation. Examples: A linear spring is linear in the sense that is produces a force proportional to the displacement it ...


3

I take the core of the question to be Is it possible to do linear stability analysis on 2nd order differential equations by finding eigen values of Jacobian matrix? The answer is yes, but first you have to convert your second-order equations into first-order ones. This is actually pretty easy to do: every time you see a second derivative, e.g. ...


3

As the first question has received sufficient exposition, I would like to make a point with regard to the second one. First thing to understand is that integrability and non-linearity of a system are two different concepts. It is true though that all linear systems in classical mechanics (i.e those that are described by systems of linear equations, be them ...


2

There is something you should be careful of regarding Liouville's theorem. If there are momentum-dependent forces, then Liouville's theorem changes because phase density is no longer incompressible. Suppose we define $f_{s}$ = $f_{s}(\mathbf{x},\mathbf{p},t)$ $\equiv$ the particle distribution function of species $s$, which is non-negative, contains a ...


2

I can think of a method, although it may require to compute $\mathbf{F}_0$ for a very large number of test points. It is based on Gauss's law for gravity $$ \frac1{m_0}\oint_{S} \mathbf{F}\cdot\mathrm d\mathbf S=-4\pi GM_S$$ where $S$ is a closed surface and $M_S$ is the total mass contained inside it. So the idea would be to use some numerical scheme to ...


2

Yes, there is investigation. Some random names on the field (more on the physics side, NO specific order): Carl Dettmann, Tamás Tél, Ott, Ying-Cheng Lai, Adilson Motter, Celso Grebogi, Holger Kantz, Alessandro Moura, Eduardo G. Altmann, etc, etc, etc. A quick search on some of these names should help you to find some recent papers on what is being done ...


2

However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with? Well, how would you know that your time series is of the same dimension as the phase space? Usually, because you already know the dynamical equations for your system (as for your pendulum). If you observe a real-life complex system, ...



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