How is it possible that linear theories, for example maxwells equations or the schroedinger equation, produce nonlinear physics?

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    $\begingroup$ I actually think it is the other way around. Linear equations follow as special cases from nonlinear equations. Take Einstein's field equations of G.R., these are a coupled set of 10, nonlinear, hyperbolic PDEs. You want Electromagnetism, stick in $F_{ab}$, you want fluid dynamics, $T_{ab} = (\mu + p) u_{a} u_{b} + p g_{ab}$. The linear theories follow from geometric simplifications to these equations. $\endgroup$ – Dr. Ikjyot Singh Kohli Oct 17 '17 at 19:21
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    $\begingroup$ How do you define "nonlinear physics"? $\endgroup$ – probably_someone Oct 17 '17 at 20:15
  • $\begingroup$ Are you assuming that, e.g., the permeability of real materials is constant? $\endgroup$ – Alfred Centauri Oct 18 '17 at 1:56
  • $\begingroup$ one of the most interesting appearance of nonlinear behavior is the one that emerges from a linear differential equation but with a free surface boundary condition, such as at the interface of air and water. $\endgroup$ – hyportnex Oct 18 '17 at 14:03
  • $\begingroup$ Can you give an example of nonlinearity you'd like explained? Right now this looks like a list question. $\endgroup$ – knzhou Oct 18 '17 at 18:45

One way of getting non-linear results from linear equations is to have interactions that are spread over time and/or space. This will require integration of the linear interaction, which will result in non-linear behavior. For example, if you integrate dx/dt = kx, you get $x=exp(kt)$, which is clearly non-linear in both k and t.

In another example, take Hooke's law $F = -k x$ (F = Force, k = constant, x = displacement from equilibrium). You can generalize it to continuous media with stress = constant X strain. This is clearly linear. However, if you calculate the deflection of a beam of a material of uniform composition and cross-section, you get non-linear effects, for example with deflection with a given force depending non-linearly on the length of the beam. This is because deformation at one point in the beam influences deformation at another point as static equilibrium is achieved. This coupling between different parts of the beam causes the non-linear term to arise.

  • $\begingroup$ How does this work quantum mechanically? If the schroedinger equation is linear and the underlying mathematics is linear, where is there room for nonlinear phenomena? $\endgroup$ – Zach Oct 17 '17 at 23:13
  • $\begingroup$ Maybe I don't understand the question, but same thing. Every location in space influences the others as it is a pde. For example, eigenvalues of hamiltonian for the hydrogen atom are spaced as n squared. Probability of tunneling is non linear in potential width, etc... $\endgroup$ – Manuel Fortin Oct 18 '17 at 0:48
  • $\begingroup$ If quantum mechanics can be "built up" do describe very complex systems, how do we get things like nonlinearity of the navier stokes equation or nonlinear optics. For situations where nonlinearity from GR is negligible, then shouldn't all phenomena be linear? $\endgroup$ – Zach Oct 18 '17 at 0:54
  • $\begingroup$ For navier stokes, look at its derivation, you will see how the non linear terms arise. It's again a distributed system where things happening at one location can influence what happens at another. $\endgroup$ – Manuel Fortin Oct 18 '17 at 1:26
  • $\begingroup$ For non linear optics, look at the second harmonic generation article on Wikipedia. It has a nice explanation. $\endgroup$ – Manuel Fortin Oct 18 '17 at 1:37

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