Classical Mechanics – Understanding Non-Linear Systems and Integrable Systems

Question

1. In general, what is meant by non-linear system in classical mechanics? Does it always concern the differential equations one ends up with (any examples would be greatly appreciated)? If so, is it considered as non-linear because of: Higher powers of system variables? ($x^2,x^3…$), or does also any function of $x$ makes the system non-linear? like $\cos(x)$, $\ln(x)$, $e^x$ etc. I am confused.

2. Furthermore, why is it that most non-linear systems are considered non-integrable? Is it due to this fact that such systems are usually considered to be unpredictable even classically? (because we can't have exact analytical solutions?).

Answer 1

(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, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k x(t) = 0$$
• The heat equation, $$\frac{\partial u(\vec x, t)}{\partial t} -\alpha \nabla^2 u(\vec x, t) = 0$$
• The wave equation, $$\frac{\partial^2 u(\vec x, t)}{\partial t^2} -c \nabla^2 u(\vec x, t) = 0$$

Non-linear systems cannot be described by a linear set of differential equations. Some examples of non-linear systems in classical mechanics:

• Aerodynamic drag, where the drag force is proportional to the square of velocity, $$F_d = \frac 1 2 \rho v^2 C_D A$$
• The Navier-Stokes equations, which are notoriously non-linear, $$\rho \left( \frac{\partial v}{\partial t} + \vec v \cdot \vec \nabla v \right) = -\vec \nabla p + \vec \nabla T + \vec f$$
• Gravitational systems, where the force is inversely proportional to the square of distance between objects, $$\vec F = -\frac {GMm}{||\vec r||^3}\vec r$$

(2) Furthermore, why is it that most non-linear systems are considered non-integrable?

That term, "non-integrable" has two very distinct meanings. One sense is that the integral cannot be expressed as a finite combination of elementary functions. The elementary functions are polynomials, rational roots, the exponential function, the logarithmic function, and the trigonometric functions. This is a rather arbitrary division. For example, the integrals $\operatorname{li}(x) = \int_0^x 1/\ln(t)\,dt$ and $\operatorname{Si}(x) = \int_0^x \sin(t)/t\,dt$ cannot be expressed in the elementary functions. These are the logarithmic and sine integrals. These "special functions" appear so often that algorithms have been devised to estimate their values. Categorizing functions as elementary versus non-elementary is a bit arbitrary.

Just because the solution to a problem can't be expressed in elementary functions doesn't mean the problem is unsolvable. It just mean it's not solvable in the elementary functions. For example, people oftentimes say the three body problem is not "solvable". That's nonsense (ignoring collision cases). In the sense of solvability in the elementary functions, even the two body problem is not "integrable". Kepler's equation, $M = E - e\sin E$, gets in the way. Just because the two body problem cannot be expressed in terms of a finite combination of elementary functions does not mean we can't solve the two body problem.

There's another sense of "integrability", which is "does the integral exist?" Going back to the n body problem, a problem exists with collisions. These collisions introduce singularities, so that one could say that the n body problem is not integrable in the case of collisions. Collisions represent one kind of singularity. Painlevé conjectured that the n body problem has collisionless singularities in when $n\ge 4$. This has been proven to be true when $n \ge 5$. Newtonian mechanics allows some configurations of gravitating point masses to be sent to infinity in finite time. This truly is an example of non-integrability.

Proving integrability (or lack thereof) in this sense is a much tougher problem than showing that a problem is (or is not) solvable in the elementary functions. There's a million dollar prize for the first person who can either prove that the Navier-Stokes equations have globally-defined, smooth solutions, or come up with a counterexample that shows that that the Navier-Stokes equations are not "integrable."