Non-linear systems in classical mechanics

(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."

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 undergoes: $F(x)=kx$ (assuming its rest position is 0). If $x$ is doubled, the reaction force $F$ is doubled too.

If you now consider a mass at the end of the spring, and you don't neglect the inertia, you get another example which involves derivatives (linear dynamics): $$m \ddot x+kx=0$$ It's linear with respect to $t\mapsto x(t)$: any linear combination of solutions is solution.

Mathematics offer very powerful tool for such systems. In linear dynamics, for example, it is quite easy to calculate modes of a given system. Then, any motion can be decomposed as a linear combination of these modes! The computations only have to be done for modes (there are as many modes as degrees of freedom) and then you can calculate any motion with no computational cost!

Nonlinear systems Again, the system is nonlinear with respect to a physical quantity.

In continuum mechanics, you can observe :

• material nonlinearities (if you double the pressure the material does not double the strains, e.g.); in 1D, an example could be: $$F(x)=kx^2$$ or: $$F(x)=\sqrt{|cos(x)|}$$ The latter is never used/exposed because no material obeys such a law. But if ever you find one, then you'll have a nonlinear system!

In 3D, see strain tensor vs Euler-Lagrange tensor e.g.

• geometrical nonlinearities (if you have large displacements).
• contact nonlinearities: if you double the position $x(t)$ of ball bouncing on the ground, you're new motion could penetrate the ground (see "unilateral constraints"), which would obviously not be a solution.

In nonlinear dynamics, modes cannot be used as in linear dynamics, because the sum of two motions is no longer a possibly motion (i.e. a solution of the system's equations).

The equations of nonlinear systems can usually not be solved analytically, unfortunately. Luckily, it is often possible to use numerical methods (such as Finite Element Methods) to solve nonlinear differential equations by approaching the exact solution, provided it exists. This is used in an extremely wide range of fields in mechanics: weather forecast, car crashes tests, modeling the combustible of rockets, cracks in concrete, thermal diffusion, acoustics, etc.

The drawback of solving equations numerically is the computational cost, which has no upper bond (the more precision you want, the smaller step you will take). And for complex problems, relevant preliminary results can take weeks...

Note: computers only solve linear problems. Numerical methods consists in making small linear step in such a way that overall the nonlinearity(ies) is(are) conserved.

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 solutions, for a linear system it's true that $ax_1(t)+bx_2(t)$, with $a$ and $b$ constants, will also be a solution, while it's not in general true for a non-linear system. For example, take the unidimensional harmonic oscilator, with potential $V(x)=\dfrac{k}{2}x^2$: $$mx''(t)+kx(t)=0,$$ it's easy to check that the linear combination of solutions is also a solution. However for the anharmonic potential $V(x)=\dfrac{\lambda}{3}x^3$, the dynamics will obey $$mx''(t)+\lambda x(t)^2=0,$$ and it's not true that the linear combinationof two solutions will in general yield another solution, since $$m(x_1(t)+x_2(t))''+\lambda(x_1(t)+x_2(t))^2=\\ (mx_1''(t)+\lambda x_1(t)^2)+(mx_2''(t)+\lambda x_2(t)^2)+2\lambda x_1(t)x_2(t)=\\ 2\lambda x_1(t)x_2(t)\neq0,$$ in general.

The problem of non-linear systems is that the theory of non-linear differential equations is much harder and lacks a lot of results that the linear theory does, so in general the systems may not be integrable given a particular differential equation and you have to study each separate case most of the times.