Non-linear systems in classical mechanics

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."

Answer 2

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.

Answer 3

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.

Answer 4

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 algebraic or differential) are integrable, but integrability does not imply linearity. Typically looking at the case of a system of differential equations describing a system, integrability is then a condition requiring analytical solvability when all the required initial and boundary conditions have been specified (the problem is then well-posed). In classical Hamiltonian systems (or the more general case being a finite system of ODEs), integrability is then specified by the Liouville-Arnold theorem . In all autonomous systems, the hamiltonian automatically becomes a constant of motion and hence in 1 dimension, all hamiltonian systems are integrable. Enlarging the phase space of dynamics, one now requires many more integrals of motion that specify iso-surfaces upon which dynamics occurs and the intersection of all these surfaces will give the required trajectory of the system, once the initial conditions are fixed.

The idea of requiring constants or invariants of an equation is a very broad one and the argument mostly goes through for most other cases too. For infinite dimensional systems (like PDEs, say), we in principle need an infinite set of constants, though it is not sufficient. The existence of such independent constants in involution with each other many times points to a deeper symmetry or degeneracy in the problem. Take for example the Kepler problem, which has three constants of motion
$$ H=\dfrac{\vec{p}^2}{2m}-\dfrac{k}{r}\quad(k>0) $$ The three constants of motion in this problem are $H,\vec{L}^2$ and $L_z$ and they all poisson commute with each other. The dynamical symmetry group of this problem turns out to be $SO(4)$ (this group contains all the canonical transformations that preserve the symmetry of the hamiltonian). An additional constant of motion is the Laplace-Runge-Lenz vector $\vec{A}$, $$ \vec{A}=\vec{p}\times\vec{L}-\dfrac{mk\vec{r}}{r} $$ which also poisson commutes with the other constants $(\{\vec{A},H\}=0,\{\vec{A},\vec{L}\}=0)$. It is these two vectors $(\vec{L}\pm\vec{A})$ modulo constant factors, that generate the group of dynamical symmetries. The fact that $\vec{A}$ is a constant implies that orbits in the Kepler problem do not precess. In the context of the hydrogen atom, this same constancy gives rise to the degeneracy of the energy levels with $\ell$ (the azimuthal quantum number). Degeneracy in $m$ (the magnetic quantum number) is demanded by the spherical symmetry of the potential. From this, we see that the existence of constants of motions that allow a system to be integrable almost always point to a deeper understanding of the problem by exposing hitherto unknown symmetries or degeneracies in the system.

There are a number of non-linear systems like the KdV equation that exhibit soliton and soliton-like solutions and this is a very active area of research in understanding the existence of solutions and integrability of nonlinear systems. One can be pretty sure that if a non-linear equation is found to be integrable, then it is definitely special and has some underlying properties that allow one to solve the equation. There is a nice write up on some of these things given in this document, that I suppose will be helpful in elucidating further details with regard to integrable systems.