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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 ...

3

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

(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, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k ... 3 As requested by the OP, I gather my points in an answer. Linear systems Linear systems are systems which are linear with respect to a physical quantity. Mathematically, their evolution can be written as a (possibly differential) equation. Examples: A linear spring is linear in the sense that is produces a force proportional to the displacement it ... 4 A linear system is one whose dynamics obeys linear differential equations, in contrast with those that are non-linear whose dynamics obeys non-linear differential equations. So if the dyanmics of the variable x(t) obeys a a differential equation$$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 ...

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