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RE: Understanding Dynamical Systems: Introduction to Chaos Theory and Its Real Life Applications
I said : F=ma does not directly specify a deterministic dynamical system since it is a second order ODE.
Observe that
mlӪ = F = -mgsinӨ. (This is according to Newton’s law of motion).
is a second order ODE. Observe that Ө' is not a dependent variable of this ODE. So you need to define it as a dependent variable then you can write the system as two first order ODEs and apply the uniqueness theorem of first order ODEs which induces flow.
I am trying to point out that if you have the equation
mlӪ = F = -mgsinӨ.
Then this does not induce a unique dynamical system since for example i could consider the system corresponding to the dependent variables defined by x=Ө , y=Ө' but also v=Ө , w=2Ө'
Is this clear?
Your explanation is clear, but I just want you to get my point. This example is simplified with assumptions to make it fit the context in which I used it above. Those two initial conditions must be specified. There are still other systems which corresponds to a two dimensional state space and two initial conditions need to be specified for determinism and the law of motion is still F=ma
In fact, in the case of nonlinear methods,if the methods are used only when determinism is strong then there would be a limitation, but it is used even when determinism is weak.
interesting intellectual argument going on here, I will just observe.
Thanks for observing, its quite interesting and love this discussion and it revealing...thanks to @mathowl
Its really intellectual. I can say that they are digging deep for understanding.
Yeah I understand it is a simplifiction.
What do you mean with
More specifically, what is strong determinism and what is weak determinism?
In using nonlinear methods (algorithms) on real (Field) data, there might be the absent of clear behavior as predicted by theoretical requirement that the data be deterministic to a very good approximation ( which has been reported in some research work I have seen). However, some qualitative information can still be obtained using these tools with some adjustments. This is what I mean by weak determinism.
(Sorry for my late response, I have been out of town)