I don't think it is.
The differential equation model of the swinging pendulum corresponds to a two-dimensional map. We assume that the pendulum is free, such that it can swing through 360° (frictionless pendulum). However, the motion of the pendulum is constrained to a circle whose radius corresponds to the length l, of the pendulum rod, while the angle of the pendulum is θ which is measured in radians.
Newton's law of motion F=ma can then be used to find the pendulum equation.
The differential equation governing the pendulum becomes:
mlӪ = F = -mgsinӨ. (This is according to Newton’s law of motion).
Therefore, the pendulum requires a two-dimensional state space.
For a solution to this differential equation (second order), two initial conditions must be defined i.e θ (0) and θ ̇(0) (first derivative). If just one is specified, then we cannot predict the future state of the pendulum.
However, if θ and θ ̇(first derivative) are specified at time t=0. We can uniquely determine θ (t) at the next instant.
You can check out:
CHAOS: Introduction to Dynamical systems. Kathleen T. Alligood, Tim D. Sauer
James A. Yorke. Springer: ISBN 0-387-94677-2
A book I read some years back during my Master’s degree.
I said : F=ma does not directly specify a deterministic dynamical system since it is a second order ODE.
Observe that
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
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?
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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.
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interesting intellectual argument going on here, I will just observe.
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Thanks for observing, its quite interesting and love this discussion and it revealing...thanks to @mathowl
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Its really intellectual. I can say that they are digging deep for understanding.
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Yeah I understand it is a simplifiction.
What do you mean with
More specifically, what is strong determinism and what is weak determinism?
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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)
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