What Do Researchers Study in Classical Mechanics and Chaos Theory?

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Discussion Overview

The discussion revolves around the research topics within classical mechanics and chaos theory, exploring the types of systems studied, methodologies employed, and applications in various fields. Participants inquire about the nature of research, including simple and many-particle systems, as well as the use of computational methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that researchers study the properties and behavior of solutions to differential equations without necessarily solving them, referencing bifurcation theory as an example.
  • Others argue that chaos can be characterized by parameters such as the Lyapunov exponent and the dimension of chaos, which are complex to calculate and require extensive computational resources.
  • A participant mentions that chaos theory is widely applicable across various fields, including applied physics and astrophysics, with specific examples like optical chaos and chaotic behavior in the solar system.
  • There is a mention of the pencil balancing problem as a simple mechanics issue that illustrates chaos theory applications.
  • Some participants discuss the historical context of chaos theory's emergence in applied mathematics, particularly in meteorology, where classical physics models are predominant.
  • One participant notes the limitations of certain methods, such as Picard's iteration, in approximating differential equations.

Areas of Agreement / Disagreement

Participants express a range of views on the applications and implications of chaos theory, with no clear consensus on specific research focuses or methodologies. Multiple competing perspectives on the relevance and application of chaos theory in different fields remain evident.

Contextual Notes

Some discussions reference specific methodologies and their limitations, such as the applicability of Picard's iteration method, and the inherent chaos in meteorological models, indicating a nuanced understanding of the subject matter.

Who May Find This Useful

This discussion may be of interest to researchers and students in physics, applied mathematics, and related fields, particularly those exploring chaos theory and its applications in various scientific domains.

neurocomp2003
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Was wondering if anyone can enlighten me on what people research, when they study classical mechanics & chaos theory? Simple systems? many particle systems? do they code?
 
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For starters, they try to find out the solutions (well, properties and the behaviour of those solutions without actually acquiring the mathematical expression for the solution) of differential equations without actually solving the diff equations. I do not know if you have ever heard of BIFURCATIONS and Bifurcation Theory
I had an intro course on this in my second year at the university.

regards
marlon
 
Any system with feedback will behave chaotically (basically, since positive feedback acts as a sort of memory).

Marlon mentioned a bottom up approach, starting with the equations, but many experimentalists study chaos from the top down, that is, making direct observations of chaos and attempting to quantify their observations and relate those quantities back to system parameters.

Chaos is commonly characterised by defining the dimension of the chaos and the Lyapunov exponent (the rate of divergence of two nearly identical trajectories in the phase space of the system). These two quantities are extremely difficult to calculate and require elaborate computations to do so.

Studying a chaotic system essentially involves the calculation of these two parameters. Chaotic systems usually have several regions of chaos, noticable changes in the behaviour of the system, depending on the amount of positive feedback. By knowing what parameters give what type of chaos, the chaos in a system can be actively controlled.

Claude.
 
coo, thanks for the replies...is chaotic theory/dynamicalsystems/bifurcation theory(marlon, yeah i know bifurcation, funny how it applies to psych) (ie using the billiard tables system in 3D environment?) used in QM or AP a lot or are there very few researchers who use it?
 
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AP = Applied Physics?

Well, if you do mean Applied Physics, then yes, chaos is widely studied. We have people at my university working on optical chaos and chaos in semiconductor lasers.

The amount of papers published on these topic was enough for SPIE to release a collection of papers in their milestone series, so yes I would say it is fairly widespread.

The thing about chaos is that it is not system specific, it turns up in just about every field of physics.

Claude.
 
ap-
astrophys
 
Ah, yes it does turn up in Astrophysics. Here are two examples I have come across.

- Bistability in organic molecular clouds.
- When one includes the gas giants when analysing the motion of the solar system, the small effect of the gas giants can induce a choatic wobble in the Earth's orbit that may be responsible for long term climate change. I think there was an article in New Scientist on this topic a while ago.

Claude.
 
marlon said:
For starters, they try to find out the solutions (well, properties and the behaviour of those solutions without actually acquiring the mathematical expression for the solution) of differential equations without actually solving the diff equations. I had an intro course on this in my second year at the university.

regards
marlon

Well you are talking about Picard's iteration method,and similar things!
But they have limited application,they can't approximate every diff. eqn.
 
They can study anything you listed. The simplest mechanics problem which has
a chaos theory application is which way a pencil will fall when balanced on its point.
 
  • #10
If I remember correctly, the mathematical study of chaos was brought into the forefront of applied maths when it was pointed out that a typical set of diff.eqs. used in meteorology was inherently chaotic.
Meteorology is a field dominated by classical physics modelling (and no discernible improvement would be found if you were to try a QM or relativistic approach).
 
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  • #11
arildno said:
If I remember correctly, the mathematical study of chaos was brought into the forefront of applied maths when it was pointed out that a typical set of diff.eqs. used in meteorology was inherently chaotic.
Meteorology is a field dominated by classical physics modelling (and no discernible improvement would be found if you were to try a QM or relativistic approach).

Yes, i once studied the application of numerical calculus in meteorology. More specifically the contribution of Lorentz. This was just an example, in my course, of how this stuff can be used in real life. Look at page 9 and chapter 1.4 of this site

marlon
 

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