TheSkySearchers-Welcome!

Just shows what can happen in incipient multiple star systems.

I guess 3-body system can be chaotic and 4-body system almost certainly is.

turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pm I guess 3-body system can be chaotic and 4-body system almost certainly is.

The history of science can be measured in terms of the number of bodies for which stable solutions exist. With Newton it’s 2. With Einstein on the one hand and Heisenberg, Schroedinger, and Dirac on the other it’s reduced to 1. I suspect that in quantum gravity it might be 0.

notFritzArgelander wrote: ↑Fri Sep 04, 2020 5:52 pm

turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pm I guess 3-body system can be chaotic and 4-body system almost certainly is.

The history of science can be measured in terms of the number of bodies for which stable solutions exist. With Newton it’s 2. With Einstein on the one hand and Heisenberg, Schroedinger, and Dirac on the other it’s reduced to 1. I suspect that in quantum gravity it might be 0.

I didn't base my thought on much anything. I just thought that 3dr order differential equation can be chaotic, and 4th order in 3 dimensional space is overdetermined, so quite small perturbance probably causes a series, maybe endless series, of reorganizations...

turboscrew wrote: ↑Fri Sep 04, 2020 5:58 pm

notFritzArgelander wrote: ↑Fri Sep 04, 2020 5:52 pm

turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pm I guess 3-body system can be chaotic and 4-body system almost certainly is.

The history of science can be measured in terms of the number of bodies for which stable solutions exist. With Newton it’s 2. With Einstein on the one hand and Heisenberg, Schroedinger, and Dirac on the other it’s reduced to 1. I suspect that in quantum gravity it might be 0.

I didn't base my thought on much anything. I just thought that 3dr order differential equation can be chaotic, and 4th order in 3 dimensional space is overdetermined, so quite small perturbance probably causes a series, maybe endless series, of reorganizations...

I was just making a little remark on how tough things get, the more you know.

But I don’t understand your remarks about order and dimensions.

AFAIK:

In Newton the gravitational n body problem is 2nd order with 6n initial conditions in a 3n dimensional configuration space.

In Hamilton’s reformulation of Newton the n body problem is 1st order with 6n initial conditions in a 6n dimension symplectic phase space.

It was just a quick idea. I somehow recall that a group of three 1st order differential equations makes it 3rd order.
If each one has a constraint, that would make three constraints (to make it stable). So 4th equation in the group would make it 4th order and add 4th constraint, but there is no orthogonal dimension where the constraints could "live" when the 4th constraint is added, so the 4th constraint would probably mess up with the other three.
I guess I'm not making much sense?

turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pm It was just a quick idea. I somehow recall that a group of three 1st order differential equations makes it 3rd order.
If each one has a constraint, that would make three constraints (to make it stable). So 4th equation in the group would make it 4th order and add 4th constraint, but there is no orthogonal dimension where the constraints could "live" when the 4th constraint is added, so the 4th constraint would probably mess up with the other three.
I guess I'm not making much sense?

Order AFAIK usually refers to the highest number of time derivatives for a dynamical system. So in Newton’s picture that’s 2 (equations involving position) and in Hamilton’s picture (equations involving position and momentum) that’s 1 only but twice as many differential equations.

So I was puzzled.

notFritzArgelander wrote: ↑Fri Sep 04, 2020 11:46 pm

turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pm It was just a quick idea. I somehow recall that a group of three 1st order differential equations makes it 3rd order.
If each one has a constraint, that would make three constraints (to make it stable). So 4th equation in the group would make it 4th order and add 4th constraint, but there is no orthogonal dimension where the constraints could "live" when the 4th constraint is added, so the 4th constraint would probably mess up with the other three.
I guess I'm not making much sense?

Order AFAIK usually refers to the highest number of time derivatives for a dynamical system. So in Newton’s picture that’s 2 (equations involving position) and in Hamilton’s picture (equations involving position and momentum) that’s 1 only but twice as many differential equations.

So I was puzzled.

I'm puzzled. How can Lorenz system be chaotic then?
I think differential equation needs to be at least of 3rd order to be chaotic.

turboscrew wrote: ↑Sat Sep 05, 2020 4:10 am

notFritzArgelander wrote: ↑Fri Sep 04, 2020 11:46 pm

turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pm It was just a quick idea. I somehow recall that a group of three 1st order differential equations makes it 3rd order.
If each one has a constraint, that would make three constraints (to make it stable). So 4th equation in the group would make it 4th order and add 4th constraint, but there is no orthogonal dimension where the constraints could "live" when the 4th constraint is added, so the 4th constraint would probably mess up with the other three.
I guess I'm not making much sense?

Order AFAIK usually refers to the highest number of time derivatives for a dynamical system. So in Newton’s picture that’s 2 (equations involving position) and in Hamilton’s picture (equations involving position and momentum) that’s 1 only but twice as many differential equations.

So I was puzzled.

I'm puzzled. How can Lorenz system be chaotic then?

It doesn't have to do with order or dimensionality. It comes from the nonlinear nature of the ODEs that describe the system. The behavior of the system is sensitive to initial conditions and the parameters of the problem. It takes a lot of work to describe it in detail. Fortunately there is a decent discussion on Wikipedia:

https://en.wikipedia.org/wiki/Lorenz_system

A related result that is more closely motivated by the classical gravity n body problem is the Kolmogorov-Arnold-Moser Theorem.

https://en.wikipedia.org/wiki/Kolmogoro ... er_theorem

It is not widely appreciated that our Solar System is not stable but qualifies as chaotic: little perturbations can add up over time to produce big changes! https://en.wikipedia.org/wiki/Stability ... lar_System

Perhaps the planet with the most unstable orbit is Mercury! https://en.wikipedia.org/wiki/Stability ... _resonance

I'm not at all surprised if solar system is chaotic. And it's, maybe, "natural selection", why our solar system is that stable. (At this point my math falls short - I don't know how to express things.) In a different, less stable "energy state", the solar system would probably have disintegrated by now.

What do you call separate subsets of a state space that the system can be in, such that the system can be pushed from one subset to another only by external perturbation?

Also, I realized that in English the expression is "system of (differential) functions", not "group of (differential) functions".
In Finnish the word is yhtälöryhmä (ryhmä = group).

Also, I sloppily talked about 3rd order differential equation when I really meant 3rd order system of differential equations. I've been in the belief that system of 3 first order differential equation can be written as one 3rd order differential equation and vice versa. Am I making any better sense?

Yes, better sense.

The stable subsets of a phase or configuration space are called orbits. The formal definition of their stability is due to Lyapunov.

https://en.m.wikipedia.org/wiki/Lyapunov_stability

notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 am Yes, better sense.

The stable subsets of a phase or configuration space are called orbits. The formal definition of their stability is due to Lyapunov.

https://en.m.wikipedia.org/wiki/Lyapunov_stability

Thanks. I also got another expression I tried to figure out earlier: "asymptotically stable".
I wonder if you can use that expression also with chaotic attractors - even if they don't have an exact orbit?

turboscrew wrote: ↑Sat Sep 05, 2020 8:50 am

notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 am Yes, better sense.

The stable subsets of a phase or configuration space are called orbits. The formal definition of their stability is due to Lyapunov.

https://en.m.wikipedia.org/wiki/Lyapunov_stability

Thanks. I also got another expression I tried to figure out earlier: "asymptotically stable".
I wonder if you can use that expression also with chaotic attractors - even if they don't have an exact orbit?

Sounds like the concept of an attractor.

notFritzArgelander wrote: ↑Sat Sep 05, 2020 10:31 am

turboscrew wrote: ↑Sat Sep 05, 2020 8:50 am

notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 am Yes, better sense.

The stable subsets of a phase or configuration space are called orbits. The formal definition of their stability is due to Lyapunov.

https://en.m.wikipedia.org/wiki/Lyapunov_stability

Thanks. I also got another expression I tried to figure out earlier: "asymptotically stable".
I wonder if you can use that expression also with chaotic attractors - even if they don't have an exact orbit?

Sounds like the concept of an attractor.

I was thinking that chaotic attractor has nothing to be approached arbitrarily close to. So I guess it can't be "asymptotically stable", but just "stable". That is, if the system is forced outside of the orbit of its attractor, and it tends to return within.

turboscrew wrote: ↑Sat Sep 05, 2020 12:54 pm

notFritzArgelander wrote: ↑Sat Sep 05, 2020 10:31 am

turboscrew wrote: ↑Sat Sep 05, 2020 8:50 am
Thanks. I also got another expression I tried to figure out earlier: "asymptotically stable".
I wonder if you can use that expression also with chaotic attractors - even if they don't have an exact orbit?

Sounds like the concept of an attractor.

I was thinking that chaotic attractor has nothing to be approached arbitrarily close to. So I guess it can't be "asymptotically stable", but just "stable". That is, if the system is forced outside of the orbit of its attractor, and it tends to return within.

An attractor can be asymptotically stable.

notFritzArgelander wrote: ↑Sat Sep 05, 2020 5:51 pm

turboscrew wrote: ↑Sat Sep 05, 2020 12:54 pm

notFritzArgelander wrote: ↑Sat Sep 05, 2020 10:31 am

Sounds like the concept of an attractor.

I was thinking that chaotic attractor has nothing to be approached arbitrarily close to. So I guess it can't be "asymptotically stable", but just "stable". That is, if the system is forced outside of the orbit of its attractor, and it tends to return within.

An attractor can be asymptotically stable.

But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?

turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pm

notFritzArgelander wrote: ↑Sat Sep 05, 2020 5:51 pm

turboscrew wrote: ↑Sat Sep 05, 2020 12:54 pm
I was thinking that chaotic attractor has nothing to be approached arbitrarily close to. So I guess it can't be "asymptotically stable", but just "stable". That is, if the system is forced outside of the orbit of its attractor, and it tends to return within.

An attractor can be asymptotically stable.

But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?

The path of the system is arbitrarily close to the attractor.

notFritzArgelander wrote: ↑Sat Sep 05, 2020 6:53 pm

turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pm

notFritzArgelander wrote: ↑Sat Sep 05, 2020 5:51 pm

An attractor can be asymptotically stable.

But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?

The path of the system is arbitrarily close to the attractor.

Arbitrarily close to what?

turboscrew wrote: ↑Sat Sep 05, 2020 7:07 pm

notFritzArgelander wrote: ↑Sat Sep 05, 2020 6:53 pm

turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pm
But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?

The path of the system is arbitrarily close to the attractor.

Arbitrarily close to what?

THAT! You're making this too hard..... Just because the set of points of an orbit is arbitrarily close to the a funny set doesn't mean that the set isn't asymptotically stable.