protoplanetary disc shredded by 3 central stars
 notFritzArgelander
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protoplanetary disc shredded by 3 central stars
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG
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Re: protoplanetary disc shredded by 3 central stars
Just shows what can happen in incipient multiple star systems.
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Re: protoplanetary disc shredded by 3 central stars
I guess 3body system can be chaotic and 4body system almost certainly is.
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
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.turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pmI guess 3body system can be chaotic and 4body system almost certainly is.
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
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...notFritzArgelander wrote: ↑Fri Sep 04, 2020 5:52 pmThe 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.turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pmI guess 3body system can be chaotic and 4body system almost certainly is.
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Re: protoplanetary disc shredded by 3 central stars
I was just making a little remark on how tough things get, the more you know.turboscrew wrote: ↑Fri Sep 04, 2020 5:58 pmI 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...notFritzArgelander wrote: ↑Fri Sep 04, 2020 5:52 pmThe 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.turboscrew wrote: ↑Fri Sep 04, 2020 5:06 pmI guess 3body system can be chaotic and 4body system almost certainly is.
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.
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
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?
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?
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
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.turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pmIt 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?
So I was puzzled.
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
I'm puzzled. How can Lorenz system be chaotic then?notFritzArgelander wrote: ↑Fri Sep 04, 2020 11:46 pmOrder 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.turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pmIt 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?
So I was puzzled.
I think differential equation needs to be at least of 3rd order to be chaotic.
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
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:turboscrew wrote: ↑Sat Sep 05, 2020 4:10 amI'm puzzled. How can Lorenz system be chaotic then?notFritzArgelander wrote: ↑Fri Sep 04, 2020 11:46 pmOrder 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.turboscrew wrote: ↑Fri Sep 04, 2020 11:00 pmIt 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?
So I was puzzled.
https://en.wikipedia.org/wiki/Lorenz_system
A related result that is more closely motivated by the classical gravity n body problem is the KolmogorovArnoldMoser 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
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
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?
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?
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
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
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
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
Thanks. I also got another expression I tried to figure out earlier: "asymptotically stable".notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 amYes, 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
I wonder if you can use that expression also with chaotic attractors  even if they don't have an exact orbit?
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Re: protoplanetary disc shredded by 3 central stars
Sounds like the concept of an attractor.turboscrew wrote: ↑Sat Sep 05, 2020 8:50 amThanks. I also got another expression I tried to figure out earlier: "asymptotically stable".notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 amYes, 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
I wonder if you can use that expression also with chaotic attractors  even if they don't have an exact orbit?
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
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.notFritzArgelander wrote: ↑Sat Sep 05, 2020 10:31 amSounds like the concept of an attractor.turboscrew wrote: ↑Sat Sep 05, 2020 8:50 amThanks. I also got another expression I tried to figure out earlier: "asymptotically stable".notFritzArgelander wrote: ↑Sat Sep 05, 2020 7:23 amYes, 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
I wonder if you can use that expression also with chaotic attractors  even if they don't have an exact orbit?
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
An attractor can be asymptotically stable.turboscrew wrote: ↑Sat Sep 05, 2020 12:54 pmI 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.notFritzArgelander wrote: ↑Sat Sep 05, 2020 10:31 amSounds like the concept of an attractor.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?
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?notFritzArgelander wrote: ↑Sat Sep 05, 2020 5:51 pmAn attractor can be asymptotically stable.turboscrew wrote: ↑Sat Sep 05, 2020 12:54 pmI 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.
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Re: protoplanetary disc shredded by 3 central stars
The path of the system is arbitrarily close to the attractor.turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pmBut chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?notFritzArgelander wrote: ↑Sat Sep 05, 2020 5:51 pmAn attractor can be asymptotically stable.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.
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG

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Re: protoplanetary disc shredded by 3 central stars
Arbitrarily close to what?notFritzArgelander wrote: ↑Sat Sep 05, 2020 6:53 pmThe path of the system is arbitrarily close to the attractor.turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pmBut chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?
 notFritzArgelander
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Re: protoplanetary disc shredded by 3 central stars
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.turboscrew wrote: ↑Sat Sep 05, 2020 7:07 pmArbitrarily close to what?notFritzArgelander wrote: ↑Sat Sep 05, 2020 6:53 pmThe path of the system is arbitrarily close to the attractor.turboscrew wrote: ↑Sat Sep 05, 2020 5:58 pm
But chaotic attractor, Like Lorenzs? What is the attractor then getting arbitrarily close to?
Scopes: Refs: Orion ST80, SV 80EDA f7, TS 102ED f11 Newts: Z12 f5; Cats: VMC110L, Intes MK66,VMC200L f9.75 EPs: KK Fujiyama Orthoscopics, 2x Vixen NPLs (406mm) and BCOs, Baader Mark IV zooms, TV Panoptics, Delos, Plossl 328mm. Mixed brand Masuyama/Astroplans Binoculars: Nikon Aculon 10x50, Celestron 15x70, Baader Maxbright. Mounts: Star Seeker III, Vixen Porta II, Celestron CG5, Orion Sirius EQG