This Week's Finds in Mathematical Physics (Week 222)


by John Baez
Tags: mathematical, physics, week
DRLunsford
#73
Oct12-06, 05:10 AM
P: n/a
Isolated quasars are a natural part of Halton Arp's "ejection"
scenario, although they are far more likely to be associated with a
nearby galaxy. Since electromagnetism scales (conformal invariance of
EM), such ejections could be similar in principle to the "coronal mass
ejections" on the Sun, only on a far more vast scale. There are cases
of quasars and clearly associated galaxies where the members have
widely varying redshifts. A sensible interpretation would conclude that
enormous electromagnetic fields are at work, and that some joint
manifestation of gravity and electromagnetism is underway.

See

http://xxx.uni-augsburg.de/abs/astro-ph/?0101538

-drl

John Baez
#74
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#75
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#76
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#77
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#78
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#79
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#80
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#81
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

John Baez
#82
Oct12-06, 05:10 AM
P: n/a
In article <slrndlehja.qdn.robert@atdotde.iu-bremen.de>,
Robert C. Helling <helling@atdotde.de> wrote:

>On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez
><baez@math.removethis.ucr.andthis.edu> wrote:


>> o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
>> resonance with Neptune. These are rare compared to plutinos, and
>> they're smaller, so they're stuck with boring names like 1996 TR66.
>> There are also a couple of Kuiper belt objects in 4:3 and 5:3
>> resonances with Neptune.


>Is there an easy way to see why these resonance orbits come about? Why
>do three body systems with a large central object, an intermediate
>planet and a small probe happen to get the probe in resonace with the
>planet? Is this just "frequency locking happens in chaotic systems"
>or is there an easy but more quantitative way to understand this?


I'm shamefully ignorant of this, so ten minutes' research on the
web was able to double my knowledge. I got ahold of this paper
online:

B. Garfinkel, On resonance in celestial mechanics: a survey,
Celestial Mech. 28 (1982), 275-290,
http://adsabs.harvard.edu/cgi-bin/np...eMec..28..275G

and while not easy to understand (I guess there's a huge body
of work on this subject), it uses Hamiltonian perturbation theory
and continued fractions to study resonance, and talks about a difference
between "shallow" and "deep" resonances.

It says that Laplace first explained the Great Inequality in the motion
of Jupiter and Saturn by means of a 5:2 resonance, which is a
"shallow resonance". I have no idea what the "Great Inequality" is,
other than a strange name for this 5:2 resonance. But, I read elsewhere
that:

The dynamics of the Sun-Jupiter-Saturn system was recognized
as problematic from the beginnings of perturbation theory.
The problems are due to the so-called Great Inequality (GI), which
is the Jupiter-Saturn 2:5 mean-motion near-commensurability.

This is from:

F. Varadi, M. Ghil, and W. M. Kaula,
The Great Inequality in a Planetary Hamiltonian Theory
http://arxiv.org/abs/chao-dyn/9311011

Somehow this shallow resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))

which is close to 2/5.

The Pluto-Neptune resonance, on the other hand, is a "deep resonance"
and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))

which starts out close to 2/3. (Recall that plutinos go around the
Sun about twice each time Neptune goes around thrice.)

>Probably related: There are people doing numerical long term stability
>analysis of the solar system. From what I know, they are not just
>taking F=ma and Newton's law of gravity, replace dt by delta t and
>then integrate but use much fancier spectral methods. Could somebody
>please point me to an introduction into these methods?


Here's a bit of stuff about that from "week107", perhaps not all
that helpful, but still pretty interesting:

....................................................................... ...

Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best long-term simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.

Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation --- the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else --- is the Earth-Moon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.

For more, try:

4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.

Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.

James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176-194, July 1986, reprinted in Lecture Notes in Physics #267 --
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.

James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C-34,
No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.

Arnold Neumaier
#83
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#84
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#85
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#86
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#87
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#88
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier


Arnold Neumaier
#89
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier

Arnold Neumaier
#90
Oct12-06, 05:10 AM
P: n/a
Jonathan Thornburg -- remove -animal to reply wrote:
> Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:
>
>>There are people doing numerical long term stability
>>analysis of the solar system. From what I know, they are not just
>>taking F=ma and Newton's law of gravity, replace dt by delta t and
>>then integrate but use much fancier spectral methods. Could somebody
>>please point me to an introduction into these methods?

>
>
> I don't do this sort of work myself, but the buzzwords you want are
> "symplectic ODE integrator". The basic idea is to use an ODE integration
> scheme which conserves energy, angular momentum, and maybe other nice
> things, up to floating-point roundoff error, rather than just up to
> finite differencing error like a standard ODE integrator would do.
>


T. Fuse,
Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune,
http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf
uses symplectic integration to study 2:3 resonances numerically.

The thesis
Time-frequency analysis based on wavelets for Hamiltonian systems
by Vela-Arevalo,
http://www.cds.caltech.edu/~luzvela/th2s.pdf
contains in Chapter 4 interesting numerical information about chaos,
resonances, and stability in the restricted 3-body problem. Other
interesting papers include:

http://users.auth.gr/~hadjidem/Asymmetric1.pdf
Symmetric and asymmetric librations in planetary and satellite
systems at the 2/1 resonance

astro-ph/0501004
Regimes of Stability and Scaling Relations for the Removal Time
in the Asteroid Belt

astro-ph/0203182
The Resonant Dynamical Evolution of Small Body Orbits
Among Giant Planets

http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf
Time–frequency analysis of the restricted three-body problem:
transport and resonance transitions

http://www.astro.auth.gr/~varvogli/varv5.ps
The “Third” Integral in the Restricted Three-Body Problem Revisited



Arnold Neumaier


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