does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18
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does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18
 
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does classical mechanics always fail to predict or retrodict for 3 or
more Newtonian gravity bodies? Rich Murray 2011.02.18 Hello Steven V Johnson, Can I have a free copy of the celestial mechanics software to run on my Vista 64 bit PC? In fall, 1982, I wrote a 200-line program in Basic for the Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 bodies in 3D space or 5 in 2D space, about 1000 steps in an hour, saving every 10th position and velocity -- I could set it up to reverse the velocities after the orbits became chaotic after 3 1/2 orbits from initial perfect symmetry as circles about the common center of gravity, finding that they always maintained chaos, never returning to the original setup -- doubling the number of steps while reducing the time interval by half never slowed the the evolution of chaos by 3 1/2 orbits -- so I doubted that there is any mathematical basis for the claim that classical mechanics predicts the past or future evolution of any system with over 2 bodies, leading to a conjecture that no successful algorithm exists, even without any close encounters. Has this been noticed by others? Rich Murray [hidden email] 505-819-7388 1943 Otowi Road, Santa Fe, New Mexico 87505 On Fri, Feb 18, 2011 at 4:30 PM, OrionWorks - "Steven V Johnson" <[hidden email]> wrote: > Just a brief side-comment... > > Some of this "lingo" is fascinating stuff to me. Having performed a > lot of theoretical computer simulation work on my own using good'ol > fashion Newtonian based Celestial Mechanics algorithms, where > typically I use "a = 1/r^2", I noticed orbital pattern behavior > transforms into something RADICALLY different, such as if I were to > change the classical algorithm to something like "a = 1/r^3". You can > also combine both of them like "a = 1/r^2 +/- 1/r^3" within the same > computer algorithm. That produces interesting side effects too. I'm > still trying to get a handle on it all. > > Regards > Steven Vincent Johnson > www.OrionWorks.com > www.zazzle.com/orionworks ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18
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Yes, the n-body system with n>2 is known to be chaotic, but subject to
the constraints of the KAM theorem (http://en.wikipedia.org/wiki/Kolmogorov–Arnold–Moser_theorem), ie there exist quasi-periodic orbits for certain initial conditions. This was certainly known stuff when I studied dynamical systems as an undergrad in the early '80s. On Fri, Feb 18, 2011 at 08:17:37PM -0700, Rich Murray wrote: > does classical mechanics always fail to predict or retrodict for 3 or > more Newtonian gravity bodies? Rich Murray 2011.02.18 > > Hello Steven V Johnson, > > Can I have a free copy of the celestial mechanics software to run on > my Vista 64 bit PC? > > In fall, 1982, I wrote a 200-line program in Basic for the > Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 > bodies in 3D space or 5 in 2D space, about 1000 steps in an hour, > saving every 10th position and velocity -- I could set it up to > reverse the velocities after the orbits became chaotic after 3 1/2 > orbits from initial perfect symmetry as circles about the common > center of gravity, finding that they always maintained chaos, never > returning to the original setup -- doubling the number of steps while > reducing the time interval by half never slowed the the evolution of > chaos by 3 1/2 orbits -- so I doubted that there is any mathematical > basis for the claim that classical mechanics predicts the past or > future evolution of any system with over 2 bodies, leading to a > conjecture that no successful algorithm exists, even without any close > encounters. > > Has this been noticed by others? > > Rich Murray [hidden email] 505-819-7388 > 1943 Otowi Road, Santa Fe, New Mexico 87505 > > On Fri, Feb 18, 2011 at 4:30 PM, > OrionWorks - "Steven V Johnson" <[hidden email]> wrote: > > > Just a brief side-comment... > > > > Some of this "lingo" is fascinating stuff to me. Having performed a > > lot of theoretical computer simulation work on my own using good'ol > > fashion Newtonian based Celestial Mechanics algorithms, where > > typically I use "a = 1/r^2", I noticed orbital pattern behavior > > transforms into something RADICALLY different, such as if I were to > > change the classical algorithm to something like "a = 1/r^3". You can > > also combine both of them like "a = 1/r^2 +/- 1/r^3" within the same > > computer algorithm. That produces interesting side effects too. I'm > > still trying to get a handle on it all. > > > > Regards > > Steven Vincent Johnson > > www.OrionWorks.com > > www.zazzle.com/orionworks > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [hidden email] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: [Vo]:does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 2011.02.18
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In reply to this post by Rich Murray-2
The only access to "the physics itself" we have with finite nervous
systems is by using digital approximations with finite number strings, processed by algorithms of finite instruction size, so there are always round-off errors, which always diverge without limit, even if there are no close encounters. So, it's a huge leap of faith to assume that the "present data" for a certain finite time interval actually allows prediction of a single future path or retrodiction of a single past path -- ie, classical mechanics probably can be proved to be incurably flawed, while allowing a certain amount of qualified estimation of probable paths forward and backward in time for the first 3 "orbits" or so... I've read that actually the 3-body problem does have exact general solutions, which involve such long, very slowly converging sequences of terms, as to be practically unworkable in practice. Probaby, it can be shown that the energy needed to run an ideal finite digital computer until a certain limit of accuracy is reached (testable by running the same problem in parallel with identical computers, watching to see at what point the results start to scatter) will grow so fast with time and accuracy as to exhaust the energy available in any universe that supports the computer... Probably someone has already studied this... It's not just that shit happens -- "happens" happens... So, in reality, the "present" interval, however brief in time and tiny in space, necessarily in complex interaction with a possibly infinite external universe or hyperverse, must be inexplicable, "causeless", ie, totally "magical"... This has in recent thousands of years been a common insight for advanced explorers of expanded awareness in many traditions. Rich Murray "lookslikeallthoughtiswrong"@godmail.com On Fri, Feb 18, 2011 at 9:50 PM, Charles Hope <[hidden email]> wrote: > I'm thinking your findings of irreversibility reflected the idiosyncrasies of floating point math represented in binary numbers, and not the physics itself. > > Sent from my iPhone. > > On Feb 18, 2011, at 22:17, Rich Murray <[hidden email]> wrote: > >> does classical mechanics always fail to predict or retrodict for 3 or >> more Newtonian gravity bodies? Rich Murray 2011.02.18 >> >> Hello Steven V Johnson, >> >> Can I have a free copy of the celestial mechanics software to run on >> my Vista 64 bit PC? >> >> In fall, 1982, I wrote a 200-line program in Basic for the >> Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 >> bodies in 3D space or 5 in 2D space, about 1000 steps in an hour, >> saving every 10th position and velocity -- I could set it up to >> reverse the velocities after the orbits became chaotic after 3 1/2 >> orbits from initial perfect symmetry as circles about the common >> center of gravity, finding that they always maintained chaos, never >> returning to the original setup -- doubling the number of steps while >> reducing the time interval by half never slowed the the evolution of >> chaos by 3 1/2 orbits -- so I doubted that there is any mathematical >> basis for the claim that classical mechanics predicts the past or >> future evolution of any system with over 2 bodies, leading to a >> conjecture that no successful algorithm exists, even without any close >> encounters. >> >> Has this been noticed by others? >> >> Rich Murray [hidden email] 505-819-7388 >> 1943 Otowi Road, Santa Fe, New Mexico 87505 >> >> On Fri, Feb 18, 2011 at 4:30 PM, >> OrionWorks - "Steven V Johnson" <[hidden email]> wrote: >> >>> Just a brief side-comment... >>> >>> Some of this "lingo" is fascinating stuff to me. Having performed a >>> lot of theoretical computer simulation work on my own using good'ol >>> fashion Newtonian based Celestial Mechanics algorithms, where >>> typically I use "a = 1/r^2", I noticed orbital pattern behavior >>> transforms into something RADICALLY different, such as if I were to >>> change the classical algorithm to something like "a = 1/r^3". You can >>> also combine both of them like "a = 1/r^2 +/- 1/r^3" within the same >>> computer algorithm. That produces interesting side effects too. I'm >>> still trying to get a handle on it all. >>> >>> Regards >>> Steven Vincent Johnson >>> www.OrionWorks.com >>> www.zazzle.com/orionworks >> > > ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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In reply to this post by Russell Standish
With particular regard to computer simulations of
celestial mechanics, Gerry Sussman wrote a paper sometime in (IIRC) the late 1970s, about the ultimate instability of the solar system (one of the classical motivations for celestial mechanics in general and the 3-body problem in particular). I could be vaguer if I tried. Lee Rudolph > Yes, the n-body system with n>2 is known to be chaotic, but subject to > the constraints of the KAM theorem > (http://en.wikipedia.org/wiki/Kolmogorov-Arnold-Moser_theorem), ie > there exist quasi-periodic orbits for certain initial conditions. > > This was certainly known stuff when I studied dynamical systems as an > undergrad in the early '80s. > > On Fri, Feb 18, 2011 at 08:17:37PM -0700, Rich Murray wrote: > > does classical mechanics always fail to predict or retrodict for 3 or > > more Newtonian gravity bodies? Rich Murray 2011.02.18 > > > > Hello Steven V Johnson, > > > > Can I have a free copy of the celestial mechanics software to run on > > my Vista 64 bit PC? > > > > In fall, 1982, I wrote a 200-line program in Basic for the > > Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 > > bodies in 3D space or 5 in 2D space, about 1000 steps in an hour, > > saving every 10th position and velocity -- I could set it up to > > reverse the velocities after the orbits became chaotic after 3 1/2 > > orbits from initial perfect symmetry as circles about the common > > center of gravity, finding that they always maintained chaos, never > > returning to the original setup -- doubling the number of steps while > > reducing the time interval by half never slowed the the evolution of > > chaos by 3 1/2 orbits -- so I doubted that there is any mathematical > > basis for the claim that classical mechanics predicts the past or > > future evolution of any system with over 2 bodies, leading to a > > conjecture that no successful algorithm exists, even without any close > > encounters. > > > > Has this been noticed by others? > > > > Rich Murray [hidden email] 505-819-7388 > > 1943 Otowi Road, Santa Fe, New Mexico 87505 > > > > On Fri, Feb 18, 2011 at 4:30 PM, > > OrionWorks - "Steven V Johnson" <[hidden email]> wrote: > > > > > Just a brief side-comment... > > > > > > Some of this "lingo" is fascinating stuff to me. Having performed a > > > lot of theoretical computer simulation work on my own using good'ol > > > fashion Newtonian based Celestial Mechanics algorithms, where > > > typically I use "a = 1/r^2", I noticed orbital pattern behavior > > > transforms into something RADICALLY different, such as if I were to > > > change the classical algorithm to something like "a = 1/r^3". You can > > > also combine both of them like "a = 1/r^2 +/- 1/r^3" within the same > > > computer algorithm. That produces interesting side effects too. I'm > > > still trying to get a handle on it all. > > > > > > Regards > > > Steven Vincent Johnson > > > www.OrionWorks.com > > > www.zazzle.com/orionworks > > > > ============================================================ > > FRIAM Applied Complexity Group listserv > > Meets Fridays 9a-11:30 at cafe at St. John's College > > lectures, archives, unsubscribe, maps at http://www.friam.org > > -- > > ---------------------------------------------------------------------------- > Prof Russell Standish Phone 0425 253119 (mobile) > Mathematics > UNSW SYDNEY 2052 [hidden email] > Australia http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 20
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"A Digital Orrery," James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, in IEEE Transactions on Computers, C-34, No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in Physics #267 -- Use of supercomputers in stellar dynamics, Springer Verlag, 1986.
But also look at: which gives you a pointer to the online copy of Sussman's text on the subject.
-- rec --
On Sat, Feb 19, 2011 at 4:53 AM, <[hidden email]> wrote: With particular regard to computer simulations of ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
Re: does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 20
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In reply to this post by lrudolph
more re eventual chaos in classical mechanics: Rich Murray 2011.02.19
from Roger Critchlow <[hidden email]> to The Friday Morning Applied Complexity Coffee Group <[hidden email]> date Sat, Feb 19, 2011 at 9:19 AM subject Re: [FRIAM] does classical mechanics always fail to predict or retrodict for 3 or more Newtonian gravity bodies? Rich Murray 20 mailing list friam_redfish.com.redfish.com Filter messages from this mailing list 9:19 AM (9 hours ago) "A Digital Orrery," James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, IEEE Transactions on Computers, C-34, No. 9, pp. 822-831, September 1985, reprinted in Lecture Notes in Physics #267 -- Use of supercomputers in stellar dynamics, Springer Verlag, 1986. But also look at: http://en.wikipedia.org/wiki/Structure_and_Interpretation_of_Classical_Mechanics which gives you a pointer to the online copy of Sussman's text on the subject. -- rec -- Rich Murray: http://mitpress.mit.edu/SICM/ <a href="http://mitpress.mit.edu/SICM/book-Z-H-43.html#%_sec_3.7">http://mitpress.mit.edu/SICM/book-Z-H-43.html#%_sec_3.7 3.7 Exponential Divergence Hénon and Heiles discovered that the chaotic trajectories had remarkable sensitivity to small changes in initial conditions -- initially nearby chaotic trajectories separate roughly exponentially with time. On the other hand, regular trajectories do not exhibit this sensitivity -- initially nearby regular trajectories separate roughly linearly with time. Consider the evolution of two initially nearby trajectories for the Hénon-Heiles problem, with energy E = 1/8. Let d(t) be the usual Euclidean distance in the x, y, px, py space between the two trajectories at time t. Figure 3.23 shows the common logarithm of d(t)/d(0) as a function of time t. We see that the divergence is well described as exponential. On the other hand, the distance between two initially nearby regular trajectories grows much more slowly. Figure 3.24 shows the distance between two regular trajectories as a function of time. The distance grows linearly with time. It is remarkable that Hamiltonian systems have such radically different types of trajectories. On the surface of section the chaotic and regular trajectories differ in the dimension of the space that they explore. It is interesting that along with this dimensional difference there is a drastic difference in the way chaotic and regular trajectories separate. For higher dimensional systems the surface of section technique is not as useful, but trajectories are still distinguished by the way neighboring trajectories diverge: some diverge exponentially whereas others diverge approximately linearly. Exponential divergence is the hallmark of chaotic behavior. ] Hello Stephen A. Lawrence, Thanks for the informative answer. It'd be impressive if the most accurate methods since this review in 1987 agree with each other far into the future and past -- how can we find out the details about results for the 3-body problem, in commonsense terms? Is this accessible for PC users? Could a business sell the program and run a collaborative blog for users? Laskar #1 In 1989, Jacques Laskar of the Bureau des Longitudes in Paris published the results of his numerical integration of the Solar System over 200 million years. These were not the full equations of motion, but rather averaged equations along the lines of those used by Laplace. Laskar's work showed that the Earth's orbit (as well as the orbits of all the inner planets) is chaotic and that an error as small as 15 metres in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time. [edit]Laskar & Gastineau Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more thorough approach by directly simulating 2500 possible futures. Each of the 2500 cases has slightly different initial conditions: Mercury's position varies by about 1 metre between one simulation and the next.[13] In 20 cases, Mercury goes into a dangerous orbit and often ends up colliding with Venus or plunging into the sun. Moving in such a warped orbit, Mercury's gravity is more likely to shake other planets out of their settled paths: in one simulated case its perturbations send Mars heading towards Earth.[14] 13. ^ "Solar system's planets could spin out of control". newscientist. Retrieved 2009-06-11. 14. ^ "Existence of collisional trajectories of Mercury, Mars and Venus with the Earth". Retrieved 2009-06-11. http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html Letter Nature 459, 817-819 (11 June 2009) doi:10.1038/nature08096; Received 17 February 2009; Accepted 22 April 2009 ARTICLE LINKS Figures and tables Supplementary info SEE ALSO News and Views by Laughlin Editor's Summary Existence of collisional trajectories of Mercury, Mars and Venus with the Earth J. Laskar 1 & M. Gastineau 1 Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar 1 Correspondence and requests for materials should be addressed to J.L. (Email: [hidden email] ). Abstract It has been established that, owing to the proximity of a resonance with Jupiter, Mercury’s eccentricity can be pumped to values large enough to allow collision with Venus within 5 Gyr (refs 1–3). This conclusion, however, was established either with averaged equations 1, 2 that are not appropriate near the collisions or with non-relativistic models in which the resonance effect is greatly enhanced by a decrease of the perihelion velocity of Mercury 2, 3. In these previous studies, the Earth’s orbit was essentially unaffected. Here we report numerical simulations of the evolution of the Solar System over 5 Gyr, including contributions from the Moon and general relativity. In a set of 2,501 orbits with initial conditions that are in agreement with our present knowledge of the parameters of the Solar System, we found, as in previous studies 2, that one per cent of the solutions lead to a large increase in Mercury’s eccentricity -- an increase large enough to allow collisions with Venus or the Sun. More surprisingly, in one of these high-eccentricity solutions, a subsequent decrease in Mercury’s eccentricity induces a transfer of angular momentum from the giant planets that destabilizes all the terrestrial planets ~3.34 Gyr from now, with possible collisions of Mercury, Mars or Venus with the Earth. Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar 1 Correspondence and requests for materials should be addressed to J.L. (Email: [hidden email] ). So, with the most accurate methods, 1% of <5x10^9 Earth orbits lead to chaos -- but also occurring in the solar system in that time are changes via civilizations, solar evolution, major meteor impacts, intra solar system gas density and temperature changes, about 20 orbits around the Galactic center, with resulting encounters with dark matter flows and the Galactic plane, and things that go bump in the night... Rich On Sat, Feb 19, 2011 at 1:59 PM, Stephen A. Lawrence <[hidden email]> wrote: > > > On 02/18/2011 10:17 PM, Rich Murray wrote: > > does classical mechanics always fail to predict or retrodict for 3 or > more Newtonian gravity bodies? Rich Murray 2011.02.18 > [ ... ] > > > In fall, 1982, I wrote a 200-line program in Basic for the > Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 > bodies in 3D space... > [ ... ] > so I doubted that there is any mathematical > basis for the claim that classical mechanics predicts the past or > future evolution of any system with over 2 bodies, leading to a > conjecture that no successful algorithm exists, even without any close > encounters. > > Has this been noticed by others? > > See, for example, > > http://en.wikipedia.org/wiki/Stability_of_the_Solar_System#Digital_Orrery > > > There are also far better algorithms than what you were using, which, I'm > sure, was a simple integrator of the nonlinear system of equations. Simply > cutting the time step doesn't do much for you if the basic algorithm isn't > very accurate. > > See, for example, > > http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-46DFTHW-8W&_user=10&_coverDate=12%2F31%2F1987&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=59646ea61335b206d3a7cea0bed0ce8d&searchtype=a > > (sorry, I don't have the full text, but the abstract sounds interesting.) On Sat, Feb 19, 2011 at 4:53 AM, <[hidden email]> wrote: > With particular regard to computer simulations of > celestial mechanics, Gerry Sussman wrote a paper > sometime in (IIRC) the late 1970s, about the > ultimate instability of the solar system (one > of the classical motivations for celestial > mechanics in general and the 3-body problem > in particular). > > I could be vaguer if I tried. > > Lee Rudolph > >> Yes, the n-body system with n>2 is known to be chaotic, but subject to >> the constraints of the KAM theorem >> (http://en.wikipedia.org/wiki/Kolmogorov-Arnold-Moser_theorem), ie >> there exist quasi-periodic orbits for certain initial conditions. >> >> This was certainly known stuff when I studied dynamical systems as an >> undergrad in the early '80s. >> >> On Fri, Feb 18, 2011 at 08:17:37PM -0700, Rich Murray wrote: >> > does classical mechanics always fail to predict or retrodict for 3 or >> > more Newtonian gravity bodies? Rich Murray 2011.02.18 >> > >> > Hello Steven V Johnson, >> > >> > Can I have a free copy of the celestial mechanics software to run on >> > my Vista 64 bit PC? >> > >> > In fall, 1982, I wrote a 200-line program in Basic for the >> > Timex-Sinclair $100 computer with 20KB RAM that would do up to 4 >> > bodies in 3D space or 5 in 2D space, about 1000 steps in an hour, >> > saving every 10th position and velocity -- I could set it up to >> > reverse the velocities after the orbits became chaotic after 3 1/2 >> > orbits from initial perfect symmetry as circles about the common >> > center of gravity, finding that they always maintained chaos, never >> > returning to the original setup -- doubling the number of steps while >> > reducing the time interval by half never slowed the the evolution of >> > chaos by 3 1/2 orbits -- so I doubted that there is any mathematical >> > basis for the claim that classical mechanics predicts the past or >> > future evolution of any system with over 2 bodies, leading to a >> > conjecture that no successful algorithm exists, even without any close >> > encounters. >> > >> > Has this been noticed by others? >> > >> > Rich Murray [hidden email] 505-819-7388 >> > 1943 Otowi Road, Santa Fe, New Mexico 87505 >> > >> > On Fri, Feb 18, 2011 at 4:30 PM, >> > OrionWorks - "Steven V Johnson" <[hidden email]> wrote: >> > >> > > Just a brief side-comment... >> > > >> > > Some of this "lingo" is fascinating stuff to me. Having performed a >> > > lot of theoretical computer simulation work on my own using good'ol >> > > fashion Newtonian based Celestial Mechanics algorithms, where >> > > typically I use "a = 1/r^2", I noticed orbital pattern behavior >> > > transforms into something RADICALLY different, such as if I were to >> > > change the classical algorithm to something like "a = 1/r^3". You can >> > > also combine both of them like "a = 1/r^2 +/- 1/r^3" within the same >> > > computer algorithm. That produces interesting side effects too. I'm >> > > still trying to get a handle on it all. >> > > >> > > Regards >> > > Steven Vincent Johnson >> > > www.OrionWorks.com >> > > www.zazzle.com/orionworks ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org |
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