| Andrew Cooke | Contents | Latest | RSS | Twitter | Previous | Next

C[omp]ute

Welcome to my blog, which was once a mailing list of the same name and is still generated by mail. Please reply via the "comment" links.

Always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Python web; Java enterprise; functional languages; GPGPU; SQL databases; etc. Based in Santiago, Chile; telecommute worldwide. CV; email.

Personal Projects

Lepl parser for Python.

Colorless Green.

Photography around Santiago.

SVG experiment.

Professional Portfolio

Calibration of seismometers.

Data access via web services.

Cache rewrite.

Extending OpenSSH.

C-ORM: docs, API.

Last 100 entries

The Best Of All Possible Worlds; Kenzaburo Oe: The Changeling; Peach Jam; Taste Test; Strawberry and Raspberry Jam; flac to mp3 on OpenSuse 42.1; Also, Sebald; Kenzaburo Oe Interview; Otake (Kitani Minoru) move Black 121; Is free speech in British universities under threat?; I am actually good at computers; Was This Mansplaining?; WebFaction / LetsEncrypt / General Disappointment; Sensible Philosophy of Science; George Ellis; Misplaced Intuition and Online Communities; More Reading About Japan; Visibilty / Public Comments / Domestic Violence; Ferias de Santiago; More (Clearly Deliberate); Deleted Obit Post; And then a 50 yo male posts this...; We Have Both Kinds Of Contributors; Free Springer Books; Books on Religion; Books on Linguistics; Palestinan Electronica; Books In Anthropology; Taylor Expansions of Spacetime; Info on Juniper; Efficient Stream Processing; The Moral Character of Crypto; Hearing Aid Info; Small Success With Go!; Re: Quick message - This link is broken; Adding Reverb To The Echo Chamber; Sox Audio Tools; Would This Have Been OK?; Honesty only important economically before institutions develop; Stegangraphy via PS4; OpenCL Mess; More Book Recommendations; Good Explanation of Difference Between Majority + Minority; Musical Chairs - Who's The Privileged White Guy; I can see straight men watching this conversation and laffing; Meta Thread Defending POC Causes POC To Close Account; Indigenous People Of Chile; Curry Recipe; Interesting Link On Marginality; A Nuclear Launch Ordered, 1962; More Book Recs (Better Person); It's Nuanced, And I Tried, So Back Off; Marx; The Negative Of Positive; Jenny Holzer Rocks; Huge Article on Cultural Evolution and More; "Ignoring language theory"; Negative Finger Counting; Week 12; Communication Via Telecomm Bids; Finding Suspects Via Relatives' DNA From Non-Crime Databases; Statistics and Information Theory; Ice OK in USA; On The Other Hand; (Current Understanding Of) Chilean Taxes / Contributions; M John Harrison; Playing Games on a Cloud GPU; China Gamifies Real Life; Can't Help Thinking It's Thoughtcrime; Mefi Quotes; Spray Painting Bike Frame; Weeks 10 + 11; Change: No Longer Possible To Merge Metadata; Books on Old Age; Health Tree Maps; MRA - Men's Rights Activists; Writing Good C++14; Risk Assessment - Fukushima; The Future of Advertising and Surveillance; Travelling With Betaferon; I think I know what I dislike so much about Metafilter; Weeks 8 + 9; More; Pastamore - Bad Italian in Vitacura; History Books; Iraq + The (UK) Governing Elite; Answering Some Hard Questions; Pinochet: The Dictator's Shadow; An Outsider's Guide To Julia Packages; Nobody gives a shit; Lepton Decay Irregularity; An Easier Way; Julia's BinDeps (aka How To Install Cairo); Good Example Of Good Police Work (And Anonymity Being Hard); Best Santiago Burgers; Also; Michael Emmerich (Vibrator Translator) Interview (Japanese Books); Clarice Lispector (Brazillian Writer); Books On Evolution; Looks like Ara (Modular Phone) is dead; Index - Translations From Chile

© 2006-2015 Andrew Cooke (site) / post authors (content).

Angular Momentum, Differential Equations

From: "andrew cooke" <andrew@...>

Date: Wed, 11 Jan 2006 10:44:21 -0300 (CLST)

A while back, on AskMe, someone asked aboutthe behaviour of the following
physical system:

 Two points, of equal mass, are stationary in an otherwise empty universe.
 They are attracted to each other via gravity.

Now to solve this directly you know that acceleration is proportional to
gravity and so inversely proportional to the separation.  And you end up
with something like
  x'' = -k / x^2
Which is rather nasty to solve (something like a tan substitution?).

Now vacapinta observed that the system is the "usual" gravitational two
body problem, so the time before the two objects collide (which was
requested by the original poster) was half the period given by Kepler's
equations.

A bit of a recap on the two body problem: it reduces to the classical
single body problem (eg planet in orbit around much more massive star)
with a simple transformation (centre of mass, major axis etc).

This made my brain explode.  Because solving the single body problem is
easy.  Yet I just said that solving the problem above was hard.

Why is the single body problem easy?  Indirectly, because angular momentum
is not zero.  So the particle always misses the centre, avoiding the nasty
infinities you get with the two points described above (lim 1/x^2 as x ->
infinity).

Another way of looking at this is to see that you get simple harmonic
motion (or something very like it; I haven't looked in detail) when you
resolve along the x/y axes.  This is because when x=0, y is non-zero and
the force is perpendicular to the x axis.

That also helps show why angular momentum doesn't affect the period of the
orbit - because it's "orthogonal" to the forces involved.

So by introducing angular momentum into the problem I first described you
end up with a system that is simpler, mathematically.  You can then look
at the limit as the angular momentum tends to zero to see the solution for
the original system.

This seems very odd to me.

And a final comment - if you think about this in terms of Lagrangians
(which I was never taught, despite having a degree in physics from
Cambridge - something that seems *very* odd in retrospect) then you get a
path integral (I think).  Now is there some relationship between this and
all that fuss about poles in the complex plane and path integrals that go
around them?  I wish I could remember all that stuff....

Andrew

Mach

From: "andrew cooke" <andrew@...>

Date: Wed, 11 Jan 2006 10:49:29 -0300 (CLST)

In an otherwise empty universe two points may not have angular momentum,
right?

(Also, replies should now appear "instantly")

Comment on this post