Andrew Cooke | Contents | Latest | RSS | 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

Choochoo Training Diary

Last 100 entries

Surprise Paradox; [Books] Good Author List; [Computing] Efficient queries with grouping in Postgres; [Computing] Automatic Wake (Linux); [Computing] AWS CDK Aspects in Go; [Bike] Adidas Gravel Shoes; [Computing, Horror] Biological Chips; [Books] Weird Lit Recs; [Covid] Extended SIR Models; [Art] York-based Printmaker; [Physics] Quantum Transitions are not Instantaneous; [Computing] AI and Drum Machines; [Computing] Probabilities, Stopping Times, Martingales; bpftrace Intro Article; [Computing] Starlab Systems - Linux Laptops; [Computing] Extended Berkeley Packet Filter; [Green] Mainspring Linear Generator; Better Approach; Rummikub Solver; Chilean Poetry; Felicitations - Empowerment Grant; [Bike] Fixing Spyre Brakes (That Need Constant Adjustment); [Computing, Music] Raspberry Pi Media (Audio) Streamer; [Computing] Amazing Hack To Embed DSL In Python; [Bike] Ruta Del Condor (El Alfalfal); [Bike] Estimating Power On Climbs; [Computing] Applying Azure B2C Authentication To Function Apps; [Bike] Gearing On The Back Of An Envelope; [Computing] Okular and Postscript in OpenSuse; There's a fix!; [Computing] Fail2Ban on OpenSuse Leap 15.3 (NFTables); [Cycling, Computing] Power Calculation and Brakes; [Hardware, Computing] Amazing Pockit Computer; Bullying; How I Am - 3 Years Post Accident, 8+ Years With MS; [USA Politics] In America's Uncivil War Republicans Are The Aggressors; [Programming] Selenium and Python; Better Walking Data; [Bike] How Fast Before Walking More Efficient Than Cycling?; [COVID] Coronavirus And Cycling; [Programming] Docker on OpenSuse; Cadence v Speed; [Bike] Gearing For Real Cyclists; [Programming] React plotting - visx; [Programming] React Leaflet; AliExpress Independent Sellers; Applebaum - Twilight of Democracy; [Politics] Back + US Elections; [Programming,Exercise] Simple Timer Script; [News] 2019: The year revolt went global; [Politics] The world's most-surveilled cities; [Bike] Hope Freehub; [Restaurant] Mama Chau's (Chinese, Providencia); [Politics] Brexit Podcast; [Diary] Pneumonia; [Politics] Britain's Reichstag Fire moment; install cairo; [Programming] GCC Sanitizer Flags; [GPU, Programming] Per-Thread Program Counters; My Bike Accident - Looking Back One Year; [Python] Geographic heights are incredibly easy!; [Cooking] Cookie Recipe; Efficient, Simple, Directed Maximisation of Noisy Function; And for argparse; Bash Completion in Python; [Computing] Configuring Github Jekyll Locally; [Maths, Link] The Napkin Project; You can Masquerade in Firewalld; [Bike] Servicing Budget (Spring) Forks; [Crypto] CIA Internet Comms Failure; [Python] Cute Rate Limiting API; [Causality] Judea Pearl Lecture; [Security, Computing] Chinese Hardware Hack Of Supermicro Boards; SQLAlchemy Joined Table Inheritance and Delete Cascade; [Translation] The Club; [Computing] Super Potato Bruh; [Computing] Extending Jupyter; Further HRM Details; [Computing, Bike] Activities in ch2; [Books, Link] Modern Japanese Lit; What ended up there; [Link, Book] Logic Book; Update - Garmin Express / Connect; Garmin Forerunner 35 v 230; [Link, Politics, Internet] Government Trolls; [Link, Politics] Why identity politics benefits the right more than the left; SSH Forwarding; A Specification For Repeating Events; A Fight for the Soul of Science; [Science, Book, Link] Lost In Math; OpenSuse Leap 15 Network Fixes; Update; [Book] Galileo's Middle Finger; [Bike] Chinese Carbon Rims; [Bike] Servicing Shimano XT Front Hub HB-M8010; [Bike] Aliexpress Cycling Tops; [Computing] Change to ssh handling of multiple identities?; [Bike] Endura Hummvee Lite II; [Computing] Marble Based Logic; [Link, Politics] Sanity Check For Nuclear Launch; [Link, Science] Entropy and Life

© 2006-2017 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