if I recall correctly, for 1/r^2 gravity there is a conserved quantity other than the energy and the angular momentum — this is why this system is “integrable". this additional quantity puts a bunch of elliptical orbits into an equivalence class with the same period. but I don’t remember how it works… can someone tell us? - Cris
On Jan 16, 2018, at 1:02 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Kepler's [third?] law, which states that the squared period is proportional to the cubed orbital radius, is derivable from dimensional analysis.
Of course, the dimensional analysis 'proof' of Kepler's third law is somewhat dissatisfying to me because it doesn't say *which* radius should be cubed -- this matters if different planets have different eccentricities.
[It happens that it's the semi-major axis, but this is by no means obvious. Does anyone know an *elegant* justification for this being the case (i.e. not involving churning through lots of coordinate calculations)?]
Best wishes,
Adam P. Goucher
Sent: Tuesday, January 16, 2018 at 7:03 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] "Baby" dimensional analysis
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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