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November 22

solving this ODE

the following ode: wolfram alpha link. So, I thought I would want to plot how orbit looks like using the usual Newton laws of motion, but trying to solve the position of the satellite given nothing but initial position and initial velocity, I seem to get this ugly ODE, I'm not very experienced with nonlinear ODEs, but I still would want to get some insight into how these odes could be solved, or how to draw a slope field from it, so that I can understand it. thanks Sistemx (talk) 05:33, 22 November 2019 (UTC)[reply]

If I understand the equation correctly, this is the equation of a repulsive inverse law, as opposed to gravity which is an attractive inverse square law. This is a central force and there is a lot of material on solving these, see Classical central-force problem and Binet equation. Nonlinear differential equations, especially of higher order, rarely have closed form solutions and some kind of numerical approach is generally the way to go. In this case the equation has a special form which is well studied so you should be able to get more information without numerical analysis. --RDBury (talk) 06:27, 22 November 2019 (UTC)[reply]

@Sistemx: Long story short is, you'll want to rewrite your equation in polar coordinates where it'll be easier to uncouple the system. --Jasper Deng (talk) 12:51, 22 November 2019 (UTC)[reply]

Axioms where 1/|ℝ| > 0?

What are the best-studied systems of axioms in which the reciprocal of the cardinality of the continuum is an infinitesimal greater than zero? An example of where such an inequality might be useful would be in describing the probability of a possibly continuously-distributed variable having a given exact value (which would be ${\frac {1}{|\mathbb {R} |}}$ when the distribution had support at that value but no discrete probability of that value, 0 when it had no support, and a real number in (0, 1] when it had discrete probability). Likewise, "almost surely but not surely" could be formalized as "with probability $1-{\frac {1}{|\mathbb {R} |}}$ ". Neon Merlin 06:20, 22 November 2019 (UTC)[reply]

I think what you're getting at is something like Loeb space. We don't have much on it since there don't seem to be many secondary sources, but a search on "non-standard measure theory" turns up some relevant papers, stack exchange posts, etc. Pretty sure you'll want a good grounding in both (standard) measure theory and non-standard analysis before seeing what happens when you smash them together. --RDBury (talk) 15:47, 22 November 2019 (UTC)[reply]

Well, the thing is, nonstandard analysis doesn't recognize a hyperreal equal to the cardinality of the reals (or the reciprocal of that cardinality). So I kind of doubt that these Loeb spaces will be a direct answer to the literal question. I can't rule out that they'd be an answer to the (not directly stated) underlying question about replacing "almost surely" by probability 1 minus an infinitesimal, but I kind of doubt that they do.

If you want to take the reciprocal of the cardinality of R, the only context I know of where that might make sense is the surreal numbers. I'm not sure it makes sense in that context, but at least it seems promising. --Trovatore (talk) 00:29, 23 November 2019 (UTC)[reply]

I think surreal numbers use ordinals rather than cardinals, so you can talk about

1 \over \omega

but not about

1 \over \aleph _{0}

. I understand that nonstandard analysis doesn't exactly match the OP's question, I doubt anything does, but Loeb space was the closest thing I could find. I'm thinking the main problem is arithmetic with cardinals has enough weirdness that if you try to form an algebraic system with them you end up with contradictory results. For example,

\aleph _{0}+\aleph _{0}=\aleph _{0}

, so cancelling you get

\aleph _{0}=0

. --RDBury (talk) 04:54, 23 November 2019 (UTC)[reply]

Well, you could take the reciprocal of the initial ordinal corresponding to the cardinality of R. That's a lot more than you can do with ordinary hyperreals (say, the ultrapower of the reals by a nonprincipal ultrafilter on N). --Trovatore (talk) 19:16, 23 November 2019 (UTC)[reply]