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You have four big problems and two small problems. The big problems are that you are initializing the Earth's and Moon's initial position and velocity incorrectly. The initial distance between the Earth and Moon is off by a factor of 1.0123, as is the initial relative velocity. The small problems are (1) an incorrect value for the Earth-Moon semi-major axis ...

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You may use PICOS for Python: "PICOS is a user friendly interface to several conic and integer programming solvers, very much like YALMIP under MATLAB." Since the version 1.0.1, it is possible to do complex semidefinite programming with Picos: http://picos.zib.de/v101dev/complex.html

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UPDATED: See below. Your NDSolve inputs seem to be doing what I would expect for a mass around a gravitational center. Using: a = 0; b = 0; traj = Table[ s = NDSolve[{x''[t] == -x[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2), y''[t] == -y[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2), x[0] == 1, y[0] == 0, x'[0] == 0, y'[0] == v}, {x, y}, {t, -20, ...

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If your potential is $\propto 1/r$, you're effectively simulating gravity. If it gives you better intuition, imagine it as the earth around the sun. As long as your numerical solver is doing a decent job, you shouldn't expect the ball to spiral in, the correct solution would be a conic section, that is it would orbit the origin in an elliptical path if the ...

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There is a normalized form, though it's properly called the dimensionless Euler equations. The way to do it is define: scale time $t_0$ scale density $\rho_0$ scale length $L_0$ and then derive the scales from these: $$v_0 = \frac{L_0}{t_0},\quad p_0=\rho_0v_0^2$$ NB: it is possible to use other combinations, but I find that these are often the ...

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I think the most powerful approach to the general problem of arbitrary domains is the approach detailed in Reviving the Method of Particular Solutions by Timo Betcke and Lloyd N. Trefethen from 2005. [doi] [pdf]. In it they describe a modern modification of the historical method of numerically finding the solution to the helmhotz equation from ...

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