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This probably isn't exactly what you're looking for, but if you're looking for the time-independent bound states of a system, the Fourier grid Hamiltonian method may be applicable. Here is an application of it to the following strange-looking potential well: Here are a few low-energy bound states: And here are some of the high-energy ...

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There are some simulation tools available online, but whether they are useful to you depends on the details of your requirements. Check out the list of quantum simulators here. Or this one.

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I'm not sure if it would necessarilly lead to such an instability, as Joce says for a first order method you'll need a really small time step to maintain any accuracy, but at the moment your acceleration is very wrong. What you effectively have here is ${\bf a} = - \frac{GM}{r^2} {\bf r}$ when you want ${\bf a} = - \frac{GM}{r^2} {\bf \hat{r}}$ (or ${\bf ... 0 The time discretisation you have chosen is an explicit Euler scheme. In order for it to be stable, you need the time step to be low enough, see e.g. wikipedia. You could use an implicit method, or increase the order of the method, but in any case there will always be a numerical drift in the orbits, proportional to the numerical accuracy. This can be ... 0 As a free and powerful software I can recommend you Cadabra. It is designed specially for the field theory calculations. Cadabra is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor computer algebra, tensor polynomial simplification including ... 1 Disclaimer: I have no engineering background, so if anything I write is in error, definitely point it out. However, if the 3-axis accelerometer only returns the proper acceleration vector$\mathbf{a}\$, then if the object is moving around and physically accelerating, it is impossible to determine the orientation of the object without additional information. ...

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