Is any software design to take into account modern physic theory such as general relativity or string theory in to mechanical simulation for celestial mechanics?
For a straight set of differential equations where n might get large, I'd recommend XMDS2. (www.xmds.org)
Once installed, there's an n-body simulation in the examples folder (gravity.xmds)
For solar system simulation general relativity is not completely irrelevant, it's employed in the form of some post-Newtonian approximation. A software realization of the two-body problem is shown in the appended link. String theory obviously makes no sense here. Software for simulating solar system dynamics mostly goes under the name of 'ephemerides software'. There is much available which you can easily find in the web. I like to program such stuff myself.
http://demonstrations.wolfram.com/TheGravitationalTwoBodyProblemInTheEinsteinInfeldHoffmannApp/
There are several advanced software packages which incorporate general relativity. In fact you cannot perform accurate orbital integrations without considering GTR. These packages are not available in general, but their outputs are and you can get code to interpolate planet positions to some epoch. See for instance the JPL software at http://ssd.jpl.nasa.gov/horizons.cgi. This you can access via a web interface. JPL has more accurate products available such as DE421. The French, Russians and Germans also have very good solar system ephemerides packages. They all incorporate GTR. I am writing such a package as I need it to do Lunar Laser Ranging data analysis. It is a non-trivial task and needs a large number of solve-for parameters in a solar system barycentre reference frame.
The General Relativity will not be relevant for most of the objects of the Solar system, because our observations are not precise enough, the objects are so far from the Sun that the effect of GR will be in the computation noise, and/or the kind of observations we have are weakly influenced by GR (astrometric observations are not very sensitive to it for example, except in extreme cases like Mercury). In the case of the Moon, especially LLR data, you indeed need to include the effects of GR, at least in the treatment of the lighttime (for the motion of the Moon itself I do not remember clearly if GR is needed).
For both JPL and IMCCE (France), the softwares are not public, only the end products, that is the ephemerides, but you can find most details on their principles in articles by people from IMCCE (look for Valery Lainey, Agnes Fienga, me (shameful self promotion, sorry)...).
As for string theory... totally irrelevant here, it is a theory which will be relevant for atomic/sub-atomic scales, not at all for Celestial Dynamics. In the same way, we do not take Quantum Mechanics into account, we do not expect a tunnel effect instead of a shock in the case of the collision of two asteroids (a bit oversimplified, but I think you get the idea).
Actually n-body problem is my main interest in this question. a software that simulate n-body problem!
For a straight set of differential equations where n might get large, I'd recommend XMDS2. (www.xmds.org)
Once installed, there's an n-body simulation in the examples folder (gravity.xmds)
I am suffering to find the exact graph of Poincare surfaces of section in the restricted three/four body problem . I have used Mathematica software but I cannot find smooth graph of Poincare surfaces of section.
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