With symplectic integrators, the aim of the game (among other things) is to try and preserve structure of the Hamiltonian - for e.g. conserved quantities like angular momentum. With specific focus on classical orbital mechanics, does anyone have experience with symplectic integration incorporating non-conserved forces? for e.g. Solar radiation, drag etc. I can't see a reason why they would have an effect on the integrator. Can anyone shed some light on this?
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asked Dec 29, 2016 at 0:24
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1$\begingroup$ AFAI understand, symplectic integrators preserve Liouville invariants, i.e. the area in 2-dimensional phase space, the 4-volume and some specific sum of 2-areas in 4-dimensional phase space etc. These are not preserved unless the Hamiltonian is conserved. See Channell and Scovel, Nonlinearity vol 3 (1990) p.231. Could you provide additional details on what you have in mind? $\endgroup$– ZeroTheHeroDec 29, 2016 at 4:14
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$\begingroup$ EDIT: I was looking at orbital models - initially with classical orbits for e.g. satellite motion. Essentially a Newtonian force model beginning with the normal two body problem then adding additional forces expressed in terms of a Hamiltonian for e.g. forces like extra celestial bodies, non-spherical geometry of the Earth. Then it occurred to me about non-conservative forces like solar radiation etc etc. and I'm not entirely sure about symplectic integration methods and non-conservative forces. $\endgroup$– RumplestillskinDec 29, 2016 at 4:20
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$\begingroup$ Possibly if you treat the non-conservative bits as perturbations but I've never seen this, Maybe this is done in atmosphere modeling... would be a place to look. $\endgroup$– ZeroTheHeroDec 29, 2016 at 4:43
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$\begingroup$ So do you reckon a model comprised of both conservative and non-conservative forces would hold up better using classical integrators rather than structure preserving integrators? I think this is quite interesting. $\endgroup$– RumplestillskinDec 29, 2016 at 4:47
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1$\begingroup$ AFAI know symplectic integrators are quite complicated. My experience is that they provide faster and more accurate integrators because they maintain important qualitative features of the systems. They work well (save time) with many-body problems (galaxies colliding etc). Orbital dynamics and associated perturbations are historically well described using Hamilton-Jacobi type formalisms. I would start there before contemplating symplectic integrators, unless you are interested in very long timescales (order of Solar system age). Just a guess I'm afraid; not an answer. $\endgroup$– ZeroTheHeroDec 29, 2016 at 4:56
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1 Answer
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Symplectic integrators are only for conservative forces and cannot, in general, be used for non-conservative forces.
However, there have been attempts at generalizing such integrators to allow for non-conservative forces. Two recent examples I was able to find were
- Tsang, Galley, Stein and Turner Slimplectic Integrators: Variational Integrators for General Nonconservative Systems (2105) (arXiv link)1
- Luo & Guo Application of Explicit Symplectic Algorithms to Integration of Damping Oscillators (2011) (arXiv link)
The abstract to the Tsang et al paper mentions a few other of the authors' previous papers on the subject as well.
1. There is a public code (in Python) that shows their method.
answered Dec 30, 2016 at 12:19
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$\begingroup$ Excellent. I'm struggling to see the importance of the integrators in real life situations where non conservative forces are abundant. Well, in particular to orbital motion. I can't see an advantage in using an SI vs a high order classical integrator except in the case of an articulate designed classroom example. Look forward to reading the papers. $\endgroup$ Dec 30, 2016 at 23:12