[Spoiler: Rationale for the poll]
It seems not everyone on this thread (and no-one at SQUAD) is really familiar with numerical integration. For the sake of everybody's sanity, I think I'll make a short series of posts roughly explaining Hamiltonian mechanics, RK and SPRK integrators, as well as the Kepler problem (which is more complicated than it looks) and the Saha & Tremaine integrator. The point of the above poll is to give me a rough idea where to start.
EDIT: Okay, I'll proceed with my explanations assuming basic physics and calculus.
[Spoiler: Recommended reading]
- For those of you who are not familiar with calculus, you can get a quick introduction in Richard Feynman's Lectures on Physics, chapter 8. Also read sections 11-4 and 11-5 of chapter 11 to see how that applies to vector-valued functions.
- Readers unfamiliar with the concepts of kinetic and potential energy should read chapter 4 of the Lectures on Physics
I am writing a series of explanatory posts in order to make the discussion easier to follow:
I'll add links to my upcoming posts on the Kepler problem, SPRK integrators etc. here.
I was asked to make a dev thread for this mod (which only exists as a prototype useless for gaming purposes at the moment), so here it is.
The source code is available on GitHub under the MIT license.
This mod aims to replace KSP's unstable physics integration with a higher-order symplectic integrator, adding N-body Newtonian gravitation in the process. Once N-body gravitation with point masses is made playable (including faithful trajectory prediction over long periods and reasonably accurate dynamically-updated trajectory predictions over a few years for vessels under thrust), further interesting physical effects can be added.
[Spoiler: list of acronyms]
ACS: Attitude Control System
AMD: Advanced Micro Devices
ARG: Ada Rapporteur Group
API: Application Programming Interface
BIPM: Bureau International des Poids et Mesures
CR3BP: Circular-Restricted 3-Body Problem
CGPM: Conférence Générale des Poids et Mesures
CIPM: Comité International des Poids et Mesures
CIL: Common Intermediate Language
CLI: Common Language Infrastructure
CLR: Common Language Runtime
CLS: Common Language Specification
CPU: Central Processing Unit
CTS: Common Type System
DFT: Discrete Fourier Transform
DLL: Dynamic-Link Library
ER3BP: Elliptic-Restricted 3-Body Problem
ETHZ: Eidgenössische Technische Hochschule Zürich
EVA: Extra-Vehicular Activity
FAR: Ferram Aerospace Research
FFT: Fast Fourier Transform
FMM: Fast Multipole Method
FPU: Floating-Point Unit
FSAL: First Same As Last
GR: General Relativity
ICC: Intel C++ Compiler
IDE: Integrated Development Environment
IEEE: Institute of Electrical and Electronics Engineers
IRC: Internet Relay Chat
JPL: Jet Propulsion Laboratory
KSO: Kerbin-Synchronous Orbit
KSP: Kerbal Space Program
LKO: Low Kerbin Orbit
MET: Mission Elapsed Time
MIT: Massachusetts Institute of Technology
NASA: National Aeronautics and Space Administration
OBE: Overcome By Events
ODE: Ordinary Differential Equation
OP: Original Post
PDF: Portable Document Format
PFCE: PlanetFactory Creator Edition
PSA: Public Service Announcement
RCS: Reaction Control System
RHS: Right-Hand Side
RSS: Real Solar System
RT2: RemoteTech 2
SAS: Stability Augmentation System
SI: Système International
SIMD: Single Instruction, Multiple Data
SOI: Sphere Of Influence
SPRK: Symplectic Partitioned Runge-Kutta
SSE: Streaming SIMD Extensions
VB: Visual Basic
VS2013: Visual Studio 2013
WIP: Work In Progress
B: number of massive bodies.
V: number of massless bodies.
[Spoiler: main integration]
The main integration refers to the integration of celestial dynamics in the absence of thrust, both for the propagation of the vessels and for the predicted trajectories of passive vessels and planets. It also includes the real-time integration of vessels under thrust and, when this is added, the integration and trajectory prediction for planned burns under timewarp (e.g. ion engines).
[Spoiler: dynamic integration]
The dynamic integration, or integration for vessels under thrust, refers to the predicted trajectory of a vessel assuming thrust is cut immediately (what happens is stock KSP when you fire your engines). This integration has to be very fast, as the initial conditions change quickly and prediction over several years may be required for planetary transfers &c.
[Spoiler: fictitious forces]
Fictitious forces in any reference frame are those proportional to the mass of the object upon which they act. This includes gravity. This convention only really makes sense in GR, which I do not currently intend to implement, but it's more convenient.
[Spoiler: geometric acceleration]
The geometric acceleration is the acceleration due to fictitious forces.
[Spoiler: proper acceleration]
The proper acceleration is the acceleration not due to fictitious forces.
Dates are ISO 8601 extended format.
The refactoring of the physics bubble has been done (and some tests have been added), as well as some cleanup in the C# adapter (including some performance improvements, a lot of the performance issues occur on the C# side since the implementation is a bit careless there).
More refactoring occurred (use of a not_null template for non-null pointers).
The next step on the path towards playability is persistence: the state of the plugin must be persisted so that we remember the states of the planets, the histories of the vessel trajectories, etc. (currently loading a save or reverting will result in either a crash or a loss of consistency due to the planets having moved around).
Persistence of our types will be achieved using protocol buffers, stored in config nodes: we don't want to use KSP's config format directly, since we have lots of highly structured data and operate far from KSP's API it would be inefficient and clumsy.
We'll probably also use protobufs for caching when implementing trajectory predictions later on.
Some work has already been done in that direction.
Here are some more screenshots, showing a flight from Kerbin to the Kerbin-Mun L4 Lagrange point (the reference frame for each screenshot is indicated in the "Traces of various description" window).
Support for intrinsic acceleration has been added. Refactoring will be needed, but it works, as demonstrated by the following plane change manœuvre:
Some work has been done on better abstractions for rendered trajectories, nothing significant yet (in particular there still is something in Õ(n) which could be in Õ(1), runs at each frame, where the constant factor involves evaluating trig functions, and n~10_000 is the number of rendered points, so that rendering trajectories in the barycentric rotating frame is slow), but some pretty pictures.
This shows a vessel near the L5 point of the Kerbin-Mun system.
In the first image, the trajectory of the vessel is displayed in the nonrotating reference frame centred on the Mun, in the second it is displayed in the nonrotating reference frame centred on Kerbin, and in all subsequent images it is displayed in the barycentric corotating reference frame of the Kerbin-Mun system.
Work on intrinsic acceleration is ongoing.
Code has been written to plot the trajectories in nonrotating body-centred reference frames (and barycentric corotating, not reviewed yet). While abstractions will have to be written to do that more cleanly and efficiently, this yields some pretty pictures.
Caveat lector: While some of the following orbits may look very wild, bear in mind they are in non-inertial reference frames. Two of these wilder trajectories are followed by the same trajectory in a more pedestrian reference frame. In the Kerbin-centric non-rotating reference frame, they would all look like two or three elliptic orbits connected by strange segments
where the perturbation by the Mun is to blame.
The last picture below (in the barycentric corotating frame of the Kerbin-Mun system, that is, the unique frame in which the Kerbin-Mun barycentre as well as the line through Kerbin and the Mun are invariant) shows a trajectory that differs strongly from what stock KSP would have yielded: in stock this would have been a circular orbit around the Mun near the edge of its SoI. Instead it is a transfer to an eccentric Kerbin orbit, which is picked up by the Mun again after a while and ends with a crash on the Mun.
With the addition of plotting, the C++ plugin has caught up with the features of the initial C# prototype. After the abstractions for plotting are written (we will add the remaining interesting reference frames in the process), the next step will be to take care of altering the trajectory when, e.g., engines are used.
It is apparent that the C++ plugin is significantly faster, much more reliable, and easier to debug than the C# prototype (it is also correct; some hasty shortcuts were made in the prototype that resulted in unreliable initial states and other problems).
Trajectory, an abstraction for a tree of trajectories that can be forked into child alternatives (useful i.e. for predictions, manoeuvre nodes, but also simply computations at different precisions), has been written (part of the physics namespace)
The plugin has returned, first as a simple orbit-freezing plugin, then as a plugin that actually integrates the motion of vessels (only on-rails for the moment). Is does not plot yet, so it is still behind the C# prototype in terms of features, but it is significantly faster.
As was previously mentioned, all calculations are done in a native assembly, called by P/Invoke via a (cdecl) interface.
It should be noted that the plugin uses glog everywhere for logging (since logging from native code is needed).
Of course, this means Principia will have its own log files (in <KSP directory>/glog/Principia, with a log of the last session in <KSP directory>/stderr.log.
Another interesting aspect is that there will be no silent exceptions as in managed plugins, instead, the game will either segfault (when dereferencing an invalid pointer) or abort (SIGABRT) if a glog CHECK macro fails (on Windows the latter yields the "KSP.exe has stopped working" message).
I have been informed by Sarbian et alii that this will result in Fun support.
I wrote the second explanatory post, on Hamiltonian mechanics (PDF).
Prerequisites are chapters 4, 8, and sections 11-4 and 11-5 of Richard Feynman's Lectures on Physics.
The next post will be on symplecticity and the leapfrog integrator.
The integrator (still a 5th order SPRK, same as in the C# prototype, the Saha & Tremaine stuff isn't here yet) has been implemented, tested and benchmarked (using a Windows port of google/benchmark) in C++, as well as the first level of abstraction above it, principia::physics::NBodySystem (header, body, test, header for test data, test data, test of test data (which is also a nice example of the use of principia::quantities and principia::geometry, benchmark (using the test data)).
A significant change of plans has occured: As Unity uses the .NET Framework v2.0, it is not possible to use plugins compiled for the Framework v4.5. This means C++/CLI projects need to be compiled using the platform toolset v90 (from VS2008), rather than VS2013's v120. Of course v90 does not support C++11 nor C++14, so it is not an option for this code that strongly depends on C++11 features.
As a result, we will keep using the C++11 codebase, compiling it to a native DLL, and using P/Invoke to call it from a C# adapter. The C# adapter will perform no calculations, only transmit data from KSP to the native plugin and apply the changes/render the trajectories returned by the plugin. A simple test P/Invoke plugin can be seen on the repository (header, body, C# adapter) and works in KSP.
This means there will be separate x86 and AMD64 builds for each target operating system (Microsoft Windows and 57 varieties of Unix), possibly more if we decide to do specific optimisations for Intel chips (though ICC is not cheap).
Moreover, this will allow a switch to clang in the near future, so that we can have saner error messages and better compliance with the standard.
I now have a collaborator on Principia, https://github.com/pleroy (my father). This means the code gets reviewed and that development is faster.
We have decided to completely switch to C++/CLI due to better test tools and useful language features. Most of the code will actually be written in standard C++, with the implementation inlined in header files, so that they are seen by the eventual C++/CLI managed assembly (If we were to use C++/CLI everywhere, we would need managed boundaries between the assemblies, which is quite inconvenient).
As an added advantage, using standard C++ enables putting performance-critical parts into a native assembly if needed at a later time, without requiring a significant rework of the code.
We have started switching to gmock, gtest and glog for mocking, testing and logging. These tools are more convenient and powerful than the Visual Studio testing framework and are open source, so that users of Visual Studio Express (which does not support Microsoft's testing framework) will be able to build and run tests if they want to.
Here is the first test to be converted to gtest: https://github.com/mockingbirdnest/P.../sign_test.cpp.
The Quantities library (C++) is pretty much done and tested.
I am currently porting the C# Geometry assembly mentioned previously (now called CTSGeometry) to C++ in order to enable its use with these quantities.
The Geometry assembly seems to be mostly feature complete, so I'll start writing tests tomorrow (Saturday) after changing a few access modifiers, replacing the ugly switch statements in Permutation by something nicer, and a few optimisations.
See here for an overview of its features.
I have investigated the feasibility of using other languages for KSP plugins:
The following languages can be used on their own to write a plugin: C#(unsurprisingly), F#, C++/CLI (with no calls to native code).
VB.NET can be used, but wrappers in one of the above languages are needed due to its case-insensitivity.
Unity does not support mixed-mode DLLs, so calls to native code have to be made using DllImports in one of the above languages.
I have started doing some refactoring since the sphaghettiness of the code was getting on my nerves.
I have written strongly typed wrappers for the numerous reference frames spawned by KSP (direct vs. indirect, rotating vs. inertial, world vs. body-centric, etc.) as I have had numerous bugs due to a misplaced xzy, rotation, translation, scaling, inertial force etc.
Of course, since KSP has the brilliant idea of using both direct and indirect reference frames, I needed distinct types for vectors, bivectors and trivectors (basically I had to strongly type Grassmann algebras; there can be no cross product, only wedge products, commutators and left and right actions of alternating forms---where is identified with through the inner product, and is identified with ).
I do not think I will implement strong typing for physical quantities yet (though I'd like to), since C# generics are not powerful enough for that; I would need C++ templates. I'll do that when I rewrite things in C++/CLI.
The next step is to implement my own versor, since Unity's Quaternion is in single-precision float and KSP's QuaternionD is broken.
The rest should be more straightforward refactoring.
It turns out I have trouble properly setting the position when off-rails (finding out where the reference frame actually is is hard).
However, there are some nicer news: I seem to have found the source of the phantom acceleration bias (not the ones arising from rotation, the bias that is removed by timewarping). The floating origin sometimes floats several km away from the ship, so that's probably just floating-point inaccuracies (the usual Kraken). If that hypothesis turns out to be true, this particular acceleration bias will be easy to fix, just reset the floating origin often enough.
I have successfully implemented the integration of proper acceleration for the active vessel when off-rails.
Further experimentation on proper acceleration has led me to the following conclusions:
- There is a bias in proper acceleration coming from some improperly initialised variable in KSP. Indeed, when loading a vessel in LKO, I observe a strong bias in proper acceleration (~20 mm s-2). This bias is observed independently of the way proper acceleration is computed (differentiating position twice, differentiating any of the numerous velocities, etc.) and geometric accelerations have been checked from first principles (the difference in geometric acceleration depending on the point of the vessel it is computed at is negligible). The bias is reduced to below 1 mm s-2 when warping then coming out of warp. It should be noted that the strong bias is not seen in Vessel.perturbation, but Vessel.perturbation consistently has a bias of 4 mm s-2. As I have attempted to compute the proper acceleration in many different ways and all were consistent with each other and inconsistent with Vessel.perturbation, I assume Vessel.perturbation is mostly nonsense.
- Accelerations below 1 mm s-2 are biased, unavoidable, unusable in stock, and should be clamped to 0. The acceleration from low-thrust engines will have to be fetched by hand.
- It had previously been mentioned that spinning the ship accelerates it. If spinning the ship with angular velocity ω produces a phantom acceleration a, then spinning it with angular velocity -ω produces a phantom acceleration -a. It seems a is either prograde or retrograde.
I have finally managed to work on this a bit over the weekend.
Extensive experimentation has failed to yield a better velocity, so we are stuck with computing the proper acceleration from the rather mysterious Vessel.obt_velocity for now (for some reason in whatever reference frame this velocity is in, the geometric accelerations are as expected, except for the Coriolis acceleration which is halved ).
This yields roughly the same results as Vessel.perturbation without the moving average.
The same experiments have further shown that a naïve low-pass filter will not be enough to remove Unity's silliness. For instance, one finds the proper acceleration is strongly correlated with the rotation rate of the ship. Smarter post-processing will be needed.
I shall now attempt to actually integrate whatever proper acceleration I have managed to grab.
I also wrote the first explanatory post, which describes ODEs and Runge-Kutta methods (PDF).
I have tried not only to explain how Runge-Kutta methods work (this can easily be found on Wikipedia) but why they have this structure.
Hopefully you will find it interesting.
I fixed a bug or two since 2014-02-05, added a few TODOs, but have not cleaned up the code to any measurable extent. I am, however, publishing it on GitHub (under the MIT license).
Caveat Compilator: As previously stated, this is just a proof of concept with a bunch of traces. Pressing the wrong buttons in the wrong order will result in lots of NullReferenceExceptions and off-rails crafts will behave weirdly. Using HyperEdit to set up initial conditions, then switching to Newtonian physics and fooling around with reference frames can be entertaining though.
You might learn something from looking at the Integrators part (which admittedly contains only one integrator), the rest was quickly hacked together, has no tests, is badly structured &c.
The 'Simulator' project probably won't compile, it's leftover from my Alternis Kerbol simulations. It's not needed.
This is currently hardly more than a proof of concept. The simulation only affects on-rails vessels, thrust is ignored (thus you can't actually play while the simulation is running) and the integrator is too slow. However, it shows a few interesting things:
- Near-unit-roundoff (using IEEE 754 binary64) errors can be achieved while sustaining 100_000x timewarp with B = 17, V < 5. Handwavy asymptotic calculations indicate that for V = 100, this integrator would slow down to about 20_000x timewarp. Using better integrators (and saner tolerances), performance could easily be increased by a couple of orders of magnitude, making the simulation usable for RSS as well as stock.
- Trajectory plotting in rotating reference frames, or reference frames centered around a body other than the primary, can be useful even for near-Keplerian orbits: it allows for visualisation of RT2 satellite coverage, easier landing prediction, etc.
- The stock integrator is terrible, and will have to be overriden at all times while in space (when near a Lagrange point, a fraction of a second under stock integration will turn a Kerbin capture into a Mun collision).
- If you think floating SOIs (or worse yet, SOIs with repulsive gravity) are a half-decent model for Lagrange points, you don't really understand how Lagrange points work.
Expect a slow development cycle, due to a combination of laziness, exams, and this actually being a complicated project.
I'll put the source on GitHub as soon as I can clean things up a bit (there are quite a few traces that were too hastily implemented for my taste). This might take a few days, see above.
Nonetheless, here are some pictures that illustrate the fun trajectories in the N-body problem. The second and third pictures have a misplaced trajectory, this bug has been fixed. Some pictures have three strange red, green and blue lines, which indicate the origin and axes of world space.
Considerations on numerical integration
The current prototype uses a straightforward fifth order SPRK method. The coefficients used are from (Atela & McLachlan 1992). The increments are accumulated using Kahan summation. The integration is performed with a constant 10 second timestep (numerical experiments suggest that the error is close to an unit roundoff with a 5 second timestep for LKO).
Regarding implementation details, the integrator is written in C# (compiled with VS2013) and uses double (IEEE 754 binary64) for all computations.
Scott Manley suggested using hybrid symplectic integrators (Chambers 1999) in order to speed things up. It should be noted that the concerns that the Chambers paper attempt to alleviate
might not be critical for gameplay purposes (the main integration can probably be done with a very small timestep compared to the ones commonly used in astronomy. Moreover, timestep reduction occurs de facto when getting out of warp, as the game cannot conceivably be played with timesteps of several seconds).
Originally Posted by Chambers 1999
It could become relevant for trajectory predictions in vessels under thrust, which has to be done in a few hundred milliseconds over several years. For those predictions, the timestep would have to be increased to durations comparable to those used in the paper, and similar concerns would arise.
The results from the papers quoted by Chambers, namely (Saha & Tremaine 1994) and (Holman & Wisdom 1991), will probably have to be used in order to make the main integrator faster. Experiments will have to be carried out in order to find out whether a higher order (> 2) integrator is worth using, and how the added cost of Keplerian evolution (which requires numerically solving the Kepler equation for all bodies, which is very costly due to the trigonometric functions) affects performance at acceptable time steps.
Open questions for the interested reader
Regarding KSP modding
The method I currently use for drawing trajectories (LineRenderers, object in layer 9, transform.parent = null, useWorldSpace=true, updated every time the GUI is drawn) has some issues: when rotating the camera in map mode, the lines lag behind. Does this have a known solution? (I guess so, RT2 doesn't have this problem). And indeed, the RT2 code contained the solution. Thanks, Cilph!
Is there a way to directly set the position and velocity of the planets and vessels in world coordinates? OBE, the API does not make sense in highly original ways.
Can anybody think of a way to mod in axial tilt (even a hacky one; remember that I'm the one moving the planets around anyway)? OBE (done in RSS, we actually have axial tilt, it's just that all planets rotate around parallel axes).
Could somebody help me with the implementation of orbital decay drag when I get to implementing that?
ferram4 told me he'd help. Thanks!
Would it be conceivable to predict the results of aerobraking/aerocapture/ballistic reentry with FAR (within some error margins, to be displayed on the predicted trajectory)? How slow would that be?
Answered by ferram4.
This might end up happening.
Regarding astrophysics and astrodynamics
The uniformly rotating reference frame around the barycenter with respect to which the two massive bodies are fixed is very useful when looking at trajectories in the CR3BP (Lagrange 1772). What would a good reference frame for the ER3BP (for bodies with highly eccentric orbits, e.g., Eeloo in stock, Pluto in RSS) be? the uniformly rotating one around the barycenter that follows the mean anomalies, the nonuniformly rotating one that follows the true anomalies, or something else entirely?
In the rotating-pulsating reference frame that fixes both bodies, the equilibrium points are fixed. Is there a way to draw things in that reference frame on the in-game map that's not too confusing? Can it still be useful to draw the potential for the parts of the equations of motion that derive from a potential, given that the independent variable is true anomaly rather than time?
ferram4 suggested a few interesting things.
Should long-term trajectory predictions for vessels under thrust be inaccurate, is there a good way of estimating the uncertainty (for instance, would predicting a few perturbed trajectories and plotting them all give a representative idea of what might happen)?
Would drawing the gravitational potential in the current orbital plane (together with the centrifugal potential if orbits are drawn is a rotating reference frame) be useful? Answered by ferram4 with visualisation suggestions. The answer is 'yes'.
How should stationkeeping be implemented? In particular, how should the user set the orbit they want to maintain?
Regarding the KSP fora
Do we have LATEX support? If that isn't the case, is it conceivable to have it in the foreseeable future?
Once the mod gets to a functional state (taking thrust into account), I shall attempt to solve the problem of trajectory prediction for vessels under thrust. In the process, the smarter integrators mentioned above will be implemented, and the most efficient one will be kept. This should significantly increase computation speed.
This will also enable instantaneous-burn maneuver nodes (the kind of maneuver node we have now).
The barycenters will be fixed in order to stabilise the Jool system. Scott Manley (illectro)'s predictions indicate this will make it stable over at least 1_000 years (Pol was not part of those predictions, its stability remains to be determined).
Speed will be further increased by rewriting the numerical integration in (unmanaged) C++, interfaced with C# through C++/CLI. The unmanaged C++ (compiled with clang) will use 80-bit extended precision 'long double' (64-bit mantissa instead of 53-bit) for added precision.
Once the mod becomes playable with N-body gravitation with point masses, the following effects should be relatively easy to add:
- Conservation of angular momentum through timewarp and outside timewarp (allowing for spin-stabilisation and thus fixed RT2 antennae if Cilph decides to implement that);
- Thrust through timewarp, for ion engines and the like, together with maneuver nodes for non-instantaneous burns;
- Orbital decay drag, with ferram4's help;
- Stationkeeping, to deal with orbital decay and unstable weakly bound orbits;
- Gravitational moment (quadrupole) for planets, enabling such amusing things as Sun-synchronous orbits;
- Insert crazy idea here...
I would like to thank (in alphabetical order of forum usernames)
- Anatid for his attempt at documenting the KSP API,
- armed_troop for his help in making the codebase clang-compatible.
- Ezriilc, Forecaster, khyperia, Payo and sirkut for their HyperEdit plugin, which is really useful for setting up initial conditions,
- Scott Manley (illectro) for his help with numerical integration of the N-body problem,
- Matt Roesle (Mattasmack) for writing an integrator which was far better than the one I used, thus prompting me to write my current SPRK integrator,
- NovaSilisko for his Alternis Kerbol mod, which piqued my interest in the simulation of the N-body problem,
- The entire KSP modding community (especially the subset which writes clean, well-commented code) for providing code examples from which one can learn the subtleties of dealing with the KSP API.
If I forgot you in the above list, please complain!
- Mathematica documentation on SPRK methods.
- [Atela & McLachlan 1992] Robert I McLachlan and Pau Atela, The accuracy of symplectic integrators, 1992.
- [Chambers 1999] John E. Chambers, A hybrid symplectic integrator that permits close encounters between massive bodies, 1999.
- [Holman & Wisdom 1991] Jack Wisdom and Matthew Holman, Symplectic maps for the N-body problem, 1991.
- [Kenniston 2002] Michael Kenniston, Dimension Checking of Physical Quantities, 2002.
- [Lagrange 1772] Joseph-Louis Lagrange, Essai sur le problème des trois corps, 1772.
- [Saha & Tremaine 1994] Prasenjit Saha and Scott Tremaine, Long-term planetary integration with individual time steps, 1994.
- [Tao 2012] Terence Tao, A mathematical formalisation of dimensional analysis, 2012.
- Eric Weisstein's World of Physics, Gravitational Moment.
- [Yoshida 1990] Haruo Yoshida, Construction of higher order symplectic integrators, 1990.
- [Yoshida 1992] Haruo Yoshida, Symplectic Integrators for Hamiltonian Systems: Basic Theory, 1992.