The Symplectic Integrator (pt 1): Traditional vs Variational Integrators

To start us off, this is more of an in-depth research post, something I had been meaning to write up for a while, describing some recent work that Leo and I have done with our friend Chad Galley at Caltech, and Alec Turner, an undergraduate student I worked with at McGill University.

This work stemmed out of previous work that Chad, Leo and I had done together extending Chad's gorgeous work on nonconservative actions. I won't discuss the nonconservative action formalism in this post, but I will provide a brief summary next week in part 2 in order to describe our use of it for the "Slimplectic" Integrator.

Traditional vs Symplectic Integration

Long-term numerical integration of physical systems are extremely important when studying the stability and evolution of planetary systems, or other astrophysical N-body systems. Traditional integration methods, such as explicit Euler or classical Runge-Kutta schemes, are unstable over thousands or millions of dynamical times, leading to errors in the constants of motion, e.g. energy or momenta, that tend to grow linearly with time. 

In a classic paper of computational astrophysics, Jack Wisdom and Matt Holman described a "symplectic map" for N-body dynamics which allowed for long-term integrations of (conservative) N-body orbits to occur, while preserving physical properties like the constants of motion. Symplectic integrators can be built by partitioning a system's Hamiltonian and then applying the BCH theorem in order to derive mappings that approximate the motion up to a given order.

Traditional integrators, which can be built to minimize the local "error" in the integration-step, can perform comparatively better over short integrations, however in the long term, the non-linear error resulting from the drifting of "constants" of motion make them far worse than symplectic methods.

Solar system evolution stability using first-order traditional and variational (symplectic) integrators. (Hairer, Lubich & Wanner 2006)

Plots of the energy and angular momentum drift from another set of integrations comparing a 4th order symplectic integrator (SI4) to 4th order Runge-Kutta (RK4) taken from Kinoshita (1991).

In order to understand why methods like symplectic integrators are able to preserve the physical constants of motion so well it is helpful to utilize the framework of Variational Integrators and Noether's Theorem.

Noether's Theorem

Of course, since we are discussing the preservation of "constants" of motion, any explanation we have should make full use of Noether's theorem. It is one of the most powerful and beautiful theorems in mathematical physics, and describes the relationship between constants of motion and the symmetries of the system.

Noether's theorem is one of the most powerful in mathematical physics.

More precisely, Noether's theorem relates (differentiable) symmetries of the action that describes the system with quantities that will be conserved by the Euler-Lagrange equations that are derived by extremizing that action.

This means that if we can perform continuous transformations to the system that leave the action invariant, then there will be a constant of motion related to that symmetry which is preserved as the system evolves. Thus, time-translation symmetry leads to energy conservation, translational symmetry leads to momentum conservation, and rotational symmetry leads to angular momentum conservation.

For traditional integration methods, like Runge-Kutta, the focus is on approximating the equation of motion. The differential equation of motion is usually discretized into a form that can be implemented directly on a computer through, for example, finite differencing. Thus, even if the original equation of motion is the Euler-Lagrange equation stemming from some particular action, the actual discrete equation of motion that is solved by the computer does not come from the action, and will not know about its symmetries, so Noether's theorem does not apply. In essence this is why traditional integration methods are not able to prevent long-term drift of the so-called "constants" of motion.

Standard integration methods (e.g. Runge-Kutta) focus on discretizing the equations of motion, leading to a disconnect with the original action. This prevents Noether's theorem from applying, and, without the appropriately constant "constants" of motion, long-term instability occurs.

Variational Integration

A simple way to understand how these integrators work is through the framework of variational integrators (see e.g. the fantastic Marsden & West 2001). Symplectic integrators can generally be written (locally) as variational integrators, and I find this approach, championed by the late, great Jerry Marsden at Caltech, to be much more intuitive than the normal symplectic approach. 

Variational Integrators work by discretizing the action integral itself, rather than the equations of motion. By varying an action that is already discretized, discrete equations of motion are obtained which can be exactly numerically implemented, which are indeed the Euler-Lagrange equations of an action that is closely related to the physical one. 

But how do you discretize the action integral? Exactly how you would discretize any integral, through numerical quadrature! By applying well-known numerical quadratures of a given accuracy order, and then varying the discrete action integral, we can obtain discrete equations of motion that are also accurate up to the same order. 

For Variational Integrators, numerical quadrature rules applied to the action integral yield integrators of the same order after the discretized action is varied. Numerical quadrature is much easier to explain to people than the BCH theorem. 

As long as the discretized action possesses the same symmetries as the original action, the discrete Euler-Lagrange equations we obtain from them will be subject to Noether's theorem, conserving the appropriate quantities (in fact Noether's theorem also tells us how to find the form for the associated conserved momenta, which may be slightly different than in the physical system due to the discretization).

The variational integrator first discretizes the action before performing the variation to obtain the discrete equations of motion. As long as the discretized action possesses the same symmetry as the original action, the discrete Euler-Lagrange equations of motion know about an action, and will exactly preserve the appropriate constants of motion, via Noether's Theorem. 

There are a few wrinkles here, of course. By performing the discretization in time, we break the time-translation symmetry in an exponentially small way. This is the reason why symplectic/variational integrators do not precisely preserve energy in the same way they preserve the other momenta. However, it can be shown that the energy error is always bounded, resulting in energy error that oscillates around zero, with an envelope that depends on the resolution. 

Next week...

Of course, because variational integration or symplectic methods involve systems where an action or Hamiltonian can be specified, they are, by and large, restricted to conservative problems. Problems that involve friction, drag, tidal dissipation, or other nonconservative effects cannot take advantage of the traditional symplectic approach. 

Next week, in part 2 of this post, I will talk about how we used our new nonconservative action principle to develop a variational integration method that applies to general nonconservative systems. This "Slimplectic" Integrator* has all the long-term computational benefits of symplectic integrators, but it can be applied to nonconservative systems where regular symplectic integrators cannot be used. 

*I call it this because phase-space volume "slims" down for dissipative systems, but is constant for conservative ones.