The Case of the Fickle Found Furby

Fast Radio Bursts

Early last week, Dusty discussed Fast Radio Bursts (FRBs, or as I prefer to call them, Furbies). To recap, they are short (~millisecond) bursts in the radio (GHz) frequencies, that seem to come from cosmological distances, as determined from their dispersion measures, related to the amount of time between arrivals of the pulse at different radio frequencies. The more free electrons between us and a radio burst, the longer this delay will be, with the time delay scaling as the inverse of the frequency squared. The column density of electrons indicate that these bursts come from either dense plasma environments, or from cosmological distances (with the electrons coming from the intergalactic medium).

The first detected FRB, known as the Lorimer burst. The $\nu^{-2}$ dispersion, large dispersion measure, and short duration define the class.

The other kind of Furbies, also radio emitters, but more easily localized, with varying optical colors.  

Since these bursts are so fast (it's in the name!), it is very difficult to figure out precisely where they are coming from. Until last week we didn't have any clear identification of a true distance or redshift. We know that they seem to be isotropically distributed in the sky (thus supporting the hypothesis that they are cosmological). Most have been discovered by the Parkes radio observatory in Australia, mostly due to survey design reasons, but several have also been seen by the Arecibo and Green Bank Radio Telescopes. One detected by GBT also showed signs of circular polarization, seeming to indicate that it was associated with a strongly magnetized/rotating object. So far about 17 have been detected, though when convolving this number with the beam size and observational strategies of the various radio surveys, this gives a rate of roughly thousands per sky per day!

A Furby Located?

Last Thursday, an extremely interesting Nature paper came out discussing the apparent localization of a Fast Radio Burst by an afterglow and the identification of a host galaxy. Keane et al. report that using their extremely impressive real time identification of FRB 150418. (most FRBs have been detected by an archival search of the data), a large number of followup telescopes were triggered with an afterglow in the radio (5 GHz and 7.5 GHz) detected by ATCA that faded over about 6 days.

From Keane et al. 2016, on the purported radio afterglow associated with FRB 150418. The detection are the two purple points and one green point on the left, with low level emission and upper limits after. The FRB was detected at time 0. 

If true this is a HUGE deal. It would be the first identification of an FRB host galaxy, providing a distance and redshift (z = 0.49). The combination of dispersion measure and redshift provide an estimate of the "missing" intergalactic baryon density, providing an estimate of the mass in the intergalactic medium (IGM) that agrees well with the cosmological results from WMAP.

The host environment, however is at odds with a magnetar being the source of FRBs. Having FRBs be associated with giant magnetar flares is, I feel, the leading candidate for their astrophysical source. However, magnetars exist in young stellar populations, while the galaxy in question is "red and dead", meaning it has an old stellar population with no star formation). Additionally the energetics of the afterglow (which contains much more energy than the initial FRB) suggest that it must come from something much more powerful than a magnetar flare. The authors even suggest that these factors indicate a NS merger as a source for the burst and afterglow. This, however, is problematic, as the rate of neutron star mergers is not nearly high enough to account for the inferred FRB rate, which is close to the supernova rate! If the NS-mergers accounted for a large fraction of the inferred rate of FRBs this would mean that 1) Short GRBs are extremely beamed 2) Initial LIGO should have probably seen a merger if they happen so often. The alternative many suggest is that there may be multiple progenitor classes (though the rates must still be comparable for it to have been picked up with so small a sample).

Keane et al. also examine the possibility of a false positive, by looking at the rate of radio transients, and conclude that there is a < 6% chance of a coincident transient detection. However, in the last few days it has emerged that only considering the radio transient rate may have been a crucial, but subtle mistake.

Not So Fast, Radio Burst!

Over the weekend Peter Williams and Edo Berger of Harvard CfA posted a preprint questioning the Keane et al. results. (They actually posted a link to their preprint on facebook on Friday morning, which is impressive turn around time indeed!) They argue that it is the rate of variable radio sources that should be considered, and show that there should be order unity of these expected in the beam of Parkes associated with the FRB. They argue that the "afterglow" then, is likely associated with AGN variability. Most compellingly (to me), they show how the evolving ratio of the 5.5 GHz to 7.5 GHz flux from the source is inconsistent with a relativistic fireball that one expects to be powering a true afterglow of a burst event. (This also jives with how it requires a lot more energy to power such an afterglow than most think would be available for a single compact object).

If the "afterglow" of FRB 150418 is indeed due to AGN variability, this would be easy enough to check with a campaign of follow-up observations, which of course Edo and Peter carried out. The result was this Astronomer's Telegram:

ATel 8752 from Williams and Berger reporting on the VLA follow up of the purported FRB host galaxy.

Using the VLA, they reported a 157 microJansky detection of the radio source, which suggests a re-brightening at the 3 sigma level. That's pretty damning of the afterglow interpretation, though there is no direct detection of variability in their two 1.5 hour observations. 

It is worth noting that radio transient/variability people that I know and respect think this closes the case, but some of my colleagues who study AGN for a living have told me that such variability from a radio quiescent AGN isn't actually expected (they said they expect such variability from a blazar source, though the optical observations make it clear it is not a blazar in this case). [EDIT - Edo writes to mention that since the source is only a few degrees from the Galactic plane, the variability seen is consistent with the level of ISM scintillation you'd expect from an AGN with no intrinsic variability. Interesting! Though pulsar people should not have missed this as they deal with scintillation all the time!] As an outsider to both fields I can't really provide a solid interpretation, but my general impression is that the radio sky (and particular quiescent radio galaxies) at these frequencies and timescales are not well studied. Regardless, the re-brightening, if held up, combined with the strange spectral evolution, does make it hard to believe the afterglow interpretation, and subsequently the association with a host galaxy and distance. 

A Furby, disassociated. 

Too bad!

Random Topics: GW GRBs, mini-JWST, Exoplanets and lunch.

So after all the excitement last week, the subsequent week has been filled with discussions and other work getting caught up. For me, the last few days have been spent getting caught up on some review work that is going to be due soon, so this will be a fairly short post.

Fermi

On Friday, for our high-energy astrophysics meeting at UMD, we had a nice round-table discussion with one of the Fermi team regarding the Fermi-signal coincident with the LIGO detection GW150914. Their arXiv (not-yet peer-reviewed) paper is here, and discusses their detection.

Fermi-GBM detected a signal with a fluence of $10^{49}$ ergs/s  between 1 keV and 10 MeV, lasting about a second. The fluence, duration, and hardness was similar to a weak short-GRB. The reason it was so poorly localized was that the Fermi spacecraft was pointed in orthogonal to the optimal direction to localize the signal source. They characterize this as a roughly 2-3 sigma detection, with a 0.0022 false alarm probability, though there are some details of this FAP calculation that I don't quite follow (particularly, why this scales with time from the GW trigger event).

It is worth noting that Fermi expects to see events of this significance randomly every few thousand seconds or so, therefore there is a possibility this is just a coincidence. SWIFT-BAT did not detect anything, though it was also pointed in the wrong direction and doesn't have as wide field of a detector, so they had to make do with observations 2 days after the fact, during which there was no detection. More interestingly, the INTEGRAL-ACS which has a much larger collecting area than Fermi-GBM, (it's a coincidence shield that surrounds the spacecraft) but also much higher background, did NOT see anything at the same time as the Fermi-GBM detection. However, they are sensitive to a different energy band, and if the short-GRB like spectrum is taken as a given, INTEGRAL is expected to miss about 50% of the bursts Fermi catches at that fluence.

This of course did not stop many many so-called "creative" ideas from flooding the arXiv about how a BH-BH merger produced an electromagnetic GRB-like counterpart. I won't get into them here, and will leave the discussion for another day.

Mini-JWST

This isn't really all that research related but I just wanted to post a picture of our new JWST model we just got from MESAtech. The James Webb Space Telescope is of course the very expensive IR space telescope that carries the hopes, dreams, and opportunity cost of the entire astronomical community into the future. It will be revolutionary for extragalactic astronomy, mapping of the local group, and exoplanetary atmosphere studies, among other things.

This 1/40th scale model was 3D printed as a reward for kickstarting MESAtech's fully robotic model for high-school outreach. The pieces were all snap together and Vicky assembled it in about 45 minutes, with just some wire snips and a few pieces of tape to help tighten up some fits. We got it for $50 during the kickstarter, but they are now available for purchase from the MESAtech site for $75. It folds/unfolds, and at 1/40th scale is just about the right size compared to LEGO minifigs.

Exoplanets and Lunch

Lastly, this week I organized the inaugural Exoplanet lunch at UMD, in order to bring together Drake's group with some other exoplanet researchers, like myself. Personally, I'm hoping that a closer interaction with the exoplanet observers will lead to some interesting projects that will help me get a better feel for the details of exoplanetary data analysis and observations. Towards that end, I'm considering running a brief tutorial over the course of several lunches exploring the use of Dan Foreman-Mackey's emcee code for doing parameter estimation using Markov-Chain Monte-Carlo methods. Jake van der Plas has a very nice tutorial he ran at the last AAS, which would be a very good basis to start from. 

The Slimplectic Integrator (pt II): The Nonconservative action principle and Variational Integration

In part I of this post I discussed how N-body variational integrators can be constructed (by discretizing the action integral itself, then finding the Euler-Lagrange equations of the discretized action), and why Noether's theorem means that this will give good long term behavior for conserved quantities like Energy, or Angular Momentum.

However, because this method relies on the existence of an action, to be discretized, it only works for conservative systems. In order to be able to apply this numerical method to systems where there is dissipation, drag or some other nonconservative (polygenic, path-dependent, or irreversible) forces we will need an equivalent of the action for such systems. Additionally, in order to guarantee good long-term behavior, we will also need a version of Noether's theorem that applies to such an action.

The Nonconservative Action Principle

In Galley (2013) and Galley, Tsang & Stein (2014), we recently developed such a Nonconservative Action Principle, adapting Hamilton's principle to apply to nonconservative physics.

Nonconservative forces can arise when a subset of the true physical degrees of freedom of a system are ignored, often through a natural separation of scales, or by experimental design. 

Even though, micro-physically, a damped harmonic oscillator is well described by a Lagrangian or Hamiltonian if one includes all the particles and/or thermal degrees of freedom in the damper, when one integrates-out those micro-physical degrees of freedom the system should look nonconservative. 

Usually we are forced to "integrate out" these inaccessible degrees of freedom at the level of the equations of motion. When integrating out at the level of the (normal) action, things tend to become acausal, where the evolution of the system depends on both the initial and final configurations of the system (and thus inconsistent with the correct equation of motion description).

The reasons for this are technical (and may be worth another post), but are related to how Hamilton's principle is traditionally constructed as boundary value problem in time, rather than an initial value problem.

Left: Hamilton's principle requires a minimization of a path between $q_i$ at time $t_i$, and $q_f$ at time $t_f$, describing a boundary value problem in time. Right: The nonconservative action principle doubles the paths, allowing the position at $t_f$  to vary, consistent with an initial value problem. The paths are varied separately, with an equality condition at the final time, and then set equal after taking the Physical Limit. 

Instead Chad constructed the nonconservative action principle to be consistent with an initial value problem. This involves doubling the degrees of freedom and varying each separately subject to a final time equality condition that closes the loop (see figure above), allowing the final time values of the system to vary freely. This is related to the closed-time-path Schwinger-Keldish formalism in non-equilibrium quantum mechanics (in fact in an appendix here we showed how this formalism is the formal classical limit of such a quantum theory).

By moving to this extended, doubled phase-space, extremizing the action over each path, and then equating the doubled variables together (we call this taking the Physical Limit), we gain the freedom to allow nonconservative effects to be included. These nonconservative effects show up in the nonconservative potential $K(q_1, q_2, \dot{q}_1, \dot{q}_2, t)$, which couples the two paths together, and can appear in the nonconservative action. By construction, inaccessible degrees of freedom can also be integrated-out in a self-consistent way, at the level of the action, with no acausal effects.

To construct the nonconservative potential, $K$, which captures the non-hamiltonian physics, one can either build it from known dissipative equations of motion through an adjoint procedure, often equivalent to the Lagrange-d'Alembert approach, or one can build it from known microphysical actions by integrating out inaccessible degrees of freedom.

The Nonconservative Noether's Thoerem

We needed two parts to make a variational integrator work, an action principle, whose action integral can be discretized, and Noether's theorem, to exploit the symmetries of the action to correctly evolve (or preserve) the Noether currents. It turns out that we can prove a generalized Noether's theorem for the nonconservative action principle. The Noether currents are still defined by the symmetries of the conservative parts of the action, but their evolution is described by derivatives of the non-conservative potential $K$.

The nonconservative version of Noether's theorem.

The Slimplectic Integrator

By applying the variational integrator procedure (discussed in part I) to this new action principle we are able to construct general nonconservative variational integrators that are no longer limited to conservative systems (see figure below). We call these “slimplectic” integrators as the phase space volume tends to slim-down for dissipative systems). Slimplectic integrators have all the long-term stability properties of symplectic integrators but can be applied to general nonconservative systems.

This allows long-term numerical evolution of orbital dynamics problems where dissipative or velocity dependent effects, such as tides, drag, or magnetic interactions, can become dynamically important.

Top: For a standard numerical integrator equations of motion are discretized using well-known methods (e.g. Runge-Kutta) in order to derive discrete equations of motion that can be solved numerically. Such algorithms tend to experience long-term instability of quantities that should be physically conserved, as these discrete equations of motion are not connected to the action, and Noether’s theorem does not hold.

Middle: Variational Integrators work by first discretizing the action integral using numerical quadrature. Varying the discretized action produces already discretized equations motion that can be exactly (up to round-off) solved numerically. If the choice of discretization preserves the symmetries of the original action in the discretized action, then Noether’s theorem guarantees the solutions of the discrete equations of motion will preserve the constants of motion (e.g. angular momentum) up to round-off. This is why variational integrators (such as symplectic integrators) have such impressive long-term conservation properties compared to other numerical methods.

Bottom: By applying the Variational Integrator procedure to the Nonconservative Action formalism, we have developed a new type of integrator that has all the long-term integration benefits of a symplectic or variational integrator, but that can be applied to nonconservative systems. By discretizing the Nonconservative Action, and applying the nonconservative variational principle, the discrete equations of motion are guaranteed by the generalized Noether’s theorem (Galley, Tsang, & Stein 2014) to evolve the Noether currents “correctly” during long-term integrations.

We have developed a simple public python-based demonstration code on github that can be used to numerically integrate any system by specifying the (nonconservative) action. Included below are links to several example of a nonconservative system evolved using the slimplectic and traditional (Runge-Kutta) numerical schemes. The slimplectic integrators have fractional energy error that remains bounded, even as the system’s energy evolves, while the traditional methods have fractional energy errors that grow linearly with time.

The slimplectic iPython notebook samples are:

A damped simple oscillator
Poynting-Robertson drag (see figures below)
Post-Newtonian gravitational-wave inspiral of neutron star binaries

In this example we model the effect of Poynting-Robertson drag on a dust grain as it orbits the sun. Poynting-Robertson drag occurs because the dust grain absorbs incoming radiation from the sun, and re-emits in the frame co-moving with the dust particle. This leads to anisotropic radiation in the observer’s frame and a net dissipative drag force. In this case the dust particle begins with an eccentricity 0.2, and semi-major axis a = 1 AU.

The RK methods have poor long term stability, with energy error growing linearly with time, while the energy error remains bounded for the slimplectic methods. However, there is significant precession in the 2nd order slimplectic method, due to the lack of conserved quantity that prevents precession. The phase errors of the slimplectic methods scale linearly with time, while the phase errors of the RK method scale quadratically. The growth at the end of the evolution for the 4th order Slimplectic method is likely due to a build-up of zeroing/round-off error due to the simple implementation.

Prescriptions for measuring and transporting local angular momenta in general relativity

I, along with my collaborators David Nichols, Justin Vines, and Éanna Flanagan, posted our latest preprint, Prescriptions for measuring and transporting local angular momenta in general relativity, on the arXiv tonight. After I tweeted the link, George informed me that I was now obligated to blog about it:

@Vagelford Let me guess, blog about it on the RemarXiv?

— Leo C. Stein (@duetosymmetry) February 8, 2016

Ok, so here goes a little self-promotion!

This paper is a refinement on David and Éanna's earlier paper. The topic at hand is angular momentum in general relativity.

So, how is angular momentum defined in GR? One quantity that everyone agrees upon is the ADM angular momentum, which is precisely the Noether charge associated with the rotational symmetry of asymptotically flat spacetime. Masha recently touched upon the ADM mass in this post.

Angular momentum is related to a symmetry, so of course we must care about it. From Emmy Noether's theorem, we know that this total angular momentum is conserved. In a sense we can say that the ADM angular momentum (and mass) "live at $$i^0$$," the point at spacelike infinity.


In this paper, we were studying a different type of angular momentum. Instead of living at $$i^0$$, we are interested in a quantity near $$\mathcal{I}^+$$ (pronounced "scri plus"), the region of future null infinity. Here we're talking in the language of conformally compactifying an asymptotically flat spacetime, so we can squeeze the whole thing into a Penrose diagram like this one. Every spacetime which is asymptotically flat has the same structure "far away" from all the curvy bits of spacetime.

Why do we want to study angular momentum near scri? One reason: it's very natural to think of angular momentum decreasing when e.g. gravitational radiation carries it away from a system—like a black hole binary, allowing the two black holes to eventually merge! But the ADM angular momentum is conserved, so what gives? To understand this, you have recognize that the ADM quantity comes from integrating along a "Cauchy surface," a hypersurface that's everywhere spacelike and makes it out to $$i^0$$, like any of the $$\Sigma_{1,2,3}$$ in the Penrose diagram.

Spacelike infinity ($$i^0$$) is not the right setting to discuss things like how much angular momentum is carried away by gravitational waves, because the ADM angular momentum can't change! Instead of talking about quantities that "live at" $$i^0$$, we want to talk about quantities that "live at" $$\mathcal{I}^+$$.

Future null infinity ($$\mathcal{I}^+$$) is much more complicated than $$i^0$$. Spacelike infinity is kind of "rigid," while $$\mathcal{I}^+$$ is comparatively "floppy," though less floppy than the interior (bulk) of the spacetime. These statements can be made mathematically precise in terms of the symmetry groups of these spaces. The symmetry group of $$\mathcal{I}^+$$ is the famous BMS group, named for Bondi, van der Burg, Metzner, and Sachs, who studied its structure in the 60s. The BMS group has enjoyed a renewed interest in recent years, when it was discovered that BMS symmetries, Weinbergesque soft theorems, and long-ranged "memories" are different faces of the same underlying physics.

So, if you want to discuss angular momentum at $$\mathcal{I}^+$$, the BMS group is going to tell you the mathematical rules you have to follow (technically: quantities must "live in" representations of the group). For a while, the literature referred to something called the BMS angular momentum "ambiguity." However, ambiguity is not really the right word. Angular momentum is not ambiguous, it just transforms in a much more complicated way (under BMS transformations) than angular momentum does in flat (Minkowski) spacetime).

Ok, enough with the longwinded background. What did we actually do in our paper?

First of all, we worked in a simplified setting: when the region of interest of $$\mathcal{I}^+$$ is approximately stationary. In this setting, angular momentum has a much simpler transformation law when you move from point to point than in the fully dynamical setting.

Let's consider what happens when you expand around stationary $$\mathcal{I}^+$$ in powers of $$1/r$$. To leading order, the only property of spacetime is the Bondi mass. When spacetime is stationary, this is constant, and every stationary asymptotically flat vacuum spacetime is identical to Schwarzschild expanded to this order. When you go to next-to-leading order, you learn one additional property of the spacetime, which is like an angular momentum of the spacetime. Here is a crucial fact we make use of: every stationary asymptotically flat spacetime vacuum is identical to the Kerr spacetime expanded to this order.

Now, in the Kerr spacetime, it just so happens that there's a pair of tensors $$(\xi^a,{}^{*}\!f^{ab})$$ which satisfy the pair of differential equations:
\[
\nabla_a \xi^b &=  -\frac{1}{4} R^{b}{}_{acd} {}^{*}\!f^{cd} \, , \\
\nabla_a {}^{*}\!f^{bc} &= -2 \xi^{[b} {\delta^{c]}}_a \, .
\]
We can interpret these equations as a rule for how to transport a linear momentum $$\xi$$ and an angular momentum $${}^{*}\!f$$ from point to point in a consistent fashion (the quantity $${}^{*}\!f$$ is not really an angular momentum, but that's another topic).

So here's hope that a bunch of observers hanging out in spacetime can locally measure what they think are the linear momentum $$P^a$$ and angular momentum $$J^{ab}$$ of the spacetime about them. And moreover, if Alice measures her quantities $$(P,J)_A$$, and Bob measures his quantities $$(P,J)_B$$; and then Bob drags his quantities over to Alice in a certain well-defined way, then their two quantities will agree, up to the required accuracy!

David Nichols has laid out the 8-step procedure for locally measuring $$(P,J)$$ up to the required accuracy, which is really impressive to me.

My contribution to the calculation was the fiber bundle approach, which really makes it quite easy to compute the holonomy of this transport law, and makes clearer what are the necessary and sufficient conditions for the existence of consistent solutions to the transport differential equations. The fiber bundle idea is represented pictorially in the first image at the top of this post.

Read the paper for all the gory details! It's only 9 pages long. Satisfaction guaranteed or your money back.

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.