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.
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| 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.
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| 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 $K$.
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| 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.
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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.
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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
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| 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. |
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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.
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