Hamilton's principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton's principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic non-conservative systems, thereby filling a longstanding gap in classical mechanics. Thus dissipative effects, for example, can be studied with new tools that may have application in a variety of disciplines. The new formalism is demonstrated by two examples of non-conservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment.Hamilton's principle of stationary action [1] is a cornerstone of physics and is the primary, formulaic way to derive equations of motion for many systems of varying degrees of complexity -from the simple harmonic oscillator to supersymmetric gauge quantum field theories. Hamilton's principle relies on a Lagrangian or Hamiltonian formulation of a system, which account for conservative dynamics but cannot describe generic non-conservative interactions. For simple dissipation forces local in time and linear in the velocities, one may use Rayleigh's dissipation function [1]. However, this function is not sufficiently comprehensive to describe systems with more general dissipative features like history-dependence, nonlocality, and nonlinearity that can arise in open systems.The dynamical evolution and final configuration of non-conservative systems must be determined from initial conditions. However, it seems under-appreciated that while initial data may be used to solve equations of motion derived from Hamilton's principle, the latter is formulated with boundary conditions in time, not initial conditions. This observation may seem innocuous, and it usually is, except that this subtlety may manifest undesirable features. Remarkably, resolving this subtlety opens the door to proper Lagrangian and Hamiltonian formulations of generic non-conservative systems.An illustrative example. To demonstrate the shortcoming of Hamilton's principle, consider a harmonic oscillator with amplitude q(t), mass m, and frequency ω coupled with strength λ to another harmonic oscillator with amplitude Q(t), mass M , and frequency Ω. The action for this system isThe total system conserves energy and is Hamiltonian but q(t) itself is open to exchange energy with Q and should thus be non-conservative. For a large number of Q oscillators the open (sub)system dynamics for q ought to be dissipative. Let us account for the effect of the Q oscillator on q(t) by finding solutions only to the equations of motion for Q and inse...

We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced order model in both parameter and physical dimensions that can be used as a surrogate for the true or fiducial waveform family. First, a set of m parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these m parameters induce the selection of m time values for interpolating a waveform time series using an empirical interpolant that is built for the fiducial waveform family. Third, a fit in the parameter dimension is performed for the waveform's value at each of these m times. The cost of predicting L waveform time samples for a generic parameter choice is of order OðmL þ mc fit Þ online operations, where c fit denotes the fitting function operation count and, typically, m ≪ L. The result is a compact, computationally efficient, and accurate surrogate model that retains the original physics of the fiducial waveform family while also being fast to evaluate. We generate accurate surrogate models for effective-one-body waveforms of nonspinning binary black hole coalescences with durations as long as 10 5 M, mass ratios from 1 to 10, and for multiple spherical harmonic modes. We find that these surrogates are more than 3 orders of magnitude faster to evaluate as compared to the cost of generating effective-one-body waveforms in standard ways. Surrogate model building for other waveform families and models follows the same steps and has the same low computational online scaling cost. For expensive numerical simulations of binary black hole coalescences, we thus anticipate extremely large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy.

A generic, noneccentric binary black hole (BBH) system emits gravitational waves (GWs) that are completely described by seven intrinsic parameters: the black hole spin vectors and the ratio of their masses. Simulating a BBH coalescence by solving Einstein's equations numerically is computationally expensive, requiring days to months of computing resources for a single set of parameter values. Since theoretical predictions of the GWs are often needed for many different source parameters, a fast and accurate model is essential. We present the first surrogate model for GWs from the coalescence of BBHs including all seven dimensions of the intrinsic noneccentric parameter space. The surrogate model, which we call NRSur7dq2, is built from the results of 744 numerical relativity simulations. NRSur7dq2 covers spin magnitudes up to 0.8 and mass ratios up to 2, includes all l ≤ 4 modes, begins about 20 orbits before merger, and can be evaluated in ∼50 ms. We find the largest NRSur7dq2 errors to be comparable to the largest errors in the numerical relativity simulations, and more than an order of magnitude smaller than the errors of other waveform models. Our model, and more broadly the methods developed here, will enable studies that were not previously possible when using highly accurate waveforms, such as parameter inference and tests of general relativity with GW observations.

We use the effective field theory (EFT) framework to calculate the tail effect in gravitational radiation reaction, which enters at the fourth post-Newtonian order in the dynamics of a binary system. The computation entails a subtle interplay between the near (or potential) and far (or radiation) zones. In particular, we find that the tail contribution to the effective action is nonlocal in time and features both a dissipative and a "conservative" term. The latter includes a logarithmic ultraviolet (UV) divergence, which we show cancels against an infrared (IR) singularity found in the (conservative) near zone. The origin of this behavior in the long-distance EFT is due to the point-particle limit-shrinking the binary to a pointwhich transforms a would-be infrared singularity into an ultraviolet divergence. This is a common occurrence in an EFT approach, which furthermore allows us to use renormalization group (RG) techniques to resum the resulting logarithmic contributions. We then derive the RG evolution for the binding potential and total mass/energy, and find agreement with the results obtained imposing the conservation of the (pseudo) stress-energy tensor in the radiation theory. While the calculation of the leading tail contribution to the effective action involves only one diagram, five are needed for the one-point function. This suggests logarithmic corrections may be easier to incorporate in this fashion. We conclude with a few remarks on the nature of these IR/UV singularities, the (lack of) ambiguities recently discussed in the literature, and the completeness of the analytic post-Newtonian framework.

Simulating a binary black hole coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic −2 Y lm waveform modes resolved by the NR code up to l ¼ 8. We compare our surrogate model to effective one body waveforms from 50M ⊙ to 300M ⊙ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases). DOI: 10.1103/PhysRevLett.115.121102 PACS numbers: 04.25.dg, 02.60.-x, 04.30.-w, 04.30.Db Since the breakthroughs of 2005 [1-3], tremendous progress in numerical relativity (NR) has led to hundreds of simulations of binary black hole (BBH) coalescences [4][5][6][7][8][9][10]. This progress has been driven partly by the data analysis needs of advanced ground-based gravitational wave detectors like LIGO [11] and Virgo [12]. Recent upgrades to these detectors are expected to yield the first direct detections of gravitational waves (GWs) from compact binary coalescences [13].Despite the remarkable progress of the NR community, a single high-quality simulation typically requires days to months of supercomputing time. This high computational cost makes it difficult to directly use NR waveforms for data analysis, except for injection studies [4,9], since detecting GWs and inferring their source parameters may require thousands to millions of accurate gravitational waveforms. Nevertheless, a first template bank for nonspinning binaries in Advanced LIGO has been recently constructed from NR waveforms [14]. Furthermore, NR waveforms have been used successfully in calibrating inspiral-merger-ringdown effective-one-body (EOB) [15][16][17][18][19][20][21] and phenomenological [22][23][24][25] models. These models have free parameters that can be set by matching to NR waveforms and are suitable for certain GW data analysis studies [26]. However, these models can have systematic errors since they assume a priori physical waveform structure and are calibrated and tested against a small set of NR simulations.In this Letter, we present an ab initio methodology based on surrogate [27,28] and reduced order modeling techniques [29][30][31][32][33] that is capable of accurately predicting the gravitational waveform outputs from NR without any phenomenological assumptions or approximations to general relativity. From a small set of specially selected nonspinning BBH simulations performed with the Spectral Einstein code (SpEC) [34][35][36], we build a surrogate model that...