60 research outputs found
The classical mechanics of non-conservative systems
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 long-standing 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.Comment: 5 pages, 1 figure. Updated to incorporate referees' comments. Matches
published versio
Caustic echoes from a Schwarzschild black hole
We present the first numerical construction of the scalar Schwarzschild Green
function in the time-domain, which reveals several universal features of wave
propagation in black hole spacetimes. We demonstrate the trapping of energy
near the photon sphere and confirm its exponential decay. The trapped wavefront
propagates through caustics resulting in echoes that propagate to infinity. The
arrival times and the decay rate of these caustic echoes are consistent with
propagation along null geodesics and the large l-limit of quasinormal modes. We
show that the four-fold singularity structure of the retarded Green function is
due to the well-known action of a Hilbert transform on the trapped wavefront at
caustics. A two-fold cycle is obtained for degenerate source-observer
configurations along the caustic line, where the energy amplification increases
with an inverse power of the scale of the source. Finally, we discuss the tail
piece of the solution due to propagation within the light cone, up to and
including null infinity, and argue that, even with ideal instruments, only a
finite number of echoes can be observed. Putting these pieces together, we
provide a heuristic expression that approximates the Green function with a few
free parameters. Accurate calculations and approximations of the Green function
are the most general way of solving for wave propagation in curved spacetimes
and should be useful in a variety of studies such as the computation of the
self-force on a particle.Comment: 18 pages, 23 figure
Deriving analytic solutions for compact binary inspirals without recourse to adiabatic approximations
We utilize the dynamical renormalization group formalism to calculate the
real space trajectory of a compact binary inspiral for long times via a
systematic resummation of secularly growing terms. This method generates closed
form solutions without orbit averaging, and the accuracy can be systematically
improved. The expansion parameter is where is the
initial time, is the time elapsed, and and are the angular
orbital frequency and initial speed, respectively, and is the binary's
symmetric mass ratio. We demonstrate how to apply the renormalization group
method to resum solutions beyond leading order in two ways. First, we calculate
the second order corrections of the leading radiation reaction force, which
involves highly non-trivial checks of the formalism (i.e. its
renormalizability). Second, we show how to systematically include
post-Newtonian corrections to the radiation reaction force. By avoiding orbit
averaging we gain predictive power and eliminate ambiguities in the initial
conditions. Finally, we discuss how this methodology can be used to find
analytic solutions to the spin equations of motion that are valid over long
times.Comment: 18 pages, 2 figure
Radiation reaction at 3.5 post-Newtonian order in effective field theory
We derive the radiation reaction forces on a compact binary inspiral through 3.5 order in the post-Newtonian expansion using the effective field theory approach. We utilize a recent formulation of Hamilton’s variational principle that rigorously extends the usual Lagrangian and Hamiltonian formalisms to dissipative systems, including the inspiral of a compact binary from the emission of gravitational waves. We find agreement with previous results, which thus provides a non-trivial confirmation of the extended variational principle. The results from this work nearly complete the equations of motion for the generic inspiral of a compact binary with spinning constituents through 3.5 post-Newtonian order, as derived entirely with effective field theory, with only the spin-orbit corrections to the potential at 3.5 post-Newtonian remaining
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