Wave dispersion in a pulsar plasma is discussed emphasizing the relevance of different inertial frames, notably the plasma rest frame K and the pulsar frame K ′ in which the plasma is streaming with speed β s . The effect of a Lorentz transformation on both subluminal, |z| < 1, and superluminal, |z| > 1, waves is discussed. It is argued that the preferred choice for a relativistically streaming distribution should be a Lorentz-transformed Jüttner distribution; such a distribution is compared with other choices including a relativistically streaming Gaussian distribution. A Lorentz transformation of the dielectric tensor is written down, and used to derive an explicit relation between the relativistic plasma dispersion functions in K and K ′ . It is shown that the dispersion equation can be written in an invariant form, implying a one-to-one correspondence between wave modes in any two inertial frames. Although there are only three modes in the plasma rest frame, it is possible for backward-propagating or negative-frequency solutions in K to transform into additional forward-propagating, positive-frequency solutions in K ′ that may be regarded as additional modes.

We consider critically the three most widely favored pulsar radio emission mechanisms: coherent curvature emission (CCE), beam-driven relativistic plasma emission (RPE) and anomalous Doppler emission (ADE). We assume that the pulsar plasma is one dimensional (1D), streaming outward with a bulk Lorentz factor γs ≫ 〈γ〉 − 1 ≳ 1, where 〈γ〉 is the intrinsic spread in the rest frame of the plasma. We argue that the formation of beams in a multi-cloud model is ineffective in the intrinsically relativistic case for plausible parameters, because the overtaking takes too long. We argue that the default choice for the particle distribution in the rest frame is a Jüttner distribution and that relativistic streaming should be included by applying a Lorentz transformation to the rest-frame distribution, rather than the widely assumed relativistically streaming Gaussian distribution. We find that beam-driven wave growth is severely restricted by (a) the wave properties in pulsar plasma, (b) a separation condition between beam and background, and (c) the inhomogeneity of the plasma in the pulsar frame. The growth rate for the kinetic instability is much smaller and the bandwidth of the growing waves is much larger for a Jüttner distribution than for a relativistically streaming Gaussian distribution. No reactive instability occurs at all for a Jüttner distribution. We conclude that none of CCE, RPE and ADE in tenable as the generic pulsar radio emission mechanism for “plausible” assumptions about the pulsar plasma.

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Wave dispersion in a pulsar plasma (a 1D, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the lowfrequency, non-gyrotropic approximation. For parallel propagation (wave angle θ = 0) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation |z| = z A ≈ 1 − 1/2β 2 A , where z = ω/k c is the phase speed and β A c is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function, z 2 W (z), which is negative for |z| < z 0 and has a sharp maximum at |z| = z m , with 1 − z m < 1 − z 0 ≪ 1. We give numerical estimates for the maximum of z 2 W (z) and for z m and z 0 for a 1D Jüttner distribution. The L and A modes reconnect, for z A > z 0 , to form the O and Alfvén modes for oblique propagation (θ = 0). For z A < z 0 the Alfvén and O mode curves reconnect forming a new mode that exists only for tan 2 θ z 2 0 − z 2 A . The L mode is the nearest counterpart to Langmuir waves in a nonrelativistic plasma, but we argue that there are no "Langmuir-like" waves in pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir-like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around θ ≈ 0. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a nonrelativistic plasma.

We investigate a variation of the simple double pendulum in which the two point masses are replaced by square plates. The double square pendulum exhibits richer behavior than the simple double pendulum and provides a convenient demonstration of nonlinear dynamics and chaos. It is also an example of an asymmetric compound double pendulum, which has not been studied in detail. We obtain the equilibrium configurations and normal modes of oscillation and derive the equations of motion, which are solved numerically to produce Poincaré sections. We show how the behavior varies from regular motion at low energies, to chaos at intermediate energies, and back to regular motion at high energies. We also show that the onset of chaos occurs at a significantly lower energy than for the simple double pendulum.