Pulsation and Precession of the Resonant Swinging Spring

When the frequencies of the elastic and pendular oscillations of an elastic pendulum or swinging spring are in the ratio two-to-one, there is a regular exchange of energy between the two modes of oscillation. We refer to this phenomenon as pulsation. Between the horizontal excursions, or pulses, the spring undergoes a change of azimuth which we call the precession angle. The pulsation and stepwise precession are the characteristic features of the dynamics of the swinging spring. The modulation equations for the small-amplitude resonant motion of the system are the well-known three-wave equations. We use Hamiltonian reduction to determine a complete analytical solution. The amplitudes and phases are expressed in terms of both Weierstrass and Jacobi elliptic functions. The strength of the pulsation may be computed from the invariants of the equations. Several analytical formulas are found for the precession angle. We deduce simplified approximate expressions, in terms of elementary functions, for the pulsation amplitude and precession angle and demonstrate their high accuracy by numerical experiments. Thus, for given initial conditions, we can describe the envelope dynamics without solving the equations. Conversely, given the parameters which determine the envelope, we can specify initial conditions which, to a high level of accuracy, yield this envelope.Comment: 33 pages, 9 eps figure

Precession and Recession of the Rock'n'roller

We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centre of mass does not coincide with the geometric centre. The symmetric case, with moments of inertia I_1=I_2, is integrable and the motion is completely regular. Three known conservation laws are the total energy E, Jellett's quantity Q_J and Routh's quantity Q_R. When the inertial symmetry I_1=I_2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (Q_R,Q_J) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett's and Routh's quantities, by identifying individual reversals as crossings of the orbit (Q_R(t),Q_J(t)) through the critical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession

Stepwise Precession of the Resonant Swinging Spring

The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the well-known three-wave equations that also apply, for example, in laser-matter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system's dynamics, namely, the stepwise precession of its azimuthal angle.Comment: 28 pages, 10 figure

Bottom-up versus Top-down Campaigning at the Scottish Independence Referendum 2014

The Scottish independence referendum of 2014 saw one of the longest political campaigns in modern UK politics. The referendum saw the formation of a hybrid Yes campaign that combined modern campaigning techniques with a large, activist grassroots that established a range of new political organisations. Not only did these organisations continue to develop after the referendum itself, but they contributed to the formation of a broad social movement in favour of independence for the first time in Scotland

Pressure on Nicola Sturgeon is intense, but there's still room to be positive about second Scottish indyref

First paragraph: There have been two dramatic stories in Scottish politics in recent days. One has seen Nicola Sturgeon, Scotland's first minister, hold talks with her UK counterpart Theresa May over the ongoing Brexit crisis; and confirming she will clarify plans for a second Scottish independence referendum in the coming weeks. The other involved the shock of seeing Sturgeon's predecessor Alex Salmond appear in court charged with attempted rape and sexual assault.https://theconversation.com/pressure-on-nicola-sturgeon-is-intense-but-theres-still-room-to-be-positive-about-second-scottish-indyref-11063