The Adiabatic Invariance of the Action Variable in Classical Dynamics

We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well-known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We present a new proof of the adiabatic invariance of this quantity and illustrate our arguments by means of explicit calculations for the harmonic oscillator. The new proof makes essential use of the Hamiltonian formalism. The key step is the introduction of a slowly-varying quantity closely related to the action variable. This new quantity arises naturally within the Hamiltonian framework as follows: a canonical transformation is first performed to convert the system to action-angle coordinates; then the new quantity is constructed as an action integral (effectively a new action variable) using the new coordinates. The integration required for this construction provides, in a natural way, the averaging procedure introduced in other proofs, though here it is an average in phase space rather than over time.Comment: 8 page

Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_{ab} and H_{ab}, represent the locally free curvature - i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of E_{ab} and H_{ab}, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E_{ab} and H_{ab} together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.Comment: New results on gravitational waves included; new references added; revised version (IOP style) to appear Class. Quantum Gra

Capital mobility and EU enlargement

F21, F32, F36,