*Europhys. Lett.*,

**45**(1), pp. 26-31 (1999)

## An expansion term in Hamilton's equations

Department of Mathematics and Applied Mathematics,
University of Cape Town, South Africa

Received:
18
September
1998

Accepted:
30
October
1998

For any given spacetime the choice of time coordinate is undetermined.
A particular choice is the absolute time associated with a preferred
vector field. Using the absolute time Hamilton's equations are
,
where
is the expansion of the vector field. Thus
there is a hitherto unnoticed term in the expansion of the preferred vector
field. Hamilton's equations can be used to describe fluid motion. In this
case the absolute time is the time associated with the fluid's co-moving
vector. As measured by this absolute time the expansion term is present.
Similarly in cosmology, each observer has a co-moving vector and
Hamilton's equations again have an expansion term. It is necessary to
include the expansion term to quantize systems such as the above by the
canonical method of replacing Dirac brackets by commutators.
Hamilton's equations in this form do not have a corresponding sympletic
form. Replacing the expansion by a particle number
and introducing the particle numbers conjugate momentum the standard
sympletic form can be recovered with two extra fields *N* and .
Briefly the possibility of a non-standard sympletic form and
the further possibility
of there being a non-zero Finsler curvature corresponding to this are
looked at.

PACS: 11.10.Ef – Lagrangian and Hamiltonian approach / 04.60.Ds – Canonical quantization / 52.60.+h – Relativistic plasma

*© EDP Sciences, 1999*