  This article focuses on an important quantity that will be called the
Rund-Trautman function. It already plays a central role in Noether's theorem
since its vanishing characterizes a symmetry and leads to a conservation law.
The main aim of the paper is to show how, in the realm of classical mechanics,
an 'almost' vanishing Rund-Trautman function accompanying an 'almost' symmetry
leads to an 'almost' constant of motion within the adiabatic assumption, that
is, to an adiabatic invariant. To this end, the Rund-Trautman function is first
introduced and analysed in detail, then it is implemented for the general
one-dimensional problem. Finally, its relevance in the adiabatic context is
examined through the example of the harmonic oscillator with a slowly varying
frequency. Notably, for some frequency profiles, explicit expansions of
adiabatic invariants are derived through it and an illustrative numerical test
is realized.
