  A secular variation of the gravitational constant modifies the structure and
evolutionary time scales of white dwarfs. Using an state-of-the-art stellar
evolutionary code and an up-to-date pulsational code we compute the effects of
a secularly varying $G$ on the pulsational properties of variable white dwarfs.
Comparing the the theoretical results obtained taking into account the effects
of a running $G$ with the observed periods and measured rates of change of the
periods of two well studied pulsating white dwarfs, G117--B15A and R548, we
place constraints on the rate of variation of Newton's constant. We derive an
upper bound $\dot G/G\sim -1.8\times 10^{-10}$ yr$^{-1}$ using the variable
white dwarf G117--B15A, and $\dot G/G\sim -1.3\times 10^{-10}$ yr$^{-1}$ using
R548. Although these upper limits are currently less restrictive than those
obtained using other techniques, they can be improved in a future measuring the
rate of change of the period of massive white dwarfs.

Solar and Stellar Astrophysics;High Energy Physics - Phenomenology