  Dark energy can be investigated in two complementary ways, by considering
either general parameterizations or physically well-defined models. Following
the second route, we explore the constraints on quintessence models where the
acceleration is driven by a slow-rolling scalar field. The analysis focuses on
cosmic shear, combined with supernovae Ia and CMB data. Using a Boltzmann code
including quintessence models and the computation of weak lensing observables,
we determine several two-point shear statistics. The non-linear regime is
described by two different mappings. The likelihood analysis is based on a grid
method. The data include the "gold set" of supernovae Ia, the WMAP-1 year data
and the VIRMOS-Descart and CFHTLS-deep and -wide data for weak lensing. This is
the first analysis of high-energy motivated dark energy models that uses weak
lensing data. We explore larger angular scales, using a synthetic realization
of the complete CFHTLS-wide survey as well as next space-based missions
surveys. Two classes of cosmological parameters are discussed: i) those
accounting for quintessence affect mainly geometrical factors; ii) cosmological
parameters specifying the primordial universe strongly depend on the
description of the non-linear regime. This dependence is addressed using wide
surveys, by discarding the smaller angular scales to reduce the dependence on
the non-linear regime. Special care is payed to the comparison of these
physical models with parameterizations of the equation of state. For a flat
universe and a quintessence inverse power law potential with slope alpha, we
obtain alpha < 1 and Omega_Q=0.75^{+0.03}_{-0.04} at 95% confidence level,
whereas alpha=2^{+18}_{-2}, Omega_Q=0.74^{+0.03}_{-0.05} when including
supergravity corrections.
