  Linear perturbation theory is appropriate to describe small oscillations of
stars, while a mild non-linearity is still tractable perturbatively but
requires to consider mode coupling. It is natural to start to look at this
problem by considering the coupling between linear radial and non-radial modes.
Radial pulsations of a spherical compact objects do not per se emit
gravitational waves but, if the coupling is efficient in driving and possibly
amplifying the non-radial modes, gravitational radiation could then be produced
to a significant level.
  In this paper we develop the relativistic formalism to study the coupling of
radial and non-radial first order perturbations of a compact spherical star.
  From a mathematical point of view, it is convenient to treat the two sets of
perturbations as separately parametrized, using a 2-parameter perturbative
expansion of the metric, the energy-momentum tensor and Einstein equations in
which $\lambda$ is associated with the radial modes, $\epsilon$ with the
non-radial perturbations, and the $\lambda\epsilon$ terms describe the
coupling. This approach provides a well-defined framework to consider the gauge
dependence of perturbations, allowing us to use $\epsilon$ order
gauge-invariant non-radial variables on the static background and to define new
second order $\lambda\epsilon$ gauge-invariant variables describing the
non-linear coupling. We present the evolution and constraint equations for our
variables outlining the setup for numerical computations, and briefly discuss
the surface boundary conditions in terms of the second order $\lambda\epsilon$
Lagrangian pressure perturbation.
