  In this paper, to fit the multi-scale perturbations, the single-parameter Lie
transform perturbation theory given in [Ann Phys, 151 1 (1983)] is generalized
to a multi-parameter case, which provides a formal solution of the new
Lagrangian differential 1-form transformed from the old one. In the new method,
the generators and their orders are appropriately chosen under the purpose of
reducing the fast variables from the new Lagrangian differential 1-form order
by order. As for the application, this multi-parameter Lie transform method is
applied to deriving the gyrokinetic model with the presentation of
low-frequency electrostatic perturbations. Compared with the conventional
model, the Finite Larmor Radius terms in the Lagrangian 1-form of the new model
are perfectly cancelled, without using any gauge function.
