  A theoretical framework has been developed for an NBI scenario to model the
hard nonlinear evolution of Global Alfven Eigenmodes (GAEs) where the adiabatic
motion of phase-space sturctures (holes and clumps), associated with the
frequency chirping, occurs in generalized phase-space of slowing down energetic
particles. The radial profile of the GAE is expanded using finite elements
which allows update of the mode structure as the mode frequency chirps.
Constants of motion are introduced to track the dynamics of energetic particles
during frequency chirping by implementing proper Action-Angle variables and
canonical transformations which reduce the dynamics essentially to 1D.
Consequently, we specify whether the particles are drifting inward/outward as
the frequency deviates from the initial MHD eigenfrequency. Using the principle
of least action, we have derived the nonlinear equation describing the
evolution of the radial profile by varying the total Lagrangian of the system
with respect to the weights of finite elements. For the choice of parameters in
this work, it is shown that the peak of the radial profile is shifted and also
broadens due to frequency chirping. The time rate of frequency change is also
calculated using the energy balance and we show that the adiabatic condition
remains valid once it is satisfied. This model clearly illustrates the
theoretical treatment to study the long range adiabatic frequency sweeping
events observed for Alfven gap modes in real experiments.
