  Using Legendre transformation, a standard theoretical approach extensively
used in classical mechanics as well as thermal dynamics, two-dimensional
non-linear auto-oscillators including spin torque nano-oscillators (STNOs) can
be equivalently expressed either in phase space or in configuration space where
all of them can be modeled by terminal velocity motion (TVM) particles. The
transformation completely preserves the dynamic information about the canonical
momenta, leading to very precise analytical predictions about the phase-locking
of a coupled pair of perpendicular to plane STNOs (PERP-STNOs) including
dynamical phase diagrams, (un)phase-locked frequencies, phase-locked angles,
and transient evolutions, which are all solved based on Newton mechanics.
Notably, the TVM model successfully solves the difficulty of the generalized
pendulum-like model [Chen \textit{et al}. \textbf{J. Appl. Phys. 130}, 043904
(2021)] failing to make precise predictions for the higher range of current in
serial connection. Additionally, how to simply search for the critical currents
for phase-locked (PL) and asynchronized (AS) states by numerically simulating
the macrospin as well as TVM model, which gets inspired through analyzing the
excitations of a forced pendulum, is also supplied here. Therefore, we believe
that the TVM model can bring a more intuitive and precise way to explore all
types of two-dimensional non-linear auto-oscillators.
