  In this work, two multi-harmonic Hamiltonian models for mean motion
resonances are formulated and their applications to first-order resonances are
discussed. For the $k_p$:$k$ resonance, the usual critical argument $\varphi =
k \lambda - k_p \lambda_p + (k_p - k) \varpi$ is taken as the resonant angle in
the first model, while the second model is characterized by a new critical
argument $\sigma = \varphi / k_p$. Based on canonical transformations, the
resonant Hamiltonians associated with these two models are formulated. It is
found that the second Hamiltonian model holds two advantages in comparison to
the first model: (a) providing a direct correspondence between phase portraits
and Poincar\'e sections, and (b) presenting new phase-phase structures where
the zero-eccentricity point is a visible saddle point. Then, the second
Hamiltonian model is applied to the first-order inner and outer resonances,
including the 2:1, 3:2, 4:3, 2:3 and 3:4 resonances. The phase-space structures
of these first-order resonances are discussed in detail and then the libration
centers and associated resonant widths are identified analytically. Simulation
results show that there are pericentric and apocentric libration zones where
the libration centers diverge away from the nominal resonance location as the
eccentricity approaches zero and, in particular, the resonance separatrices do
not vanish at arbitrary eccentricities for both the inner and outer
(first-order) resonances.
