  In this paper, we make a review of the controlled Hamiltonians (CH) method
and its related matching conditions, focusing on an improved version recently
developed by D.E. Chang. Also, we review the general ideas around the Lyapunov
constraint based (LCB) method, whose related partial differential equations
(PDEs) were originally studied for underactuated systems with only one
actuator, and then we study its PDEs for an arbitrary number of actuators. We
analyze and compare these methods within the framework of Differential
Geometry, and from a purely theoretical point of view. We show, in the context
of underactuated systems defined by simple Hamiltonian functions, that the LCB
method and the Chang's version of the CH method are equivalent stabilization
methods (i.e. they give rise to the same set of control laws). In other words,
we show that the Chang's improvement of the energy shaping method is precisely
the LCB method. As a by-product, coordinate-free and connection-free
expressions of Chang's matching conditions are obtained.
