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In this paper, we make a review of the controlled Hamiltonians (CH) methodand its related matching conditions, focusing on an improved version recentlydeveloped by D.E. Chang. Also, we review the general ideas around the Lyapunovconstraint based (LCB) method, whose related partial differential equations(PDEs) were originally studied for underactuated systems with only oneactuator, and then we study its PDEs for an arbitrary number of actuators. Weanalyze and compare these methods within the framework of DifferentialGeometry, and from a purely theoretical point of view. We show, in the contextof underactuated systems defined by simple Hamiltonian functions, that the LCBmethod and the Chang's version of the CH method are equivalent stabilizationmethods (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 preciselythe LCB method. As a by-product, coordinate-free and connection-freeexpressions of Chang's matching conditions are obtained.

On the relationship between the energy shaping and the Lyapunov constraint based methods