# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021022
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## Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

 1 Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina 2 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina 3 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina

Received  September 2018 Revised  June 2021 Early access August 2021

Fund Project: S. Grillo and L. Salomone thank CONICET for its financial support

In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the kinetic and potential equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the $C^{\infty}$ category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.

Citation: Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021022
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