A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint

Simone Fiori ()
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Simone Fiori: Department of Information Engineering, Marches Polytechnic University, 60131 Ancona, Italy

Mathematics, 2024, vol. 12, issue 3, 1-12

Abstract: The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d’Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on the kinetic energy and the covariant acceleration energy, as well as a potential energy function that accounts for conservative forces. In addition, the present paper introduces the notion of Rayleigh differential form to account for non-conservative forces. The corresponding fourth-order equation of motion is derived, and an interpretation of the obtained terms is provided from a system and control theoretic viewpoint. A specific form of the Rayleigh differential form is introduced, which yields non-conservative forcing terms assimilable to linear friction and jerk-type friction. The general theoretical discussion is complemented by a brief excursus about the numerical simulation of the introduced differential model.

Keywords: fourth-order dynamical system on manifold; smooth manifold; covariant derivation; covariant jerk; manifold curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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