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The paper deals with a generalization of Helmholtz' conditions for the existence of a first‐order kinetic potential related to a given set of 2nd order ordinary differential equations. The extension affects the consideration of such ODE systems which are not self‐adjoint due to dissipation. Necessary and sufficient conditions are given for the simultaneous existence of two state functions – a Lagrangian and a dissipation function – both of the first‐order such that the given set of 2nd order ODE results from the well‐known Lagrangean approach.

The paper presents a generalization of the Helmholtz conditions for the existence of a first-order kinetic potential (Lagrangian) in that cases where the motion equations of mechanical, electrical, or electromechanical systems are given in terms of nonholonomic velocities. It is assumed that the transformation between holonomic and nonholonomic velocities is known. It is shown how the generalized Helmholtz conditions for the simultaneous existence of a first-order Lagrangian and a first-order dissipation function follow from the generalized Helmholtz conditions for systems with holonomic velocities. Moreover, the classical Helmholtz conditions in case of nondissipative systems can be directly generalized to motion equations which include nonholonomic velocities. In both cases the Boltzmann-Hamel equations are used. The approaches are demonstrated by two examples: a heavy gyroscope and a 3-dimensional rotating electrical machine.

In this paper, the classical augmented PD control method is extended to a passivity‐based control of Euler‐Lagrange systems with a non‐quadratic Lagrangian. It is assumed that the systems are fully actuated. The problem is treated with two geometrically different approaches. The first approach is dealing with state space control in the configuration space by introducing the kinetic energy via the Legendre transformation of the kinetic co‐energy. The second approach considers the system's dynamics in the event space endowed with a Finsler metric based on the kinetic co‐energy. It is shown that the control laws of both approaches achieve uniformly asymptotically stable trajectory tracking.

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