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For the study of highly nonlinear, conservative dynamic systems, findingspecial periodic solutions which can be seen as generalization of thewell-known normal modes of linear systems is very attractive. However, thestudy of low-dimensional invariant manifolds in the form of nonlinear normalmodes is rather a niche topic, treated mainly in the context of structuralmechanics for systems with Euclidean metrics, i.e., for point masses connectedby nonlinear springs. Newest results emphasize, however, that a very richstructure of periodic and low-dimensional solutions exist also within nonlinearsystems such as elastic multi-body systems encountered in the biomechanics ofhumans and animals or of humanoid and quadruped robots, which are characterizedby a non-constant metric tensor. This paper discusses different generalizationsof linear oscillation modes to nonlinear systems and proposes a definition ofstrict nonlinear normal modes, which matches most of the relevant properties ofthe linear modes. The main contributions are a theorem providing necessary andsufficient conditions for the existence of strict oscillation modes on systemsendowed with a Riemannian metric and a potential field as well as aconstructive example of designing such modes in the case of an elastic doublependulum.

One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds