This paper examines the extent to which nonlinear dimension reduction techniques from machine learning can be exploited to determine dynamically optimal motions for high degree of freedom systems. Using the Gaussian Process Latent Variable Model (GPLVM) to learn the low-dimensional embedding, and a density function that provides a nonlinear mapping from the low-dimensional latent space to the full-dimensional pose space, we determine optimal motions by optimizing in the latent space, and mapping the optimal latent space trajectory to the pose space. The notion of variance tubes are developed to ensure that kinematic and other constraints are appropriately satisfied without sacrificing naturalness or richness of the motions. Case studies involving a 62-dof humanoid performing two sports motions—a golf swing and throwing a baseball—demonstrate that our method can be a highly effective and computationally efficient method for generating dynamically optimal motions.