Principal component analysis (PCA) is a major statistical analysis and machine learning tool for dimensional reduction and visualization of high-dimensional datasets [1]. Classical PCA in the Euclidean space is to find the eigenvectors associated with the top eigenvalues of the covariance matrix. Geometrically, PCA can be interpreted as finding the orthogonal directions that maximize the projected data variance to the linear subspace spanned by those directions. Recently, efforts for extending the Euclidean PCA to capture variations for a collection of probability measures have been made [2, 3, 4]. Since the Wasserstein space is an infinite-dimensional curved space, one challenge is to define a proper notion of principal mode of variations in the space of probability measures. In this paper, we take a variational and dynamical perspective of the Euclidean PCA that has robust generalization to the Wasserstein geometry. Specifically, given input data points x1,...,xn in the Euclidean space Rm, performing the standard PCA to find the first principal mode of variation gt = xn +tv passing through the mean xn = n 1 Pn i=1 xi can be reformulated as minimizing the residuals by projecting each data point in the direction v: ˆv1 = argmin