The latest version of the Variational Principle (VP) [16] generates unique non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl whose solutions belongs a diffeomorphic Lie algebra. This was theoretically used for the deformation field of an optimal control approach to non-rigid image registration [18] (referred to as the VP-control method). In order to build a deep-learning optimization mechanism compatible with the idea of the VP-control method, several fundamental issues need to be addressed. In this paper, a Lagrange-Multiplier formulation of VP (referred to as LM-VP) is introduced, for three purposes: (1) arguing that optimizing over JD and curl indeed computationally reconstructs a given ground truth grid; (2) bypassing the necessity of computing the control functions of JD and curl in VP; (3) presenting a deep-learning loss function for image registration based on the VP-control method via a similar Lagrange-Multiplier formulation. These purposes are demonstrated with the model descriptions, computational algorithms and preliminary examples. This work strongly supports a parallel research that emphasizes performance of the proposed loss function for deep-learning image registration problem.