Lagrangian identification and response prediction from discrete data within the first-kind Lagrangian framework

Identifying explicit Lagrangians from discrete state data of dynamical systems is of both scientific significance and practical value. Existing methods are predominantly established within the second-kind Lagrangian framework, where the use of generalized coordinate reduces system dimensionality at the cost of increased functional complexity. In contrast, the first-kind Lagrangian framework formulated in physical coordinates, involves higher dimensionality but features simpler functional structures. The intrinsic simplicity renders it particularly suitable for data-driven identification. This work addresses this problem by developing a data-driven method for identifying explicit Lagrangians in terms of physical coordinates. We first embed the first-kind Lagrangian framework into dynamical systems of concern, eliminate all Lagrange multipliers, expand the Lagrangian over a set of prespecified basis functions, and identify the explicit Lagrangian by sparse optimization. A well-posed problem is then constructed by combining the identified Lagrangian with full set of constraint equations and initial conditions, enabling the prediction of state responses beyond the training data. Four representative examples are provided to demonstrate the application, effectiveness and accuracy of the proposed method. In addition, we conduct a comparison with existing methods, and discuss its robustness to data noise as well as the non-uniqueness and equivalence of Lagrangians identified.