Consistent definition of partial derivatives of matrix functions in dynamics of mechanical systems

The partial derivatives of scalar functions, vector functions, and matrix functions with respect to a vector variable have many practical applications in the study of dynamics and control of mechanical systems. The paper firstly presents an overview of the matrix calculus related to Kronecker products. Secondly, two theorems establishing the relationship between the time derivative of a matrix and its partial derivative with respect to a vector, and the partial derivative of the product of two matrices with respect to a vector will be presented. Finally, a new matrix form of Lagrangian equations is developed to demonstrate the usefulness of these theorems in the study of dynamics of mechanical systems.