We know that the lagrangian function of a holonomic system subject to fixed constraints has the form $$\mathcal{L}(\mathbf{q,\dot{q}})=\frac{1}{2} \langle \mathbf{\dot{q},A(q)\dot{q}} \rangle - U(\textbf{q}) $$ where $\mathbf{q}$ is the vector of lagrangian coordinates of the system, $\mathbf{A(q)}$ is the mass matrix of the system and $U$ is its potential. Computing Eulero-Lagrange equations for this system, if we suppose the mass matrix does not depend on $\mathbf{q}$, yields the equation of motion $$\mathbf{A\ddot{q}}=-\nabla U(\mathbf{q}).$$
However, I wonder what the form of the equation of motion is when $\mathbf{A(q)}$ does depend on the coordinates. Indeed, the professor whose lectures I am attending keeps on using the equation above even then the mass matrix is not costant; I have tried to compute the equation in such case, but I do get a different equation. Does anybody know how to help me?
