(i) Consider $N$ particles moving in 3 dimensions. The Cartesian coordinates of these particles are $x^{A}(t), A=1, \ldots, 3 N$. Now consider an invertible change of coordinates to coordinates $q^{a}\left(x^{A}, t\right), \quad a=1, \ldots, 3 N$, so that one may express $x^{A}$ as $x^{A}\left(q^{a}, t\right)$. Show that the velocity of the system in Cartesian coordinates $\dot{x}^{A}(t)$ is given by the following expression:

$$\dot{x}^{A}\left(\dot{q}^{a}, q^{a}, t\right)=\sum_{b=1}^{3 N} \dot{q}^{b} \frac{\partial x^{A}}{\partial q^{b}}\left(q^{a}, t\right)+\frac{\partial x^{A}}{\partial t}\left(q^{a}, t\right)$$

Furthermore, show that Lagrange's equations in the two coordinate systems are related via

$$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=\sum_{A=1}^{3 N} \frac{\partial x^{A}}{\partial q^{a}}\left(\frac{\partial L}{\partial x^{A}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{x}^{A}}\right)$$

(ii) Now consider the case where there are $p<3 N$ constraints applied, $f^{\ell}\left(x^{A}, t\right)=$ $0, \ell=1, \ldots, p$. By considering the $f^{\ell}, \ell=1, \ldots, p$, and a set of independent coordinates $q^{a}, a=1, \ldots, 3 N-p$, as a set of $3 N$ new coordinates, show that the Lagrange equations of the constrained system, i.e.

$$\begin{gathered} \frac{\partial L}{\partial x^{A}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{A}}\right)+\sum_{\ell=1}^{p} \lambda^{\ell} \frac{\partial f^{\ell}}{\partial x^{A}}=0, \quad A=1, \ldots, 3 N \\ f^{\ell}=0, \quad \ell=1, \ldots, p \end{gathered}$$

(where the $\lambda^{\ell}$ are Lagrange multipliers) imply Lagrange's equations for the unconstrained coordinates, i.e.

$$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=0, \quad a=1, \ldots, 3 N-p .$$