Consider a Lagrangian system with Lagrangian $L\left(x_{A}, \dot{x}_{A}, t\right)$, where $A=1, \ldots, 3 N$, and constraints

$$f_{\alpha}\left(x_{A}, t\right)=0, \quad \alpha=1, \ldots, 3 N-n .$$

Use the method of Lagrange multipliers to show that this is equivalent to a system with Lagrangian $\mathcal{L}\left(q_{i}, \dot{q}_{i}, t\right) \equiv L\left(x_{A}\left(q_{i}, t\right), \dot{x}_{A}\left(q_{i}, \dot{q}_{i}, t\right), t\right)$, where $i=1, \ldots, n$, and $q_{i}$ are coordinates on the surface of constraints.

Consider a bead of unit mass in $\mathbb{R}^{2}$ constrained to move (with no potential) on a wire given by an equation $y=f(x)$, where $(x, y)$ are Cartesian coordinates. Show that the Euler-Lagrange equations take the form

$$\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}=\frac{\partial \mathcal{L}}{\partial x}$$

for some $\mathcal{L}=\mathcal{L}(x, \dot{x})$ which should be specified. Find one first integral of the EulerLagrange equations, and thus show that

$$t=F(x)$$

where $F(x)$ should be given in the form of an integral.

[Hint: You may assume that the Euler-Lagrange equations hold in all coordinate systems.]