Derive the Lagrange equations from the principle of stationary action

$$S[q]=\int_{t_{0}}^{t_{1}} \mathcal{L}\left(q_{i}(t), \dot{q}_{i}(t), t\right) d t, \quad \delta S=0$$

where the end points $q_{i}\left(t_{0}\right)$ and $q_{i}\left(t_{1}\right)$ are fixed.

Let $\phi$ and $\mathbf{A}$ be a scalar and a vector, respectively, depending on $\mathbf{r}=(x, y, z)$. Consider the Lagrangian

$$\mathcal{L}=\frac{m \dot{\mathbf{r}}^{2}}{2}-(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

and show that the resulting Euler-Lagrange equations are invariant under the transformations

$$\phi \rightarrow \phi+\alpha \frac{\partial F}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla F,$$

where $F=F(\mathbf{r}, t)$ is an arbitrary function, and $\alpha$ is a constant which should be determined.