How are Lagrangian mechanics equivalent to Newtonian mechanics? I didn't study Lagrangian mechanics yet but I did study Newtonian mechanics, and someone said to me that later we would study analytic mechanics (which contain Lagrangian mechanics) and that it contain some equations that are equivalent to Newton's laws  but are more fundamental. I want just one single example of this without complicated maths if possible, but I will study the core and pure thing later on.
 A: The Euler-Lagrange equation: $$\frac{\mathrm d\mathcal{L}}{\mathrm dx}=\frac{\mathrm d}{\mathrm dt}\frac{\mathrm d\mathcal{L}}{\mathrm d\dot{x}}$$ is equivalent to Newton's second law of motion which states: $$\mathbf{ F}=m\vec a.$$

I will not go into the pure maths way since I will only present the procedures to show that our first equation is equivalent to Newton's second law. The Lagrangian (denoted $\mathcal L$) is simply the kinetic energy of a body minus its potential energy which can be written as follows, $\mathcal L=1/2mv^2-\mathrm V(x)$. Now if you take the derivative of this Lagrangian w.r.t $x$ you will simply get the derivative of the potential energy $\mathrm{V}(x)$ w.r.t $x$, which is a force. So the LHS of the first equation is equivalent to the LHS of the second. 
$\dot{x}$ just means the derivative of $x$ w.r.t time, so if you differentiate the Lagrangian with respect to $\dot{x}$, which is velocity, you will get $m\dot{x}$ since $$\frac{\mathrm d\mathcal{L}}{\mathrm d\dot{x}}=\frac{\mathrm d}{\mathrm d\dot{x}}\frac12 mv^2-\mathrm{V}(x)=\frac{\mathrm d}{\mathrm d\dot{x}}\frac12 m\dot{x}^2=m\dot{x}.$$
And the derivative of the latter w.r.t time will be $m\ddot{x}$ which is of course $m\vec{a}$. So the RHS of the first equation is equivalent to the RHS of the second equation. Therefore: 
$$\frac{\mathrm d\mathcal{L}}{\mathrm dx}=\frac{\mathrm d}{\mathrm dt}\frac{\mathrm d\mathcal{L}}{\mathrm d\dot{x}}\iff \mathbf{ F}=m\vec a.$$
A: Here is a simple example since you requested one in your post. Consider a pendulum with length $\ell$ and bob mass $m$. Then the energies of the system are
\begin{align}
V &= mg\ell(1 - \cos(\theta))\\
T &= \frac{1}{2}m\dot{x}^2
\end{align}
Now to use the Euler Lagrange equations we need a generalized coordinate. In this case we will take $\theta$ as our generalized coordinate and assume a small angle approximation. Then with the small angle approximation $\sin(\theta) \approx\theta$ since $\sin(\theta) = \sum_{n=0}^{\infty}(-1)^n\frac{\theta^{2n+1}}{(2n+1)!}$. Therefore, $\sin(\theta)\approx\theta = \frac{x}{\ell}$ so $x = \ell\theta$ and our potential energy is 
$$
T = \frac{1}{2}m(\ell\dot{\theta})^2
$$
From the first answer, we know that 
$$
\mathcal{L} = T - V = \frac{1}{2}m(\ell\dot{\theta})^2 - mg\ell(1 - \cos(\theta))
$$
Now let's apply the EL equations
\begin{align}
\frac{\partial\mathcal{L}}{\partial\theta} &= -mg\ell\sin(\theta)\\
\frac{\partial\mathcal{L}}{\partial\dot{\theta}} &= m\ell^2\dot{\theta}\\
\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}} &= m\ell^2\ddot{\theta}\\
m\ell^2\ddot{\theta} + mgl\sin(\theta) &= 0
\end{align}
Again applying our small angle approximations, we have 
$$
m\ell^2\ddot{\theta} + mgl\sin(\theta) \approx m\ell^2\ddot{\theta} + mgl\theta = \ddot{\theta} + \frac{g}{\ell}\theta =0
$$
Now work out the solution in Newtonian Mechanics.
A: I would say that the Lagrangian approach is indeed more fundamental than Newton's laws. The reason for this is that the Euler-Lagrange equations of motion do indeed reduce to the ones that one gets from Newton's laws; they are also simpler and in general, easier to work out, but that is not the point. The point is that the Euler-Lagrange equations come from the principle of minimal action, which is a fundamental one, equivalent to the principle that dictates that every physical system will tend to minimize its free energy. Furthermore, it is in this approach (Euler-Lagrange) that through a theorem by Emmy Noether we know that the presence of symmetries leads to conserved quantities, which in turn, has led to fundamental discoveries in quantum field theory (read gauge theories).
