Symmetry and conservation law
Like I said in the edit, it's the translation of coordinates that doesn't change the Lagrangian. I suppose, a concrete example would be better. Say, I have two masses, m_{1} and m_{2} at locations x_{1} and x_{2} connected by harmonic potential.
[tex]L(x_1,x_2,\dot{x}_1,\dot{x_2}) = \frac{1}{2}(m_1 \dot{x}_1^2+m_2 \dot{x}_2^2)  \frac{k}{2}(x_2  x_1)^2[/tex]
Now, let's say I introduced new coordinates, x'_{1}=x_{1}+c, x'_{2}=x_{2}+c.
[tex]L(x'_1,x'_2,\dot{x}'_1,\dot{x}'_2) = \frac{1}{2}(m_1 \dot{x}\prime_1^2+m_2 \dot{x}\prime_2^2)  \frac{k}{2}(x'_2  x'_1)^2 = \frac{1}{2}(m_1 \dot{x}_1^2+m_2 \dot{x}_2^2)  \frac{k}{2}(x_2 + c  x_1  c)^2 = L(x_1,x_2,\dot{x}_1,\dot{x_2})[/tex]
So the Lagrangian is exactly the same after coordinate got shifted. Not just the total action, but the actual value of Lagrangian at every point.
