Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point.\\
(1) If $x _ { 1 } = - 1$ , find $a$ ;\\
(2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .