Prove it is not a closed Curve

Question

I wanna prove that $(cos(t^3+t),sin(t^3+t))=γ(t)$ which is a reparametrization of a circle .Is not a closed curve like the circle. WHat i did is take this problem to the Complex plane so $$γ(t)--->e^{i(x^3+x)}$$ .So now i wanna prove that this exponentail function is not a closed curve.I can see the graph that it is not but i can find arguments on how to say it rigorously. The definition the book has for a closed Curve is That if a curve is T-periodic and not constant then it is closed.(using Andrew Pressleys Diff. Geometry)

Answer 1

Use a proof by contradiction. Suppose that the curve is $T$-periodic. Then there exists a $T \neq 0$ such that for all $t$, $$ \exp\left[i[(t+T)^3 + t + T\right] = \exp\left[i (t^3 + t)\right] \Rightarrow \exp\left[i(3tT^2 + 3 t^2 T + T^3 + T) \right] = 1.$$

Since this looks like a homework-type problem, I'll leave it here, but if this is just for your own edification and you can't figure out how to proceed, then let me know in the comments and I'll carry the proof further.