  We discuss on the uncertainty relation (UR) for a closed one dimensional
system (circle). In such a system, we cannot use the angle along the circle as
a position variable. Otherwise we meet difficulties about the definition of the
average position and the standard deviation (SD), and Hermitian property of
angular momentum. From these reasons, we define the position variable as
Cartesian variable $(X,Y)$ that have the periodic property for angle $\phi$. In
the same way we define a SD by using that variables. Then we obtain two URs. We
also discuss the minimum wave packet (MWP) on the circle. MWPs are expressed by
von Mises distribution functions. Next we construct total URs by combining two
URs for $X$ and $Y$. Furthermore, we extend the variables to $(X_n, Y_n)$ \;
with $n= 1,2,3,\cdots$ and we have infinite series of total URs. We consider
the meaning of such extended URs.
