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DNA's shape mostly lies on its total energy $F$. Its correspondingequilibrium shape equations can be got by classical variation method: lettingthe first energy variation $\delta^{(1)}F=0$. Here we not only provide thefirst variation $\delta^{(1)}F$ but also give the second variation$\delta^{(2)}F$ in planar case. Moreover, the general shape equations of DNAare obtained and a mistake in [Zhang, \emph{et al}. {\it Phys. Rev. E} {\bf 70}051902 (2004)] is pointed out.

The First and Second Variation of the Total Energy of Closed Duplex DNA in Planar Case