The purpose of this article is two fold. First, we show how quasi-periodic solutions for theN-body problem can be constructed by variational methods. We illustrate this by constructing uncountably many quasi-periodic solutions for the fourand six-body problems with equal masses. Second, we show by examples that a system of N masses can possess infinitely many simple or multiple choreographic so...

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...