Periodic solutions with prescribed energy for some Keplerian \(N\)-body problems (Q1904222)

The existence of periodic solutions with prescribed energy \(h\) of the system \(\ddot x+ v'(x)= 0\), \({1\over 2} |\dot x|^2+ v(x)= h\) with \(x= (x_1, x_2,\dots, x_N)\), \(x_j\in \mathbb{R}^k\), \(v(x)= {1\over 2} \sum_{i\neq j} v_{ij}(x_i- x_j)\), \(v_{ij}(\xi)= - |\xi|^{-a}\), \(a> 0\), is proved for a class of Keplerian-like \(N\)-body problem, employing critical point theory with boundary conditions.