In this paper, we first survey variational approaches to potential systems to show the existence of periodic solutions. As simple examples, we consider a periodically forced pendulum and the Keplertype problem. Next we focus on the n-body problem and show the existence of symmetric periodicsolutions. To show the existence of perodic solutions by variational method, the most difficult part is to eliminate the possibility that an obtained minimizer has collisions. We introduce known methods for it. As recent progresses, we show the existence of orbits realizing given symbolic sequences in the n-body and the n-center problem. We also discuss the stability of minimizing periodic solutions.

The basic reproduction number R₀ in structured population dynamics is defined as the spectral radius of the generation evolution operator induced by the integral kernel of the renewal integral equation. If the basic population dynamics is described by the evolutionary system associated with a nonautonomous differential equation, the generation evolution operator is calculated from the infinitesimal generator of the evolution semigroup induced from the evolutionary system. Using the basic reproduction number defined by the generation evolution operator, we can examine existence and stability of total orbits of nonlinear nonautonomous system, in which the total orbit is given as a fixed point of the evolution semigroup. Thus the idea of R₀ is uniquely extended to the threshold value for extinction and persistence of population in time-heterogeneous environments.