报告摘要：

       The periodic eigenvalue problem is resulted from the variational treatment to the critical points of the $p$-th kinetic energy under the constant constraint to $p$-th potential energy of periodic motions in the Euclidean spaces $\R^d$. This is an Euler-Lagrangian equation on $\R^d$. In dimension $d=2$, it has been known that the problem admits two different sequences of eigenvalues. A problem proposed by Man\'asevich and Mawhin 20 years ago is that if these are all periodic eigenvalues.

       In this talk, I will show that for any exponent $p\ne 2$, the $p$-Laplacian on the plane will actually admit infinitely many different sequences of periodic eigenvalues. They are constructed using a notion of scaling angular momenta we will introduce. The whole proof is based on: (1) the complete integrability of the equivalent Hamiltonian system, (2) a tricky reduction to $2$-dimensional dynamical systems, and (3) a number-theoretical distinguishing between different sequences of eigenvalues.

       Some numerical simulations and further problems will be given.