The perihelion precession rate and the time derivative of the orbitaleccentricity are joined into the derivative of a complex variable beingrepresentative of the Runge-Lenz vector. The integration of the lineardifferential equation system so obtained yields the evolution of theperihelion and the eccentricity of all the planets. Each eccentricityreaches a maximum, and in the case of giant planets also a minimum.The variation of each semimajor axis is shown to be very small.Since the semimajor axes are bounded and the planetary orbits neverintersect, the stability of the solar system is proven.

This paper examines the eigenvalues and eigenfunctions of the Laplace operatorassociated with a set of mathematically defined surfaces which can be producedexperimentally by attaching two equally sized rings to opposite poles of a soapbubble and separating the rings. The shapes produced are called unduloids. Thesecalculations show 1) for a range of ring sizes, as a function of ring separation,the first indexed eigenvalue has a minimum, pointing to an ``optimum'' shape,and 2) given the eigenfunctions in the form of their differential equations anda preference for symmetry, the underlying unduloid geometry may be deduced.

The nonlinear Dirac equation is a magical equation that hides manymysteries of nature, so it is worth systematic study. In this paper,we give a detailed analysis for the effects of various covariantpotential terms on the energy eigenstates and the representationspace as well as the solvability of the spinor equation, and weobtain the specific potential energy terms that lead to the standingwave solutions. By simplifying the spinor equation, we find that thenonlinear pseudo scalar potential causes the oscillation anddivergence of the radial wave function, so the nonlinear weakinteraction between spinors may be the dynamical reason of Pauliexclusion principle. The nonlinear potentials of a spinor has somewonderful properties, such as the confinement of negative energystate, the generation of negative pressure and celestial magneticfield, etc. We also present a calculating method for solving theDirac equation with complicated potentials, which may be a usefultool for further uncovering the mysteries of elementary particles.