EXACT ZERO-ENERGY SOLUTI ON FOR A NEW FAMI LY OF ANHARMONI C POTENTI ALS Yoël Lana-Renault Departamento de Física Teórica Universidad de Zaragoza. 50009 Zaragoza. Spain e-mail: yoel@kepler.unizar.es Abstract. An explicit, oscillatory solution of the type r = [r o + A o ·sin( ϖ ·t + φ o )] a is found for the zero-energy one-dimensional motion of a particle under a specific family of anharmonic potentials. Key words: Anharmonic Potentials. Oscillatory Motion. I. The Potential In classical mechanics there are very few non-trivial potentials that admit an explicit analytical solution. Our aim in this paper is to present a new family of one-dimensional anharmonic potentials represented by the function: ( ) 1 V( ) r . . . . . 1 2 ma 2 ϖ 2 r 2 1 . . 2r o r 1 a . r o 2 A o 2 r 2 a for which one is able to find an analytical solution. In (1), r is the position of a point-like particle of mass m; r o and A o are two independent parameters, fulfilling r o > A o . As usual, ϖ will represent the angular frequency of the oscillatory solutions; -a- is a free non-null parameter, and we will study the equation of the oscillation of zero-energy around the minimum of the potential. 1