Critical number in scattering and escaping problems in classical mechanics Salvador Addas-Zanata* and Clodoaldo Grotta-Ragazzo † Instituto de Matema ´tica e Estatı ´stica, Universidade de Sa ¯ o Paulo, R. do Mata ˜ o 1010, CEP 05508-900, Sa ˜ o Paulo-SP, Brazil Received 17 January 2001; revised manuscript received 16 July 2001; published 24 September 2001 Scattering and escaping problems for Hamiltonian systems with two degrees of freedom of the type kinetic plus potential energy arise in many applications. Under some discrete symmetry assumptions, it is shown that important quantities in these problems are determined by a relation between two canonical invariant numbers that can be explicitly computed. DOI: 10.1103/PhysRevE.64.046216 PACS numbers: 05.45.-a I. INTRODUCTION Invariant tori are very important in the global dynamics of two degree of freedom Hamiltonian systems. They split the phase space in unconnected components. For several physi- cally relevant systems, the existence of such invariant tori, and therefore the understanding of the phase-space structure, can only be achieved through numerical investigation. In this paper, an analytic criterion for the existence of certain fami- lies of invariant tori is presented. These families are impor- tant in some problems of escaping from and scattering off a potential well. The class of systems such that our results apply can be described as follows. The Hamiltonian function is of the type kinetic plus potential energy, H = 1 2  p x 2 + p y 2  +V  x , y  , 1 where V has two critical points: a minimum P m and a saddle- point P s . The energy of the saddle point, which will be called critical energy and will be denoted E cr , is V ( P s ) =E cr . For energy values below E cr the corresponding energy-level sets have two distinct components, one bounded and one unbounded. The bounded component projects to the configuration space ( x , y ) inside what will be called the po- tential well see Fig. 1 for a topological representation of the level curves of V . For H =E cr , these two components touch at the equilibrium corresponding to P s and for H E cr , the two components merge into a single unbounded component. Notice that E cr plays an important role in the dynamics. For energy values below E cr , there is always a large quantity in a measure sense of bounded orbits trapped inside the poten- tial well, while for energy values above E cr , this may not occur. An important example that belongs to the above class is the Hamiltonian system for the motion of a charged particle in the field of a magnetic dipole see, for instance 1, also called the Stormer system. Due to the rotation symmetry of the magnetic field, this system can be reduced to two degrees of freedom. For a given value of angular momentum and in convenient time and length scales, its Hamiltonian function is H = 1 2  p x 2 + p y 2  +V  x , y  , V  x , y  = 1 2  1 x - x  x 2 + y 2  3/2 2 , 2 where x is the radial coordinate and y is the coordinate along the dipole axis. It is well known that V satisfies the above properties, so the level curves of the Stormer potential V are topologically as in Fig. 1: V has two critical points, a mini- mum P m and a saddle-point P s , such that V ( P s ) =E cr =1/32 and for energy values below E cr , the corresponding energy-level sets have two distinct components, one bounded and one unbounded. For H =E cr , these two components touch at the equilibrium corresponding to P s and for H E cr , the two components merge into a single unbounded component. In this paper, two different problems will be considered for systems similar to the Stormer system, contained in the class defined below expression 1. The first is the so-called ‘‘escaping problem’’ or ‘‘escaping from a potential well,’’ that is: for a given distribution of initial conditions inside the potential well with fixed energy E E cr to determine the amount of solutions that remain inside the potential well af- ter time t 0. This type of escaping problem had been re- *Current address: Department of Mathematics, Princeton Univer- sity, Fine Hall-Washington Road, Princeton, NJ 08544-1000. Email address: szanata@math.princeton.edu † Email address: ragazzo@ime.usp.br FIG. 1. Topological representation of the level sets of a typical potential function for the class we are considering. The shaded re- gion corresponds to an energy level set below the energy of the saddle point P s . PHYSICAL REVIEW E, VOLUME 64, 046216 1063-651X/2001/644/04621611/$20.00 ©2001 The American Physical Society 64 046216-1