Mathematical Sciences: Some Questions in Nonlinear          Differential Equations
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Models for the motion of a finite number of particlies where               no frictional forces are present are usually given by means of                  differential equations called Hamiltonian systems.  During the                  past decade there has been intensive research focusing on the                   existence of periodic solutions of such systems.  The work                      sponsored by this award is in part an outgrowth of a more general               development of variational methods, expecially minimax and Morse                theoretic methods, and their application to nonlinear                           differential equations.  It illustrates the effectiveness which                 can be achieved when the power of abstract mathematical research                is driven by external stimuli.                                                       One focus of this work concerns the Hamiltonian itself, a                  function of two (vector) variables which contains the position                  and velocity information about the particles.  Traditionally one                assumes that the Hamiltonians are continuously differentiable.                  Many interesting problems, however, involve singular Hamitonians.               The classical n-body problem of celestial mechanics is an                       example.  Using minimax arguments, one can show that there are                  infinitely many periodic solutions with distinct periods.  The                  present work will seek to clarify whether or not multiple                       periodic solutions can exist with the same period.  In addition,                with the advent of singular Hamiltonians, the possibility arises                for collision orbits.  Work will be done in determining                         conditions where one can distinguish between regular and                        collision orbits.  Efforts will also be made to find periodic                   solutions with prescribed energy.                                                    Branching away from periodic solutions, research will begin                on investigations into the next simplest class, those orbits                    which tend to equilibrium solutions as times increases                          indefinitely.  These are known as connecting orbits, for they                   join period ones.  Some results are available for so-called                     superquadratic Hamiltonian systems, but the area is still in its                infancy.
{'FirstName': 'Paul', 'LastName': 'Rabinowitz', 'PI_MID_INIT': 'H', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Paul H Rabinowitz', 'EmailAddress': 'rabinowi@math.wisc.edu', 'NSF_ID': '000197135', 'StartDate': '05/15/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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