Dr Geoff Prince


Department of Mathematics and Statistics
Position: Reader, Associate Professor and Head of Department.
Office Location: Room 308C, Physical Sciences 2, La Trobe University, Victoria 3086, Australia.
Phone: (03) 9479 2601
Fax: (03) 9479 2466
Email: G.Prince@latrobe.edu.au


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Research Interests

My interests lie in the applications of differential geometry to ordinary and partial differential equations. At the moment I am working on the inverse problem in the calculus of variations (“when can a system of second order O.D.E's be rewritten as Euler-Lagrange equations?”), the classification of planar flows according to their perpendicular expansion and tangential rotation, the singularity theory of second order ordinary differential equations, the symmetry of exterior differential systems. Most of this work uses a differential geometric formulation of ordinary differential equations which brings together material from Riemannian and Finsler geometry, tangent and cotangent bundle geometry and the theory of contact manifolds.

Click HERE to read about recent work on the singularity theory of differential equations by Michael Jerie and Geoff Prince.

I am responsible for the Dimsym linear differential equation and symmetry determination software package originally written by James Sherring.

Service roles and Society memberships

  • Australian Mathematical Sciences Institute AMSI - I was AMSI Deputy Director (2004), Executive and Acting Director (2005). Co-ordinator of the ICE-EM Access Grid Room project in 2006.
  • Australian Centre of Excellence for Risk Analysis ACERA - As AMSI’s representative I am a member of the ACERA Board.
  • Australian Mathematical Society AustMS - I am a Fellow of the Australian Mathematical Society and a member of the Council from 2006 to 2009. I was Director of the 2007 Annual Meeting of the Australian Mathematical Society, held at La Trobe from September 25 to 28.
  • I am a member of the American Mathematical Society (AMS).

Graduate Students

Current
  • Paul Martin, MSc. 'The Geometry of Flows'.
Past 
  • Susan Godfrey, PhD. 1992. 'Reduction of Order Techniques for Classical Orbit Problems'.
  • James Sherring, PhD. 1993. 'Symmetry and Computer Algebra Techniques for Differential Equations'.
  • Georgia Stathopoulos, MSc. 1998. 'Geometry of Planar Flows'.
  • Michael Barco, PhD. 2000. 'Symmetry and exterior differential systems'.
  • Jon Aldridge, PhD, 2004. 'The Inverse Problem in the Calculus of Variations'
  • Michael Jerie, PhD, 2005. 'The Geometry of Second Order O.D.E.'s'.

Collaborators

Publications

Papers

  1. G. E. Prince, P.G.L. Leach, T.M.Kalotas, C.J. Eliezer and R.M. Santilli, The Lie and Lie-admissible Symmetries of Dynamical Systems, Hadronic J. 3, 390-439, (1979).
  2. G. E. Prince and C.J. Eliezer, Symmetries of the Time-Dependent N-dimensional Oscillator, J. Phys. A: Math.Gen. 13, 815-823, (1980).
  3. G. E. Prince and P.G.L. Leach, The Lie Theory of Extended Groups in Hamilton Mechanics, Hadronic J. 3, 941-961, (1980).
  4. G. E. Prince and C.J. Eliezer, On the symmetries of the Classical Kepler Problem, J. Phys. A: Math. Gen. 14, 587-596, (1981).
  5. G. E. Prince, Toward a Classification of Dynamical Symmetries in Classical Mechanics, Bull. Aust. Math. Soc. 27, 53-71, (1983).
  6. G. E. Prince, Reflections on the symmetry conservation law duality and the Runge-Lenz vector, J. Phys. A: Math.Gen. 16, L105-L108, (1983).
  7. G. E. Prince, Toward a Classification of Dynamical Symmetries in Lagrangian Systems, Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, June 7-11, 1982 Torino, Italy. Att. D.Accad. d. Sc. d. Torino, Suppl. 117, 687-691 (1983).
  8. G. E. Prince, Homothetic Killing Tensors, Physics Letters A 97A, 133-136, (1983).
  9. G. E. Prince and M. Crampin, Projective differential geometry and geodesic conservation laws in general relativity, I: projective actions, Gen. Rel. Grav. 16, 921-942, (1984).
  10. G. E. Prince and M. Crampin, Projective differential geometry and geodesic conservation laws in general relativity, II: conservation laws, Gen. Rel. Grav. 16, 1063-1075, (1984).
  11. M. Crampin and G. E. Prince, The geodesic spray, the vertical projection, and Raychaudhuri's equation, Gen. Rel. Grav. 16, 675-689, (1984).
  12. M. Crampin, G. E. Prince and G. Thompson, A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics, J. Phys. A: Math.Gen. 17, 1437-1447, (1984).
  13. M. Crampin and G. E. Prince, Equivalent Lagrangian and dynamical symmetries: some comments, Phys. Lett. 108A, 191-194, (1985).
  14. M. Crampin and G. E. Prince, Generalizing gauge freedom for spherically symmetric potentials, J. Phys. A: Math. Gen. 18, 2167-2175, (1985).
  15. G. E. Prince, A complete classification of dynamical symmetries in classical mechanics, Bull. Austral. Math. Soc. 32, 299-308 (1985).
  16. M. Crampin and G. E. Prince, Alternative Lagrangians for spherically symmetric potentials, J. Math. Phys. 29,1551-1555 (1988).
  17. W. Sarlet, G. E. Prince and M. Crampin, Adjoint symmetries for time-dependent second-order equations, J. Phys. A: Math. Gen. 23, 1335-1347, (1990).
  18. S. E. Godfrey and G. E. Prince, A canonical reduction of order for the Kepler problem, J. Phys. A: Math. Gen. 24, 5465-5475, (1991).
  19. J. Sherring and G. E. Prince, Geometric aspects of reduction of order, Trans. Am. Math. Soc. 334, 433-453, (1992).
  20. M. Crampin, W. Sarlet, E. Martinez, G. B. Byrnes and G. E. Prince, Toward a geometrical understanding of Douglas's solution of the inverse problem in the calculus of variations, Inverse Problems 10, 245-260, (1994).
  21. G. E. Prince, G. B. Byrnes, J. Sherring and S. E. Godfrey, A generalization of the Liouville-Arnol'd theorem, Math. Proc. Camb. Phil. Soc. 117, 353-370, (1995).
  22. J. Sherring, A.K. Head and G.E. Prince, DIMSYM and LIE: symmetry determination packages, Mathl. Comput. Modelling 25, 153-164 (1997).
  23. J.E.R. O'Connor and G.E. Prince, Finding Collineations of Kimura Metrics, Gen. Rel. Grav. 30, 69-82, (1998).
  24. J.E. Aldridge and G.E. Prince, Computer Algebra Solution of the Inverse Problem in the Calculus of Variations, Comp. Physics Comm. 115, 489-509, (1998).
  25. M. Jerie, J.E.R. O'Connor and G.E. Prince, Computer Algebra Determination of Symmetries in General Relativity, Comp. Physics Comm. 115 , 363-380, (1998).
  26. G.E. Prince, J.E. Aldridge and G.B. Byrnes, A Universal Hamilton-Jacobi Equation for Second Order ODE's, J. Phys. A: Math. Gen. 32, 827-844, (1999).
  27. M. Jerie, J.E.R. O'Connor and G.E. Prince, Spacetime symmetries for the Kerr metric Class. Quantum Grav. 16, 2885-2887, (1999).
  28. M. Crampin, G.E. Prince, W. Sarlet and G. Thompson, The inverse problem of the calculus of variations: separable systems, Acta Appl. Math. 57, 239-254, (1999).
  29. G. E. Prince, J. Aldridge, S. E. Godfrey and G. B. Byrnes, The Separation of the Hamilton-Jacobi Equation for the Kerr Metric. Proceedings of the First Australian General Relativity Workshop, September 26-30, 1994, Canberra, J. Aust. Math. Soc. B 41, 248-259 (1999).
  30. G.E. Prince, Reply to 'Some remarks on the symmetries of the Kerr spacetime', Class. Quantum Grav. 17, 2027-2028, (2000).
  31. M. Jerie and G.E. Prince, A general Raychaudhuri's equation for second order differential equations, Corrected May 23, 2000. J. Geom. Phys. 34, 227-241, (2000).
  32. G.E. Prince, Quadratic forms associated with planar endomorphisms, Bull. Austral. Math. Soc. 62, 459-465, (2000).
  33. M. Barco and G.E. Prince, Solvable symmetry structures in differential forms applications, Acta Appl. Math. 66, 89-121, (2001).
  34. M. Barco and G.E. Prince, New symmetry solution techniques for first order non-linear PDEs, Appl. Math. Comput. 124, 169-196, (2001).
  35. G. Thompson and G.E. Prince, The inverse problem for flat kinetic minus potential Lagrangians, Extr. Math. 16, 241-248, (2001).
  36. W. Sarlet, G. Thompson and G.E. Prince, The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas's analysis, Trans. Am. Math. Soc. 354, 2897-2919 (2002). (PDF File)
  37. M. Jerie and G.E. Prince, Jacobi fields and linear connections for arbitrary second order ODE's. J. Geom. Phys. 43, 351-370, (2002). (PDF File)
  38. G.E. Prince and M.Jerie, Generalising Raychaudhuri's equation Proceedings of the 8th International Conference on Differential Geometry and its Applications, Opava, 27-31 August 2001, Part II. (Mathematical Publications 3, Silesian University at Opava). (PDF File)
  39. D. Krupka, O. Krupková, G. Prince, and W. Sarlet, Contact symmetries and variational sequences. Proc. 9th Internat. Conf. on Diff. Geom. Appl., Prague, 2004, Charles University, Prague, 599-609 (2005). (PDF File)
  40. J. E. Aldridge, G. E. Prince, W. Sarlet and G. Thompson, An EDS approach to the inverse problem in the calculus of variations, J. Math. Phys. 47, 103508:1-22 (2006). (PDF File)
  41. D. Krupka, O. Krupková, G.E. Prince and W. Sarlet, Contact Symmetries of the Helmholtz Form, Diff. Geom. Appl. 25, 518-542 (2007). (PDF File)
  42. O. Krupková and G. Prince, Lepage forms, closed two-forms and second order ordinary differential equations, Izv. Vyssh. Uchebn. Zaved. Mat. (Russian Mathematics) (N.I.Lobachevskii anniversary volume) 51, 1-16 (2007).
  43. G.E. Prince and S.P. Dubois, Mathematical Models for Motion of the Rear Ends of Vehicles, in Mathematical and Computer Modelling (to appear). (PDF File)
  44. G.E. Prince, Comment: "Period function and normalizers of vector fields in Rn with n−1 first integrals" by D. Peralta-Salas, JDE 244, (2008) 1287-1303. J. Diff. Eqns (to appear).

Book Chapters

  1. G.E. Prince, The Inverse Problem in the Calculus of Variations and its Ramifications in Geometric Approaches to Differential Equations, edited by P. Vassiliou and I. Lisle, Lecture Notes of the Australian Mathematical Society, CUP (March, 2000). (Introductory section only is available on the current page.) (PDF File)
  2. G.E.Prince, A Note on the Integrability and Closure Conditions in the Inverse Problem in the Calculus of Variations, in Applied Differential Geometry and Mechanics: Volume in Honour of the 60th Birthday of Michael Crampin, edited by W.Sarlet and F.Cantrijn, Gent, Academia Press (2003). (PDF File)
  3. G.E. Prince and D.M. King, The inverse problem in the calculus of variations: nonexistence of Lagrangians pp 131-140, in Differential Geometric Methods on Mechanics and Field theory: Volume in Honour of Willy Sarlet, edited by F.Cantrijn and B. Langerock, Gent, Academia Press (2007). (PDF File)
  4. O. Krupková and G. Prince, Second Order Ordinary Differential Equations in Jet Bundles and the Inverse Problem of the Calculus of Variations pp 837-904, in Handbook of Global Analysis, edited by D. Krupka and D. Saunders, Elsevier (2007). (PDF File)
  5. G. E. Prince, On the inverse problem for autoparallels pp 395-406 in "Variations, Geometry and Physics" in honour of Demeter Krupkas sixty-fifth birthday, O. Krupková and D.J. Saunders (Editors) Nova Science Publishers (2008). (PDF File)

Reviews

  1. G.E. Prince, Reduce - a User's View. Math. Scientist, 12, 91-96 (1987).
  2. G.E. Prince, Computer Algebra and Differential equations. Gaz. Aust. Math. Soc., 18, 83-85 (1991).

Submitted

  1. G.E. Prince, G. Stathopoulos and P. Martin, The Geometry of Planar Flows, (PDF File)

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