Research activity of Raffaele Vitolo


Last updated: 23/09/2017



List of Publications: see below.
Field of research: Geometric Methods in Mathematical Physics

Integrable Systems

Hamiltonian formalism for PDEs and Integrable Systems [22a,25a,26a,13c,29a,33a,34a,37a,38a,40a], differential equations which are uniquely characterized by their symmetry group [16a,17a,9c,10c,11c,15c,30a], generalized symmetries and applications [32a,35a,36a]. I contribute to the contents of the website http://gdeq.org/, which is a community of researchers who are interested in geometry and differential equations. I developed software for the research of integrability operators (Hamiltonian, symplectic and recursion operators) [1s,2s]. Together with A.C. Norman I wrote a technical manual on REDUCE internals and programming [3s].

Classical and quantum mechanics

Spherically symmetric solutions in Galilei relativity [2a], existence and classification of quantizations [4a,7a,3c], covariant symmetries in mechanics [2c,8a,15a,28a], classical and quantum mechanics of the rigid body [18a,19a,5c], covariant quantization and geometric quantization [6c,31a], geometry of polarizations [39a]. I recently published a paper of historical interest on an early contribution of Levi-Civita to the correspondence between symmetries and conserved quantities [27a].

Geometry of calculus of variations

I have studied variational sequences, on which I published a chapter of the Handbook of Global Analysis [2b]. These are exact sequences of spaces of differential forms on jets where one of the morphisms of the sequence is the Euler--Lagrange operator. In particular I devoted myself to variational sequences of finite order [3a,5a,6a,9a,10a,11a,12a,13a,14a,15a,20a,1c,4c,7c,12c,Tesi] and to variational multivectors, which are duals to forms in the variational sequence [8c,22a,13c,29a].

Classical field theory

Covariant Lagrangians for Einstein-Yang-Mills equations [1a,1b], inverse problem for Yang--Mills equations [21a].



Publications by Raffaele Vitolo

NOTE: most publications are either in arXiv.org or available here for download in PDF.

  1. Articles in journals
  2. Books and parts of books
  3. Conference proceedings
  4. Ph. D. thesis (in Italian)
  5. Editorial activity
  6. Software
  7. General publications

A. Articles in Journals

1a G. Giachetta, M. Mangiarotti, R. Vitolo: The Einstein-Yang-Mills equations, J. Gen. Rel. Grav. 23, n. 1 (1991), 641-659.
2a R. Vitolo: Spherical symmetry in classical and quantum Galilei general relativity, Ann. Inst. `H. Poincaré', 64, n. 2 (1996) 177--203, PDF
3a R. Vitolo: A new infinite order formulation of variational sequences, Arch. Math. Un. Brunensis, 34, n. 4 (1998), 483-504; EMS server. PDF
4a R. Vitolo: Quantum structures in Galilei general relativity, Ann. Inst. 'H. Poincaré' 70 n. 3 (1999) 239--258. PDF
5a R. Vitolo: Finite order variational bicomplexes, Math. Proc. of the Camb. Phil. Soc., 125, n. 2 (1999) 321-333; arXiv.org. PDF
6a R. Vitolo: On different geometric formulations of Lagrangian formalism, Diff. Geom. and its Appl., 10, n. 3 (1999), 225--255. Updated version here: PDF
7a R. Vitolo: Quantum structures in Einstein general relativity, Lett. Math. Phys. 51 (2000), 119--133, PDF
8a D. Saller, R. Vitolo: Symmetries in covariant classical mechanics, J. Math. Phys. 41, n. 10, October 2000, 6824-6842, arXiv.org PDF
9a M. Palese, R. Vitolo: On a class of polynomial Lagrangians, Proc. of the winter school `Geometry and Physics', Srni (Czech Rep) 2000, Rend. Mat. Palermo Serie II, Suppl. 66 (2001), 147-159, arXiv.org PDF
10a M. Francaviglia, M. Palese, R. Vitolo: Symmetries in finite order variational sequences, Czech. Math. J. 52 (127) (2002), 197-213. PDF
11a R. Vitolo: Finite order formulation of Vinogradov C-spectral sequence, Acta Appl. Math. 70 1-2, (2002) 133-154; arXiv.org Updated version here: PDF
12a I. Kolár, R. Vitolo: On the Helmholtz operator for Euler morphism, Math. Proc. of the Camb. Phil. Soc. 135 (2003), 277-290. PDF
13a G. Saccomandi, R. Vitolo: Null Lagrangians for nematic elastomers, Fundamental'naya i Prikladnaya Matematika 10 (2004) n.1, 17-28, issue on ``Geometry of integrable models'', A. V. Kiselev (ed.) (Russian); English translation in Journal of Mathematical Sciences 136 n. 6 (2006), 4470-4477. Updated version here: PDF
14a M. Francaviglia, M. Palese, R. Vitolo: A geometric formulation of Hessian and Jacobi tensors for higher order Lagrangians, Differential Geometry and its Applications 22, n. 1 (2005), 105-120. PDF
15a G. Manno, R. Vitolo: Relativistic mechanics, contact manifolds and symmetries, Note di Matematica 23, n. 2, 2004/2005, 157-171. PDF
16a G. Manno, F. Oliveri, R. Vitolo: On differential equations determined by the group of point symmetries, Jour. Math. Anal. Appl. 332 (2007), 767-786. PDF
17a G. Manno, F. Oliveri, R. Vitolo: Differential equations uniquely determined by algebras of point symmetries, Theoret. Math. Phys. 151 n. 3 (2007), 843-850 PDF
18a M. Modugno, R. Vitolo: The geometry of Newton's law and rigid systems, Archivum Math. Univ. Brunensis 43, n. 2 (2007) 197-229; arXiv.org PDF
19a M. Modugno, C. Tejero Prieto, R. Vitolo: A covariant approach to the quantisation of a rigid body, J. Phys. A: Math. theor. 41 (2008) 035304, see arXiv.org. PDF
20a G. Manno, R. Vitolo: Geometric aspects of higher order variational principles on submanifolds, Acta Appl. Math. 101 (2008), 215-229. PDF.
21a G. Manno, J. Pohjanpelto, R. Vitolo: Symmetries, conservation laws and variational principles for gauge theories, J. Geom. Phys. 58 (2008), 996-1006. PDF.
22a P.H.M. Kersten, I.S. Krasil'shchik, A.M. Verbovetsky, R. Vitolo: Integrability of Kupershmidt's deformation, Acta Appl. Math. 109 no. 1 (2010), 75-86. arXiv.org PDF.
23a J. Janyska, M. Modugno, R. Vitolo: An algebraic approach to physical scales , Acta Appl. Math. 110, no. 3 (2010), 1249-1276. arXiv.org. PDF.
24a I. Kolar, R. Vitolo: Absolute contact differentiation on submanifolds of Cartan spaces , Differential Geometry and its Applications, Volume 28, Issue 1 (2010), 19--32. PDF.
25a P. Kersten, I.S. Krasil'shchik, A.M. Verbovetsky, R. Vitolo: On integrable structures for a generalized Monge-Ampere equation , Theor. Math. Phys. 128, no. 2 (2012), 600-615. arXiv.org.
26a I.S. Krasil'shchik, A.M. Verbovetsky, R. Vitolo: A unified approach to computation of integrable structures , Acta Appl. Math. 120, no. 1 (2012), 199-218. arXiv.org.
27a G. Saccomandi, R. Vitolo: A Translation of the T. Levi-Civita paper: Interpretazione Gruppale degli Integrali di un Sistema Canonico, Rend. Acc. Lincei, s. 3a, vol. VII, 2o sem. 1899, pp. 235--238 , Regular and Chaotic Dynamics Volume 17, Issue 1, pp. 105-112 (2012). arXiv.org. Download here the original paper by Levi-Civita.
28a J. Janyska, R. Vitolo: On the characterization of infinitesimal symmetries of the relativistic phase space J. Phys. A: Math. Theor. 45 (2012) 485205 (28pp). PDF
29a E.V. Ferapontov, M.V. Pavlov, R.F. Vitolo: Projective-geometric aspects of homogeneous third-order Hamiltonian operators, J. Geom. Phys. 85 (2014), 16--18 PDF
30a G. Manno, F. Oliveri, G. Saccomandi, R. Vitolo: Ordinary differential equations described by their Lie symmetry algebra, J. Geom. Phys. (2014), 85 (2014), 2--15 PDF
31a C. Tejero Prieto, R. Vitolo: On the geometry of the energy operator in quantum mechanics, International Journal of Geometric Methods in Modern Physics, Vol. 11, No. 07 (2014): 1460027. PDF
32a G. Saccomandi, R. Vitolo: On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics, J. Math. Phys. 55 (2014), 081502, PDF
33a M.V. Pavlov, R.F. Vitolo: On the Bi-Hamiltonian Geometry of WDVV Equations, Lett. Math. Phys. 105 no. 8 (2015), 1135--1163. arXiv.org, PDF
34a E.V. Ferapontov, M.V. Pavlov, R.F. Vitolo: Towards the classification of homogeneous third-order Hamiltonian operators , Int. Math. Res. Notices 22 (2016), 6829-6855, arXiv.org, PDF
35a E. Pucci, G. Saccomandi, R. Vitolo: Bogus Transformations in Mechanics of Continua, Int. J. of Eng. Sci. 99 (2016), 13--21, PDF
36a A. Sergyeyev, R. Vitolo: Symmetries and conservation laws for the Karczewska--Rozmej--Rutkowski--Infeld equation, Nonlinear Analysis: Real World Applications 32 (2016) 1-9. arxiv.org PDF
37a M.V. Pavlov, R.F. Vitolo: Remarks on the Lagrangian representation of bi-Hamiltonian equations, J. Geom. Phys. 113C (2017), 239-249. DOI: http://dx.doi.org/10.1016/j.geomphys.2016.10.013. arxiv.org PDF
38a P. Lorenzoni, A. Savoldi, R. Vitolo: Bi-Hamiltonian structures of KdV-type, arxiv.org PDF
39a C. Tejero Prieto, R. Vitolo: The geometry of real reducible polarizations in quantum mechanics, J. Phys. A: Theor. Math. 50 no. 10 (2017), 105205; DOI: https://doi.org/10.1088/1751-8121/aa58c7 arxiv.org PDF
40a E.V. Ferapontov, M.V. Pavlov, R.F. Vitolo: Systems of conservation laws with third-order Hamiltonian structures, arxiv.org PDF

B. Books and parts of books

1b G. Giachetta, M. Mangiarotti, R. Vitolo: Some results on the Einstein-Yang-Mills equations, in `Modern Problems in Theoretical Physics', ed. O. Obukhov, P. Pronin, World Scient., Singapore 1991.
2b R. Vitolo: Variational sequences, in `Handbook of Global Analysis', ed. D. Krupka, D. Saunders, Elsevier 2007. Some lecture notes on the subject can be found here in PDF.
2b J. Krasil'shchik, A. Verbovetsky, R. Vitolo: The symbolic computation of integrability operators for partial differential equations, Springer, to appear in the series Texts and Monographs in Symbolic Computation (2017). See also the manual of the Reduce package CDE (ref. [2s] below.

C. Conference proceedings

1c R. Vitolo: Some aspects of first-order variational sequences in mechanics, Proc. of the VI Conf. in Diff. Geom. and its Appl., Brno 1995, 487--494; EMS server PDF
2c M. Modugno, R. Vitolo: Quantum Connection and Poincar\'e-Cartan form, Proc. of the conference in honour of A. Lichnerowicz, Frascati, october 1995; ed. G. Ferrarese, Pitagora, Bologna 1996, 237-279; PDF
3c R. Vitolo: Quantum structures in general relativistic theories, Proc. of the XII National. Conf. of Gen. Rel. and Grav. Phys., Roma (1996), World Scientific, 375-359; PDF
4c M. Francaviglia, M. Palese, R. Vitolo: Superpotentials in variational sequences, Proc. of the VII Conf. on Diff. Geom and Appl., Brno 1998, 469-480; EMS server. PDF
5c R. Vitolo: Quantising the rigid body, Proc. of the VII Conf. on Diff. Geom and Appl., Brno 1998, 653-664; EMS server. PDF
6c M. Modugno, C. Tejero Prieto, R. Vitolo: Comparison between Geoometric Quantisation and Covariant Quantum Mechanics, Proceedings of Lie Theory and Its Applications in Physics III (World Scientific, 2000), edited by H.-D. Doebner, V.K. Dobrev and J. Hilgert, 155-175; arXiv.org PDF
7c G. Manno, R. Vitolo: Variational sequences on finite order jets of submanifolds, Proc. of the VIII int. conf. on Diff. Geom. and its Appl., Opava (Czech Rep.), 2001, 435-446; PDF
8c S. Igonin, A. Verbovetsky, R. Vitolo: Variational Multivectors and Brackets in the Geometry of Jet Spaces, V Int. Conf. on on Symmetry in Nonlinear Mathematical Physics, Kyiv 2003 Inst. of Mathematics, Nat. Acad. of Sciences of Ukraine, 1335-1342; PDF
9c F. Oliveri, G. Manno, R. Vitolo: On an Inverse Problem in Group Analysis of PDE's: Lie--Remarkable Equations, Proceedings XIII International Conference on Waves and Stability in Continuous Media, World Scientific Co., Singapore, 2006, 420-431; PDF
10c F. Oliveri, G. Manno, R. Vitolo: Lie--Remarkable PDEs, in "Asymptotic methods in nonlinear wave phenomena", in honour of the 65th birthday of Antonio Greco, World Scientific Co., Singapore, 2007, 119-131; PDF
11c F. Oliveri, G. Manno, R. Vitolo: On the correspondence between differential equations and symmetry algebras, Symmetry and Perturbation Theory, proceedings of the conference in Otranto, 2/6--9/6/2007, World Scientific 2007, 164-171. PDF
12c R. Vitolo: Finite order variational sequences: a short review, Proceedings of the conference in honour of D. Krupka, O. Krupková, D. Saunders eds., Nova Science Publisher 2008 (Olomouc, Czech Rep.), 117--136. PDF
13c P.H.M. Kersten, I.S. Krasil shchik, A.M. Verbovetsky, R. Vitolo: Hamiltonian structures for general PDEs, in B. Kruglikov, V. Lychagin, E. Straume: Differential Equations -- Geometry, Symmetries and Integrability, Proceedings of the 2008 Abel Symposium, Springer, 187-198. PDF
14c M. Modugno, C. Tejero Prieto, R. Vitolo: Geometric aspects of the quantization of a rigid body, in B. Kruglikov, V. Lychagin, E. Straume: Differential Equations -- Geometry, Symmetries and Integrability, Proceedings of the 2008 Abel Symposium, Springer, 275-285. PDF
15c F. Oliveri, G. Manno, R. Vitolo: Differential equations and Lie symmetries, Proceedings XIV International Conference on Waves and Stability in Continuous Media, N. Manganaro, R. Monaco, S. Rionero eds., World Scientific Co., Singapore, 2008, 459-468; PDF

D. Ph. D. thesis

Ph. D. th. R. Vitolo: Bicomplessi lagrangiani ed applicazioni alla meccanica classica e quantistica, Ph.D. thesis (in Italian), Firenze 1996; PDF

E. Editorial activity

1e L. Martina, G. Saccomandi, R. Vitolo guest editors of Note di Matematica vol. 23 n. 2, devoted to symmetries of PDE, online at http://siba2.unile.it/notemat.
2e G. Gaeta, R. Vitolo, S. Walcher: Symmetry and Perturbation Theory, proceedings of the conference, Otranto (Italy), 2/6--9/6/2007; World Scientific (2008) ISBN:9812776168. http://www.worldscibooks.com/chaos/6650.html
3e S. Rionero, T. Ruggeri, G. Saccomandi, R. Vitolo: Special issue of Note di Matematica in occasion of the XVI conference Waves and Stability in Continuous Media, Brindisi (Italy), 12/6--18/6/2011; Vol. 32n.1(2012).
4e S. Rionero, T. Ruggeri, G. Saccomandi, R. Vitolo: Special issue of Acta Applicandae Mathematicae in occasion of the XVI conference Waves and Stability in Continuous Media, Brindisi (Italy), 12/6--18/6/2011; to appear (2012).
5e B. Konopelchenko, G. Landolfi, L. Martina, R. Vitolo: Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) 22-29 June 2013, Gallipoli, Italy. Journal of Physics: Conference Series 482, IOP Publishing 2014 (47 papers).

F. soFtware

1s R. Vitolo: CDIFF: a REDUCE package for computations in geometry of differential equations, a user guide to computations in geometry of differential equations. Software developed by P. Gragert, P. Kersten, G. Post, G. Roelofs, University of Twente, The Netherlands, freely available at the Geometry of Differential Equations website
2s R. Vitolo: CDE: a REDUCE package for integrability of PDEs, a REDUCE package for integrability of PDEs}, software, user guide and examples, freely available at http://gdeq.org/Category:Software, 2014.
3s A.C. Norman, R. Vitolo: Inside REDUCE, a guided tour to REDUCE internals for programmers. Included in the documentation of the REDUCE distribution at Sourceforge: http://sourceforge.net/projects/reduce-algebra/, 2014.

G. General publications

1f R. Vitolo: La forma dell'Universo (in Italian), Idee (Journal of Philosophy of the University of Salento, Milella publ.) 61 (2006), 53-59. PDF.
2f F. Paparella, R. Vitolo: Analisi numerica con GNU Octave (in Italian), Linux & C. (Italian magazine on Linux and Free Software) 52 (2006), 44-49. PDF.
3f G. Saccomandi, R. Vitolo: La mobilità accademica nella proposta di riforma (in Italian), Lavoce.info 25 November 2010.

Last updated: 23/09/2017

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