Papers of Enzo Tonti

His publications may be divided into three groups:

1 - Mathematical structure of physical theories

The author, who has been fascinated since his student days by the analogies between different physical theories, has always asked himself: why is this so ?

For many years he has analysed the geometrical, algebraic and analytical structure which is common to different physical theories in an attempt to answer this question. In some of his early works he restricted himself to summarising in operator language common mathematical properties: [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [13].

In 1974 he pointed out that these common mathematical properties stem from the fact that physical integral variables are naturally associated with geometrical elements in space (points, lines, surfaces and volumes) and with time elements (instants and intervals) [14], [15], [16], [17].

The result has been an explanation of analogies in physics: since in every physical theory there are integral variables associated with space and time elements it follows that there is a correspondence between the quantities and the equations of two physical theories in which the homologous quantities are those associated with the same space-time elements. The homologous quantities might be of a different mathematical nature, they might be scalars, vectors, tensors and so on: what they have in common is not their physical meaning or mathematical nature but their association with space and time elements.

This principle has permitted a scheme of classification of quantities and equations of each physical theory to be built, known in literature as the Tonti diagram.

2 - Variational formulation.

The problem which the author has addressed is the following:
  1. Given a system of differential equations, be they linear or non-linear, what are the conditions for variational formulation?
  2. If these conditions do not exist, is it possible to reduce the system to another equivalent system which permits variational formulation?
Vito Volterra had already given a clear answer to the first question in 1887 but this was more or less unknown both in mathematics and physics and the result was rediscovered by Vainberg in 1964 and became known as Vainberg's theorem.

The author showed how an a linear or non-linear operator can act as a vector field v =N(u) in a function space. Thus we can ask: what condition must be met for the vector field to be conservative? The condition is that the circulation of the vector N(u) along any reducible closed line in function space must be null. This can be expressed stating that the Gateaux derivative of the operator satisfies the condition:

which expresses the symmetry of the derivative of Gateaux of the operator N. This is the condition found by Volterra. If the operator is linear, it is denoted L, the condition expresses the symmetry of the operator

Once the condition has been met the functional can be found via the formula

which for linear operators is reduced to the formula


To give variational formulations to evolution equations with linear operators and initial conditions the author has shown that it is possible to use bilinear forms different from the traditional form. This made it possible to give Fourier's heat equation a variational formulation [11],[12].
The functional is stationary but not necessarily minimum in correspondence with the solution.

The problem of giving variational formulation to equations which do not permit it, known as the inverse problem of calculus of variations, is a subject the author has worked on at length. When the operator does not meet the condition of Volterra (1) the author has shown that the problem N(u)-f=0 can be transformed into an equivalent problem, that is with the same domain and the same solution, with the introduction of a linear operator K, symmetric and invertible K and writing the equivalent problem in the form

The functional is


An extended formulation can thus be given to any non linear operator. If the operator K is also defined as positive the functional is minimum in correspondence with the solution [18], [19], [20], [21].

3 - Finite formulation of physical laws.

The association of integral physical variables with space and time elements permits to give a direct finite formulation to field equations of classical physical theories. It has been shown that using a cell complex and its dual, rather than a coordinate system, it is possible to associate with a complex the configuration variables of the theory and with the dual complex the source variables. This completely discrete description draws on domain functions rather than point functions and uses notions from algebraic topology such as incidence numbers, chains, cochains, coboundary process. Cochains are the algebraic equivalent of external differential forms whilst the coboundary process is the algebraic equivalent of the external differential.

The first theory described in this way was electromagnetism [22], [23], followed by solid mechanics and fluids. The direct algebraic description thus obtained has given rise to interest for its numerical applications in that it is no longer necessary to carry out discretization of differential equations.

The equations obtained to some extent coincide with those of the Finite Elements Method (FEM), with those of the Finite Volume Method (FVM) and of Finite Integration Theory (FIT) in electromagnetism.

  1. Tonti E., Principi variazionali nell'elastostatica, Rend. Acc. Lincei, vol.XLII, 1967, pp. 390 - 394.
  2. Tonti E., Variational Principles in Elastostatics, Meccanica, Vol. II, 1967, pp.1- 8.
  3. Tonti E., Variational Formulation for Linear Equations of Mathematical Physics, Rend. Acc. Lincei, vol. XLIV, 1968, pp. 75 - 82.
  4. Tonti E., Variational Principles in Electromagnetism, Rend. Ist. Lomb., Vol. 102, 1968, pp. 845 - 861.
  5. Tonti E., Gauge Transformations and Conservation Laws, Rend. Acc. Lincei, Vol. XLV, 1968, pp. 293 - 300.
  6. Tonti E., Variational Formulations of Nonlinear Differential Equations, Bull. Acad. Roy. Belg., Vol. LV, 1969, pp. 137 - 165, 262 - 278.
  7. Tonti E., On the Formal Structure of Continuum Mechanics, part. I: Deformation Theory, Meccanica, Vol. V, 1970, pp. 22 - 30.
  8. Tonti E., Sulla struttura delle teorie lineari di campo, Rend. Ist. Lomb. vol. 104, 1970, pp. 161 - 187.
  9. Tonti E., On the mathematical structure of a large class of physical theories , Accademia Nazionale dei Lincei, estratto dai Rendiconti della Classe di Scienze fisiche, matematiche e naturali, Serie VIII, Vol. LII, fasc. 1, Gennaio 1972 (.pdf, 324 KB)
  10. Tonti E., A mathematical model for physical theories , Accademia Nazionale dei Lincei, estratto dai Rendiconti della Classe di Scienze fisiche, matematiche e naturali, Serie VIII, Vol. LII, fasc. 2-3, Febbraio-Marzo 1972 (.pdf, 324 KB)
  11. Tonti E., On the variational formulation for linear initial value problems , Annali di Matematica pura ed applicata (IV), Vol. XCV,1973, pp. 331-360 (.pdf, 1 MB)
  12. Tonti E., A Systematic Approach to the Search for Variational Principles, Proceedings of the International Conference on Variational Principles in Engineering, Southampton Univ. Press, 1973, pp. 1.1 - 1.12.
  13. Tonti E., On the Formal Structure of the Relativistic Gravitational Theory, Rend. Acc. Lincei, Vol. LVI, 1974, pp. 228 - 236.
  14. Tonti E., The Algebraic - Topological Structure of Physical Theories, Conference on Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary (Canada), 1974, pp.441-467.
  15. Tonti E., On the formal structure of physical theories , monograph of the Italian National Research Council, 1975 (.pdf, 5,4 MB)
  16. Tonti E., The Reason for Analogies between Physical Theories , Appl. Math. Modelling, Vol. I, 1976, pp. 37 - 50.
  17. Tonti E., Sulla Struttura Formale delle Teorie Fisiche, Rendiconti Seminario Matematico e Fisico di Milano, Vol. XLVI, 1976, pp. 163 - 257.
  18. Tonti E., A General Solution of the Inverse Problem of the Calculus of Variations, Hadronic Journal, Vol. 5, N.4, 1982, pp. 1404 - 1450.
  19. Tonti E., Variational Formulation for Every Nonlinear Problem , Int. J. Engng Sci, Vol 22, No 11/12, 1984 , pp. 1343-1371(.pdf, 1,3 MB)
  20. Tonti E., Inverse Problem: Its General Solution , in Topology - Differential geometry, Calculus of variations and their applications, T. M. Rassias e G. M. Rassias editor, Marcel Dekker, 1985, pp. 497-510.
  21. Tonti E., Extended Variational Formulation , in VESTNIK, Mathematical Series, Russian Peoples' Friendship University, 2 (2) 95, pp 148-162.
  22. Tonti E., On the Geometrical Structure of the Electromagnetism , Gravitation, Electromagnetism and Geometrical Structures, for the 80th birthday of A. Lichnerowicz, Edited by G. Ferrarese, 1995, Pitagora Editrice Bologna, 281-308. (pdf, 560 KB)
  23. Tonti E., Algebraic Topology and Computational Electromagnetism, Fourth International Workshop on the Electric and Magnetic Fields: from Numerical Models to Industrial Applications, Marseille, 1998, pp. 284-294.
  24. Tonti E., Finite Formulation of the Electromagnetic Field , Progress in Electromagnetics Research, PIER 32 (Special Volume on Geometrical Methods for Comp. Electromagnetics), 1-44, (pdf, 1.3 MB)
  25. Tonti E., . A Direct Discrete Formulation for the Wave Equation , Journal of Computational Acoustics, Vol.9, No.4 (2001) 1355-1382. (.pdf, 1.5 MB)
  26. Tonti E., A Direct Discrete Formulation of Field Laws: The Cell Method , CMES - Computer Modeling in Engineering & Sciences, vol.2, no.2,2001, pp. 237-258. (.pdf, 604 kb)
  27. Tonti E., Finite Formulation of the Electromagnetic Field , International Compumag Society Newsletter, (2001). Vol 8, No.1, 5-11. (pdf, 370 KB)
  28. Tonti E., Finite Formulation of the Electromagnetic Field , IEEE Transactions on Magnetics, v.38, n. 2, March 2002, 333-336.
  29. Tonti E., Zarantonello F., Algebraic Formulation of Elastostatics: the Cell Method , Computational Modeling in Engineering & Science, v.39, n. 3, 2009, 201-236.
  30. Tonti E., Zarantonello F., Algebraic Formulation of Elastodynamics: the Cell Method , Computational Modeling in Engineering & Science, v.64, n. 1, 2010, 37-70.