Analysis

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                                                                                                                                  Book

 D. H. Delphenich (ed.), Selected papers on geodesic fields, Neo-classical Press, Spring Valley, OH, 2013.

(Download consists of two files: I and II, Table of Contents below)

                                                                                                                              Articles
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D. H. Delphenich, “On the variational formulation of systems with non-holonomic constraints,” Ann. Phys. (Berlin) 18 (2009), 649-670.

D. H. Delphenich, “Integrability and the variational formulation of non-conservative mechanical systems,” Ann. Phys. (Berlin) 18 (2009), 45-56

D. H. Delphenich, “A generalization of Noether's theorem based on the virtual work functional,” arXiv:1109.0461

D. H. Delphenich, "On the role of integrability in a large class of physical systems," arXiv:1210.4976.


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                                                                                                                        TRANSLATIONS
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       Calculus of variations

C. G. J. Jacobi, "On the theory of the calculus of variations and differential equations," J. f. d. reine u. angew. Math. 17 (1837), 68-82.

E. Beltrami, "On the theory of geodetic lines," Rendiconti del Reale Istituto Lombardo (2) 1 (1868), 708-718; Opera matematiche, v. I, 366-373.

A. Mayer, "On the criteria for maxima and minima of simple integrals," J. reine angew. Math. 69 (1868), 238-263.

A. Mayer, "History of the principle of least action," Inaugural lecture, von Veit, Leipzig, 1877.

A. Mayer, "The criteria for the maxima and minima of simple integrals in isoperimetric problems," Math. Ann. 13 (1878), 55-68.

G. Frobenius, "On adjoint linear differential expressions," J. reine angew. Math. 85 (1878), 185-213.

B. Imchenestsky, "On the transformation of a differential equation of even order to the form of an isoperimetric equation," Bull. Acad. imp. Sci. St.-Petersbourg (3) 31 (1887), 283-291.

K. Zorawski, "On integral invariants of continuous transformation groups," Bull. int. de l'Acad. des Sci. de Cracovie (1889-1900), 127-130.

V. Volterra, "On the differential equations that are deduced from questions in the calculus of variations," Rend. Acc. dei Lincei (4) 6 (1890), 43-54; Opere matematiche, v. I, 454-463.

V. Volterra, "On an extension of the Jacobi-Hamilton theory in the calculus of variations," Rend. Acc. Lincei (5) 6 (1890), 127-138; Opere matematiche, v. I, 464-476.

P. Staeckel, "On the motion of a point in an n-fold manifold," Math. Ann. 42 (1893), 537-563.

P. Staeckel, "On a class of problems in dynamics," C. R. Acad. Sci. Paris 116 (1893), 485-487.

P. Staeckel, "On the integration of Hamilton’s differential equation," C. R. Acad. Sci. Paris 121 (1895), 489-492.

A. Mayer, "Lagrange's method of multipliers and the most general problem in the calculus of variations for one independent variable," Ber. Verh. Kon. Sachs. Ges. Wiss. Leipzig 47 (1895), 129-144.

A. Mayer, "The existence condition for a kinetic potential," Ber. Kön. Ges. Wiss. Leipzig 48 (1896), 519-529.

G. Darboux, "On the auxiliary equation," from Leçons sur la théorie générale des surfaces, Gauthier-Villars, Paris, 1896, Partie IV, Note XI, pp. 505-516.

S. Lie, "The theory of integral invariants is a corollary to the theory of differential invariants," Leip. Ber. (1897); Ges. Abh., V. IV, art. XXVII.

S. Lie, "On integral invariants and their use in the theory of differential equations," Leip. Ber. 4 (1897), 369-410; Ges. Abh., v. VI, art. XXVIII, pp. 664-701.

A. Hirsch, "On the characteristic property of the differential equations of the calculus of variations," Math. Ann. 2 (1897), 49-72.

S. Lie, "The theory of integral invariants is a corollary to the theory of differential invariants," Leipziger. Berichte (1897), Heft III, submitted on 14-7-1897, pp. 342-357.  Presented at the session on 3-5-1897; Gesammelte Abhandlungen, v. IV, art. XXVII, pp. 649-663.

G. von Escherich, "The second variation of a simple integral," Sitz. Kaiserliche Akad. Wiss. Wien, Math.-Naturwiss. Klasse 107 (1898), 1191-1250.

A. Hirsch, "The existence conditions for the generalized kinetic potential," Math. Ann. 50 (1898), 429-441.

K. Boehm, "The existence conditions for a kinetic potential that depends upon the first and second differential quotients of the coordinates," J. reine angew. Math. 121 (1900), 124-140.

A. Voss, "On the principles of Hamilton and Maupertuis," Gottinger Nachrichten (1900), 322-327.

A. Kneser, The Calculus of variations, Vieweg and Sons, Braunschweig, 1900: Part I, Part II.

Th. De Donder, Study of the Integral Invariants, Gauthier-Villars, Paris, 1901.

L. Koenigsberger, Principles of Mechanics. Mathematical Investigations, B. G. Teubner, Leipzig, 1901.

Th. De Donder, "Study of integral invariants (Part Two)," Rend. Circ. mat. di Palermo (1) 16 (1902), 155-179.

G. Morera, "On the Lagrange dynamical equations," Atti Reale Accad. sci. Torino 38 (1902-03), 121-134.

G. Morera, "On Hamilton's dynamical equations," Atti della R. Acc. delle sci. 39 (1903-04), 342-355.

G. Morera, "On the transformation of Hamilton’s differential equations. Notes I and II," Rend. Real. Accad. Lincei (5) 12 (1903), 113-122; pp. 149-152.

G. Hamel, "On the geometries in which the lines are shortest," Math. Ann. 57 (1903), 231-264.

O. Bolza, "On the second variation for isoperimetric problems," Math. 57 (1903), 44-47.

T. Yoshiye, "Applications of the calculus of variations to partial differential equations with two independent variables," Math. Ann. 57 (1903), 185-194.

L. W. Thome, "On an application of the theory of linear differential equations to the calculus of variations," J. reine angew. Math. 125 (1903), 1-27.

H. Hahn, "Remarks concerning the calculus of variations," Math. Ann. 58 (1904), 148-168.

L. W. Thome, "On an application of the theory of linear differential equations to the calculus of variations (supplement)," J. reine angew. Math. 128 (1905), 33-44.

H. Hans, "Remarks concerning the calculus of variations," Math. Ann. 58 (1904), 148-168.

T. Levi-Civita, "On the integration of the Hamilton-Jacobi equation by separation of variables," Math. Ann. 59 (1904), 383-397.

M. Frechet, "On an extension of the Jacobi-Hamilton method," Annali di mat. pura ed appl. (3) 11 (1904), 187-199.

J. Kurschak, "The existence conditions for the generalized kinetic potential," Math. Ann. 62 (1906), 148-155.

A. Mayer, "On Hilbert's independence theorem in the theory of the maximum and minimum of a simple integral," Math. Ann. 58 (1904), 235-248; "..., Part II," ibid. 62 (1906), 335-350.

D. Hilbert, "On the calculus of variations," Math. Ann. 62 (1906), 351-370.

D. Egorov, "The sufficient conditions for the extremum in the theory of the Mayer problem," Math. Ann. 62 (1906), 371-380.

E. Goursat, "On integral invariants," J. math. pures et appl. (6) 4 (1908), 331-365.


H. Hahn, "On the connection between the theories of the second variation and Weierstrass's theory of the calculus of variations," Rend. Circ. Matem. Palermo (1909), 49-78.

H. Vergne, "On certain properties of systems of differential equations," Ann. sci. de l’E.N.S. (3) 27 (1910), 543-563.

R. G. D. Richardson, "The Jacobi criterion in the calculus of variations and the oscillatory properties of second-order linear differential equations," Math. Ann. 68 (1910), 279-304.

Th. De Donder, "Generalization of Poisson's theorem," C. R. Acad. Sci. Paris 148 (1910), 610-612.

Th. De Donder, " On Poisson's theorem and Lie's differential invariants," C. R. Acad. Sci. Paris 151 (1910), 371-373.

Th. De Donder, "On relative integral invariants and their applications to mathematical physics," Bull. Acad. roy. Belgique, classe de sci. (5) 1 (1911), 50-70.

O. Bolza, "On Hilbert's independence theorem for the Lagrange variational problem," Rend. Circ. matem. Palermo (1) 31 (1911), 257-272.

J. Radon, "Some questions concerning the theory of maxima and minima of multiple integrals," Monats. Math. Phys. 22 (1911), 53-63.

Th. De Donder, "On the canonical Hamilton-Volterra equations," Mem. Acad. roy. Belg., Cl. de Sc. (2) 3, fasc. IV, 1911.

F. A. Dall'Acqua, "The Hamilton-Jacobi equations that can be integrated by separation of variables," Rend. Circ. mat. Palermo (1) 33 (1912), 341-351.

Th. De Donder, "On Hilbert's independence theorem," C. R. Acad. Sci. Paris 156 (1913), 609-611, 868-870,

E. Goursat, "On some points regarding the theory of integral invariants," J. math. pures et appl. (7) 1 (1915), 241-259.

G. Prange, The Hamilton-Jacobi Theory for Double Integrals, Dietrich, Gottingen, 1915.

A. Kneser, "Contributions to the theory of the calculus of variations: The connection between the Weierstrass and Jacobi-Hamilton methods and a theory of integration of Cauchy," Archiv der Math. und Phys. 24 (1915), 26-57.

E. Noether, "On invariant variational problems," Gottinger Nachrichten (1918), 235-257.

E. Bessel-Hagen, "On the conservation laws of electrodynamics," Math. Ann. 84 (1921), 258-276.

E. Cartan, Lessons on integral invariants, Hermann and Co., Paris, 1922.

C. Caratheodory, "On the envelopes of the extremals of a field in a multidimensional space," 4 (1923), 23-31.

C. Caratheodory, "The method of geodetic equidistants and the Lagrange problem," Acta Math. 47 (1925), 199-236.

A. Haar, "On adjoint variational problems and adjoint extremal surfaces," Math. Ann. 100 (1928), 481-502.

C. Caratheodory, On the calculus of variations for multiple integrals," Acta Szeged Sect. Sci. Math. 4 (1929), 193-216.

P. V. Paquet, "The exterior differential forms Wn in the calculus of variations," Bull. Acad. roy. Belg., Cl. sc. (5) 27 (1941), 65-84.

P. Dedecker, "On Bateman’s method in the inverse problem in the calculus of variations," Bull. Acad. roy. Belgique (Cl. des Sc.) (5) 35 (1949), 774-792.

P. Dedecker, "On an inverse problem in the calculus of variations," Bull. Acad. roy. Belgique, classe de sciences 38 (1950), 63-70.

P. Dedecker, On the circulation theorem of V. Bjerknes and the theory of integral invariants, Institut royale Météorologique de Belgique, Brussells, 1951.

H. Boerner, " Caratheodory's approach to the calculus of variations," Jber. Deutsch. Math. Ver. 56 (1953), 31-58.

W. Velte, "On the calculus of variations for multiple integrals," Math. Zeit. 60 (1954), 367-383.

P. Dedecker, "The Helmholtz-Cartan theorem for a simple integral of higher order," Study group on solitons, partial differential equations, and spectral methods, (16 – 27 July 1979), International Centre for Theoretical Physics, Trieste, Italy.

        Differential equations

G. Monge, "Integrating ordinary differential equations by integrating partial differential equations," Hist. Acad. Roy. Sci. (1784), 502-576.

G. Monge, "On the integration of first-order partial difference equations in three variables," Supplement to Application de l'Analyse a la Geometrie,5th ed., Bachelier, Paris, 1850.

A. Mayer, "On the Jacobi-Hamilton method for integrating first-order partial differential equations," Math. Ann. 3 (1871), 436-452.

G. Morera, "The Pfaff method for integrating first-order partial differential equations," Rend. Reale Istituto Lombardo di scienze (2) 16 (1883), 637-644, 691-699.

O. Biermann, "On n simultaneous differential equations of the form (...) = 0," Zeit. Math. Phys. 30 (1885), 234-244.

E. Picard, "On linear differential equations and algebraic groups of transformations," Ann. fac. sci. Toulouse (1) 1 (1887), pp. A1-A15.

E. Picard, "On the groups of transformations of linear differential equations," Math. Ann. 46 (1895), 161-166.

E. Picard, "On the extension of Galois’s ideas to the theory of differential equations (Extract from a latter addressed to Mr. Klein)," Math. Ann. 47 (1896), 161-162.


P. Painleve, Integrating the Differential Equations of Mechanics, Hermann, Paris, 1895. Chaps. 1-12, Chaps. 13-17.

J. Beudon, "On the characteristics of partial differential equations," Bull. Soc. Math. France 25 (1897), 108-120.

E. Goursat, "On a partial differential equation," Bull. Soc. Math. France 25 (1897), 36-48.

G. Morera, "Canonical systems of total differential equations in the theory of groups of transformations," Atti Reale Accad. sci. di Torino 38 (1902-03), 940-953.

G. Morera, "On systems of first-order partial differential equations in involution," Rend. Reale Istituto Lombardo di scienze (2) 36 (1903), 775-790.

E. Vessiot, "Ordinary differential equations. Elementary integration methods," Encykl. math. Wiss., B. G. Teubner, Leipzig, 1899-1916, Art. II A 4b., pp. 232-293.

E. von Weber, "Partial differential equations," Encycl. math. Wiss., II A 5, ed. H. Burkhardt, W. Wirtinger, and R. Fricke, B. G. Teubner, Leipzig, 1899-1916.

M. Bocher, "Boundary-value problems for ordinary differential equations," from Encyklopadie der mathematischen Wissenschaften, II A 7.a, eds. H. Burkhardt, W. Wirtinger, and R. Fricke, B. G. Teubner, Leipzig, 1899-1916, pp. 437-463.

M. Bocher, Lectures on Sturm's Method in the Theory of Linear Differential Equations and their Modern Developments, Gauthier-Villars, Paris, 1917.

T. Levi-Civita, Characteristics of differential systems and wave propagation, Alcan, Paris, 1932.

        Contact transformations

A. Mayer, "On Lie's contact transformations," Gott. Nachr. 13 (1874), 317-331.

E. Mathieu, "On the canonical differential equations of mechanics," J. pures appl. (2) 19 (1874), 265-306.

S. Lie, "Foundations of an invariant theory of contact transformations," Math. Ann. 8 (1875), 215-303.

S. Lie, "Perturbation theory and contact transformations," Arch. for Math. Christiania 2 (1877), 129-156; Gesammelte Abhandlungen, art. XX, pp. 296-319.

S. Lie, “The infinitesimal contact transformations of mechanics,” Berichte über die Verhandlunen Königlich Sächsisschen Gesellschaft der Wissenschaft zu Leipzig, Mathematische-Physische Classe, v. 41 (1889), 145-146.

S. Lie, "Contributions to the general theory of transformations," Leip. Ber. (1895); Ges. Abh., v. VI, art XXIII.

S. Lie, "The infinitesimal contact transformations of optics," Leip. Ber. (1896); Ges. Abh., v. VI, art. XXIV.

E. Vessiot, “On the mechanical interpretation of infinitesimal contact transformations,” Bull. Soc. Math. de France 34 (1906), 230-269.

E. Vessiot, “On the theory of multiplicities and the calculus of variations,” Bull. Soc. Math. France 40 (1912), 68-139.

F. Engel, “Lie’s theory invariant of contact transformations and its extensions,” from the Jahresbericht der deutschen Mathematiker-Vereinigung, v. 5, H. Liebmann and
F. Engel, Die Berührungstransformationen, Geschichte und Invariantentheorie, B. G. Teubner, Leipzig, 1914.

H. Liebmann, “The development of the theory of contact transformations,” from the Jahresbericht der deutschen Mathematiker-Vereinigung, v. 5, H. Liebmann and F. Engel, Die Berührungstransformationen, Geschichte und Invariantentheorie, B. G. Teubner, Leipzig, 1914.

J. A. Schouten, "On the differential geometry of the group of contact transformations: I. Doubly-homogeneous treatment of contact transformations." Proc. Kon. Akad. Wet. Amsterdam 40 (1937), 100-107.

.J. A. Schouten, "On the differential geometry of the group of contact transformations: II. Normal form and main theorems for doubly-homogeneous contact transformations." Proc. Kon. Akad. Wet. Amsterdam 40 (1937), 236-245.

J. A. Schouten, "On the differential geometry of the group of contact transformations: III.  Infinitesimal doubly-homogeneous contact transformations and their relationship to mechanics and electrodynamics." Proc. Kon. Akad. Wet. Amsterdam 40 (1937), 470-480.

J. A. Schouten and J. Haantjes, "On the differential geometry of the group of contact transformations: IV. Covariant derivatives in K2n-1." Proc. Kon. Akad. Wet. Amsterdam 41 (1938), 577-584.

E. Hölder, “The infinitesimal contact transformations of the variational calculus,” Jber. Deutsche Math.-Verein 49 (1939), 162-178.

        The Pfaff and Monge problems

C. G. J. Jacobi, “On the Pfaffian method of integrating an ordinary linear differential equation in 2n variables by means of a system of n equations,” J. f. Reine und Angew. Math. 2 (1827), 347-357.

J.-A. Serret, "On the integration of the equation dx2 + dy2 + dz2 = ds2," J. math. pures appl. 13 (1848), 353-368.

O. Bonnet, "On a theorem of Jacobi that relates to the integration of first-order partial difference equations," C. R. Acad. Sci. Paris 45 (1857), 581-585.
A. Mayer, "On unrestricted integrable systems of linear total differential equations and the simultaneous integration of linear partial differential equations," Math. Ann. 5 (1872), 448-470.

A. Mayer, “On an extension of Lie’s integration method,” Math. Ann. 8 (1875), 313-318.

A. Mayer, “On Weiler’s method of integrating partial differential equations of first order,” Math. Ann. 9 (1875), 347-370.

A. Mayer, “On Pfaff’s solution of the Pfaff problem.” Math. Ann. 17 (1880), 523-530.

G. Darboux, "On the Pfaff problem," Bull. sci. math. astron. (2) 6 (1882), 14-36,
                      "On the Pfaff problem (cont.)," Bull. sci. math. astron. (2) 6 (1882), 49-62.

A. Voss, "Geometrical interpretation of the differential equation P dx + Q dy + R dz = 0," Math. Ann. 16 (1880), 556-559.

G. Morera, "The Pfaff method for integrating first-order partial differential equations," Rend. Reale Istituto Lombardo di scienze (2) 16 (1883), 637-644, 691-699.

A. Voss, "On the theory of general point-plane systems," Math. Ann. 23 (1884), 45-81.

G. Darboux, "On the solution of the equation dx2 + dy2 + dz2 = ds2 and some analogous equations," J. Math. pure appl. (4) 3 (1887), 305-325.

F. Engel, "On the invariant theory of systems of Pfaff equations," Ber. Verh. Sachs. Ges. Wiss. 41 (1889), 157-176.

F. Engel, "On the invariant theory of systems of Pfaff equations (second communication)," Ber. Verh. Sachs. Ges. Wiss. 42 (1890), 192-207.

E. von Weber, "On the invariant theory of systems of PFAFF equations," Ber. Verh. Kon. Ges. Wiss. 50 (1898), 207-229.

S. Lie, "On the geometry of a Monge equation," Ber. Verh. Kon.Ges. Wiss. Leipzig 50 (1898), 1-3.

E. Cartan, "On certain differential expressions and the Pfaff problem," Ann. sci. de l'E.N.S. (3) 16 (1899), 239-332.

E. Cartan, "On the integration of systems of total differential equations," Ann. sci. de l'E.N.S. (3) 18 (1901), 241-311.

E. Goursat, "On the Monge problem," Bull. Soc. Math. 33 (1905), 2101-210.

M. Botasso, "On a solution to the Monge problem that relates to the equation f (dx1, dx2, …, dxn) with variable coefficients," C. R. Acad. Sci. Paris 140 (1905), 1579-1582.

D. Hilbert, "On the concept of the class of a differential equation," Math. Ann. 73 (1913), 95-108.

E. Cartan, "On the integration of certain undetermined systems of differential equations," J. reine angew. Math. 145 (1915), 86-91.

E. Vessiot, "On a new theory of general integration problems," Bull. Soc. math. France 52 (1924), 336-395.

E. Vessiot, "On the integration of sheaves of infinitesimal transformations in the case where the degree of the derived sheaf is n + 1, if the degree of the sheaf is n,"  Ann. sci. de l'E.N.S. (3) 45 (1928), 189-253.

E. Goursat, "On a Monge problem with several independent variables," C. R. Acad. Sci. Paris 186 (1928), 1469-1472.

P. Zervos, "The Monge Problem," Mem. Math. Sci., fasc. LIII, Gauthier-VIllars, Paris, 1932.

E. Kaehler, Introduction to the theory of systems of differential equations, B. G. Teubner, Leipzig, 1934.

T. Levi-Civita, "On the stationary solutions of Pfaffian systems," Rend. Acc. Lincei (6) 19 (1934), Nota I, pp. 261-267; Nota II, pp. 470-477.

          Backlund transformations

A. V. Backlund, "On surface transformations," Math. Ann. 9 (1875), 297-320.

A. V. Backlund, "On the theory of surface transformations," Math. Ann. 19 (1882), 387-422.

J. Clairin, "On the Backlund transformations," Ann. sci. de l'E.N.S. (3) 19 (1902), 3-63.

E. Goursat, "The Backlund Problem,: Mem. Sci. Math., fasc. VI, Gauthier-Villars, Paris, 1925.

            Book:Selected papers on geodesic fields, translated and edited by D. H. Delphenich, 2011

                       (File consists of two parts: I and II)

            Contents

A. Clebsch, “On the second variation of multiple integrals,” J. de Crelle, 56 (1859), 122-149.

J. Hadamard, “On a question in the calculus of variations,” Bull. Soc. Math. de France 30 (1902), 253-256.

J. Hadamard, “On some questions in the calculus of variations,” Bull. Soc. Math. de France 33 (1905), 73-80.

D. Hilbert, “On the calculus of variations,” Math. Ann. 69 (1906), 351-370; Ges. Abh. v. III, pp. 38-55.

C. Carathéodory, “On the calculus of variations for multiple integrals,” Acta Szeged 4 (1929), 193-216.

H. Weyl, “Observations on Hilbert’s independence problem and Born’s quantization of field equations,” Phys. Rev. 46 (1934), 505-508.

H. Weyl, “Geodesic fields in the calculus of variations for multiple integrals,” Ann. Math. 36 (1935), 607-629.

H. Boerner, “On extremals and geodesic fields in the calculus of variations for multiple integrals,” Math. Ann. 112 (1936), 187-220.

J. Géhéniau, “Generalization of the Weierstrass excess formula, as deduced from the Hilbert-de Donder independence theorem,” Bull. Acad. Roy. Belg. 21 (1935), pp. 385 and 504.

Th. Lepage, “On the geodesic fields of the calculus of variations,” Bull. Acad. Roy. Belg. 22 (1936), 716-729.

R. Debever, “Mayer fields in the calculus of variations of multiple integrals,” Bull. Acad. Roy. Belg. 23 (1937), 809-815.

E. Hölder,”The infinitesimal contact transformation of the variational calculus,” Jber. d. deutsche Math.-Verein. 49 (1939), 162-178.

H. Boerner, “Calculus of variations from Stokes’s theorem,” Math. Zeit. 46 (1940), 709-719.

H. Boerner, “On the Legendre condition and field theory in the calculus of variations for multiple integrals,” Math. Zeit. 46 (1940), 720-742.

Th. Lepage, “On the geodesic fields of multiple integrals,” Bull. Acad. Roy. Belg. 27 (1941), 27-46.

Th. Lepage, “Stationary fields, geodesic fields, and integrable forms,” Bull. Acad. Roy. Belg. 28 (1942), 73-94.

L. van Hove, “On the de Donder-Weyl fields and their construction by the method of characteristics,” Bull. Roy. Acad. Belg. 31 (1945), 278-285.

L. van Hove, “On the Carathéodory fields and their construction by the method of characteristics,” Bull. Roy. Acad. Belg. 31 (1945), 625-638.

P. Dedecker, “Calculus of variations, differential forms, and geodesic fields,” Coll. Intl. du C.N.R.S., Strasbourg, 1953.