WAVE THEORY
_______________________________________________________________________________________________________________________________________________________
PERSONAL RESEARCH
________________________________________________________________________________________________________________________________________________________
Book
D. H. Delphenich, Continuum-mechanical Models for Quantum Mechanics, Neo-classical Press, Spring Valley, 2017.
In three files: Intro-Chap V, Chap VI - Epilogue, Appendices
Published articles
D. H. Delphenich, “On the pre-metric foundations of wave mechanics I: massless waves,” Ann. Phys. (Berlin) 18 (2009), 206-230
Conference presentations
D. H. Delphenich, “Possibilities for a Causal Interpretation for Quantum Mechanics,” presented at the Vigier IV conference, Paris, 2003, arXiv:quant-ph/0401105
arXiv.org uploads
D. H. Delphenich, "On the equations of diffracted geodesics," arXiv:2008.03113
D. H. Delphenich, "On geodesics of gradient-index optical metrics and the optical-mechanical analogy," arXiv:2002.04390 .
D. H. Delphenich, “Generalized Madelung transformations for quantum wave equations I: generalized spherical coordinates for field spaces,” arXiv:0802.0305
D. H. Delphenich, “The Geometric Origin of the Madelung Potential,” arXiv:gr-qc/0211065
D. H. Delphenich, “Foliated cobordism and motion,” arXiv:gr-qc/0209091
D. H. Delphenich, "The relativistic Pauli equation," http://arxiv.org/abs/1207.5752
D. H. Delphenich, "A strain tensor that couples to the Madelung stress tensor," arXiv.org/1303.3582.
_______________________________________________________________________________________________________________________________________________________
TRANSLATIONS
_______________________________________________________________________________________________________________________________________________________
Contact transformations and wave motion
S. Lie, “The infinitesimal contact transformations of mechanics,” Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaft zu Leipzig, Mathematische-Physische Classe, v. 41 (1889), 145-146.
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, “Essay on the propagation of waves,” Ann. Sci. de l’E. N. S., 3rd series, 29 (1909), 405-448.
E. Vessiot, “On the theory of multiplicities and the calculus of variations,” Bull. Soc. Math. France 40 (1912), 68-139.
E. Vessiot, “On the propagation of waves and the Mayer problem,” J. de Math. (6) 9 (1913), 39-76.
E. Vessiot, "On the integral invariants of wave propagation," Bull. Soc. math. France 42 (1914), 142-167.
G. Juvet, Analytical Mechanics and Wave Mechanics, Mem. Sci. math., fasc. 83, Gauthier-VIllars, Paris, 1937.
Characteristics and bicharacteristics
J. Beudon, "On the characteristics of partial differential equations," Bull. Soc. Math. France 25 (1897), 108-120.
T. Levi-Civita, Characteristics of differential systems and wave propagation, Alcan, Paris, 1932.
Nonlinear waves
H. Hugoniot, “Memoir on the propagation of motion in an indefinite fluid (part one),” J. de Math. (4th series), t. III (1887), 477-492.
J. Hadamard, Lessons on the propagation of waves and the equations of hydrodynamics, Hermann and Co., Paris, 1903, Part I, Part II.
H. Weyl, "The asymptotic distribution law for the eigen-oscillations of an elastic body of arbitrary form," Rend. Circ. Mat. Palermo 34 (1914), 1-49.
G. Petiau, "On a nonlinear generalization of wave mechanics and the properties of the corresponding wave functions," Supp. Nuov. Cim.
9 (1958), 542-568.
J. van Mieghem, Contribution to the theory of Huygens's enveloping-wave principle, Palais des Academies, Brussels, 1936.
G. Boillat, “The propagation of waves,” Traité de physique théorique et de physique mathématique , t. XXIII, Gauthier-Villars, Paris, 1965.
G. Petiau, "On the quantum theory of fields that are associated with some simple models of nonlinear field equations," Nuov. Cim. 40 (1965), 84-101.
E. Hölder, “Historical overview of the mathematical theory of discontinuity waves since Riemann and Christoffel,” from E. B. Christoffell: the Influence of his Work on Mathematics and the Physical Sciences, ed. P. L. Butzer and F. Feher, Birkhäuser, Basel, 1981.
Geometrical optics
E. Malus, “Optics," J. Ec. poly. 7 (1808), 1-44.
Ch. Dupin, "On the paths that are followed by light and elastic bodies in general under the phenomena of reflection and refraction," from Applications de geometrie et de mecanique..., Courcier, Paris, 1822.
M. Gergonne, "Optics. Purely geometric proof of a fundamental principle in the theory of caustics," Ann. de Math. pures et appl. 16 (1825-1826), 307-314.
J. Plucker, "Discussion of the general form for light waves," J. f. reine u. angew. Math. 19 (1839), 1-44.
J. Bertrand, "Memoir on the theory of surfaces," J. Math. pures et appl. 9 (1844), 133-154.
J. Bertrand, "Note on the wave surface," Comptes rendus 47 (1858), 817-819.
F. Brioschi, "On the line of curvature of the wave surface," Ann. mat. pura ed appl. 2 (1859), 135-136.
E. Combescure, "On the lines of curvature of the wave surface," Ann. di. mat. pura ed. appl. 2 (1859), 278-285.
E. Kummer, "General theory of rectilinear ray systems," J. f. reine u. angew. Math. 57 (1860), 189-230.
E. Kummer, (no title: rectilinear systems of light rays), Monatsber. d. konig.-preuss. Akad. d. Wiss. zu Berlin, phys-math. Klasse, 30 Juli 1860, pp. 469-474.
E. E. Kummer, "Three models of general, infinitely-thin, rectilinear ray sheaves that are composed of filaments, Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1860), 469-474.
E. E. Kummer, "On atmospheric ray refraction," J. f. d. reine u. angew. Math. 61 (1863), 263-275.
E. E. Kummer, "On the fourth-degree surfaces with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 246-270.
E. E. Kummer, "On the ray systems whose focal surfaces are surfaces of degree four with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 495-499.
R. Meibauer, "Theory of rectilinear systems of light rays," C. Luderitz, Berlin, 1864.
E. E. Kummer, "On algebraic ray systems; in particular, on those of orders one and two," Abh. d. Kon. Akad. Wiss. zu Berlin (1866), 1-120.
A. Levistal, "Research in geometrical optics," Ann. de l'E.N.S. (1) 4 (1867), 195-254.
M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes," J. f. reine u. angew. Math. 76 (1873), 156-169.
M. Blasendorff, "On the relations between two general ray systems...," Inaugural Dissertation, Berlin, 1883.
M. Blasendorff, "On optical ray systems," J. f. d. reine u. angew. Math. 97 (1884), 172-176.
S. Lie and F. Klein, "On the principal tangent curves of the fourth-degree Kummer surface with 16 nodes," Math. Ann. 23 (1884); F. Klein, Ges. math. Werke, art. VI.
G. Darboux, "On the wave surface," Ann.sci. de l'E.N.S (3) 6 (1889), 379-388.
G. Darboux, Chapter XIII: "Normal lines to a surface," from Lecons sur la theorie des surfaces, etc., Book IV, Gauthier-Villars, Paris, 1889.
F. Klein, “On the recent English papers on mechanics,” Jber. d. Deutscher Mathematiker-Vereinigung 1 (1891/92).
H. Bruns, The Eikonal, S. Hirzel, Leipzig, 1895.
G. Darboux, Note VIII: "On the asymptotic lines and lines of curvature of the Fresnel wave surface," from Lecons sur la theorie des surfaces, Part IV, Gauthier-Villars, Paris, 1896, pp. 466-488.
F. Haussdorff, "Infinitesimal maps in optics," Ber. Verh. Kön. Sächs. Ges. Wiss. zu Leipzig, math.-phys. Klasse 48 (1896), 79-130.
T. Levi-Civita, "Complements to the Malus-Dupin theorem. I," Rend. R. Accad. dei Lincei 9 (1900), 185-189.
F. Klein, “On the Bruns Eikonal,” Zeit. f. Math. u. Phys. 46 (1901); Ges. math. Abh., v. 2, art. LXXI, pp. 601.
F. Klein, "Spatial collineations in optical instruments," Zeit. Math. Phys. 46 (1901), 376-382.
L. Bianchi: Chapter X: "Systems of (infinity-squared) rays or rectilinear congruences," from Lezioni di geometria differenziale, 2nd ed., v. I, Enrico Spoerri, Pisa, 1902.
R. Straubel, "On a general theorem of geometrical optics and some applications," Phys. Zeit. 4 (1902-03), 114-117.
E. Study, "On Hamilton's geometrical optics and its relationship to the theory of contact transformations," Jber. d. Deutschen Math.-Verein. 14 (1905), 424-438.
K. Zindler, "The development and present state of differential line geometry," Jahres. der deutschen Math.-Verein. 15 (1906), 185-213.
W. Voigt, "Remarks on the theory of conical refraction," Ann. Phys. (Leipzig) (4) 19 (1906), 14-21.
A. Garbasso and G. Fubini, "On the most general problem in optics," Atti della R. accad. sci. Torino 44 (1908/1909), 159-170.
A. Sommerfeld and J. Runge, "The application of vector calculus to the foundations of geometrical optics," Ann. Phys. (Leipzig) 35 (1911), 277-298.
H. Schlusser, The focal theory of linear ray congruences, Hoffmann, Leipzig, 1912.
B. Caldonazzo, "Trajectories of light rays and material points in a gravitational field," Nuovo Cim. (5) 5 (1913), 267-300.
R. Dontot, "On integral invariants and some points of geometrical optics," Bull. Soc. math. France 42 (1914), 53-91.
T. De Donder, "On the integral invariants of optics," Bull. Soc. math. France 42 (1914), 91-95.
T. Levi-Civita, "Einstein's theory and Fermat's principle," Nuovo Cim. 16 (1918), 105-114.
M. v. Laue, "Theoretical remarks on some recent optical observations about the theory of relativity," Phys. Zeit. 21 (1920), 659-662.
G. Prange, "W. R. Hamilton's significance to geometrical optics," Jber. d. Deutschen Math.-Verein. 30 (1921), 69-82.
J. Le Roux, "Special relativity and the geometry of wave systems," J. math. pures appl. 1 (1922), 205-253.
J. Hadamard, "Huygens's principle," Bull. Soc. math. France 52 (1924), 610-640.
C. Caratheodory, "On the connection between the theory of absolute optical instruments and a theorem in the calculus of variations," Sitz. der math.-phys. der Abt. Bay. Akademie de Wiss. zu Munchen 1 (1926), 1-18.
C. Caratheodory, Geometrical Optics, Springer, Berlin, 1937.
P. M. Quan, "Electromagnetic inductions in general relativity and Fermat's principle," J. Rat. Mech. Anal. 1 (1957), 54-80.
J. Ehlers, "On the transition from wave optics to geometrical optics in the general theory of relativity," Z. Naturforschg. 22a (1967), 1328-1333.
V. C. Bouslaev, "Generating integral and Maslov's canonical operator for the W. K. B. method," Complement 2 in V. P. Maslov, Theorie des perturbations et methodes asymptotiques, Dunod, Gauthier-Villars, Paris, 1972.
H. Knorrer, "The Fresnel wave surface," Lecture four of Mathematische Miniaturen 3: Arithmetik und Geometrie. Vier Vorlesungen, Birkhauser, Basel, 1986.
Quantum waves
P. Ehrenfest, "Does the angle of aberration measure the wave velocity in the case of a dispersive ether?" Ann. Phys. (Leipzig) 338 (1910), 1571-1576.
E. Schrodinger, "Hertz's mechanics and Einstein's theory of gravitation," Unpublished notes from around 1918.
L. de Broglie, Waves and Motions, Gauthier-Villars, Paris, 1926.
G. Wentzel, "A generalization of the quantization conditions for the purpose of wave mechanics," Zeit. Phys. 38 (1926), 518-529.
H. A. Kramers, "Wave mechanics and half-integer quantization," Zeit. Phys. 39 (1926), 828-840.
L. Brillouin, "Schrodinger's wave mechanics: a general method of solution by successive approximations," C. R. Acad. Sci. 183 (1926), 24-26.
W. Gordon, "The Compton effect according to Schrodinger's theory," Zeit. Phys. 40 (1926), 117-133.
V. Fock, "On the invariant form of the wave equation and the equations of motion for a charged mass-point," Zeit. Phys. 39 (1926), 226-232.
L. Flamm, "The foundations of wave mechanics," Phys. Zeit. 27 (1926), 600-617.
P. Ehrenfest and G. E. Uhlenbeck, "Graphical illustration of De Broglie's phase waves in O. Klein's five-dimensional universe," Zeit. Phys. 39 (1926), 495-498.
F. London, "Quantum-mechanical interpretation of Weyl’s theory," Zeit. Phys. 42 (1927), 375-389.
L. de Broglie, "The five-dimensional universe and wave mechanics," J. Phys. Radium (2) 8 (1927), 65-73.
O. Klein, "On the article by L. de Broglie 'The five-dimensional universe and wave mechanics'," J. Phys. Radium (2) 8 (1927), 242-243.
L. Rosenfeld, "Five-dimensional wave mechanics," Bull. Acad. roy. de Belg. 13 (1927), 304-325, 447-458, 573-579, 661-682.
O. Klein, "Electrodynamics and wave mechanics from the standpoint of the correspondence principle," Zeit. Phys. 41 (1927), 407-442.
E. Madelung, “Quantum theory in hydrodynamical form,” Zeit. Phys. 40 (1927), 322.
A. Lande, Optics, Mechanics, and Wave Mechanics, from Handbuch der Physik, v. 20, L. Grebe, et al. (eds.), Springer, Berlin, 1928, pp. 317-452.
P. Frank, "The fundamental concepts of analytical mechanics as the foundations of quantum and wave mechanics," Phys. Zeit. 30 (1929), 209-228.
V. Fock, “On the concept of velocity in the Dirac theory of the electron,” Zeit. Phys. 55 (1929), 127-140.
D. Iwanenko, “Remark on quantum-mechanical velocity,” Zeit. f. Phys. 55 (1929), 141-144.
A. Buhl, " Vortices, corpuscles, waves," Ann. fac. Sc. de Toulouse (3) 24 (1932), 1-48.
R. Zaycoff, "On the extension of wave mechanics," Zeit. Phys. 83 (1933), 338-440; ibid. "...(Second communication)," 84 (1933), 264-267.
J.-P. Vigier, Structure of micro-objects in the causal interpretation of quantum theory, Gauthier-Villars, Paris, 1956.
Y. B. Rumer, Studies in 5-optics, State Publishing Office, Moscow, 1956.
L. de Broglie, The theory of measurement in wave mechanics, Gauthier-Villars, Paris, 1957.
G. Petiau, "On some types of nonlinear wave equations and their solutions," 27 (1957-58), 1-25.
G. Petiau, "On a nonlinear generalization of wave mechanics and the properties of the corresponding wave functions," Supp. Nuov. Cim. 9 (1958), 542-568.
F. Bopp, "Elementary processes in quantum mechanics, from a stochastic standpoint," Ann. d. Phys. (7) 17 (1966), 407-414.
_______________________________________________________________________________________________________________________________________________________
PERSONAL RESEARCH
________________________________________________________________________________________________________________________________________________________
Book
D. H. Delphenich, Continuum-mechanical Models for Quantum Mechanics, Neo-classical Press, Spring Valley, 2017.
In three files: Intro-Chap V, Chap VI - Epilogue, Appendices
Published articles
D. H. Delphenich, “On the pre-metric foundations of wave mechanics I: massless waves,” Ann. Phys. (Berlin) 18 (2009), 206-230
Conference presentations
D. H. Delphenich, “Possibilities for a Causal Interpretation for Quantum Mechanics,” presented at the Vigier IV conference, Paris, 2003, arXiv:quant-ph/0401105
arXiv.org uploads
D. H. Delphenich, "On the equations of diffracted geodesics," arXiv:2008.03113
D. H. Delphenich, "On geodesics of gradient-index optical metrics and the optical-mechanical analogy," arXiv:2002.04390 .
D. H. Delphenich, “Generalized Madelung transformations for quantum wave equations I: generalized spherical coordinates for field spaces,” arXiv:0802.0305
D. H. Delphenich, “The Geometric Origin of the Madelung Potential,” arXiv:gr-qc/0211065
D. H. Delphenich, “Foliated cobordism and motion,” arXiv:gr-qc/0209091
D. H. Delphenich, "The relativistic Pauli equation," http://arxiv.org/abs/1207.5752
D. H. Delphenich, "A strain tensor that couples to the Madelung stress tensor," arXiv.org/1303.3582.
_______________________________________________________________________________________________________________________________________________________
TRANSLATIONS
_______________________________________________________________________________________________________________________________________________________
Contact transformations and wave motion
S. Lie, “The infinitesimal contact transformations of mechanics,” Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaft zu Leipzig, Mathematische-Physische Classe, v. 41 (1889), 145-146.
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, “Essay on the propagation of waves,” Ann. Sci. de l’E. N. S., 3rd series, 29 (1909), 405-448.
E. Vessiot, “On the theory of multiplicities and the calculus of variations,” Bull. Soc. Math. France 40 (1912), 68-139.
E. Vessiot, “On the propagation of waves and the Mayer problem,” J. de Math. (6) 9 (1913), 39-76.
E. Vessiot, "On the integral invariants of wave propagation," Bull. Soc. math. France 42 (1914), 142-167.
G. Juvet, Analytical Mechanics and Wave Mechanics, Mem. Sci. math., fasc. 83, Gauthier-VIllars, Paris, 1937.
Characteristics and bicharacteristics
J. Beudon, "On the characteristics of partial differential equations," Bull. Soc. Math. France 25 (1897), 108-120.
T. Levi-Civita, Characteristics of differential systems and wave propagation, Alcan, Paris, 1932.
Nonlinear waves
H. Hugoniot, “Memoir on the propagation of motion in an indefinite fluid (part one),” J. de Math. (4th series), t. III (1887), 477-492.
J. Hadamard, Lessons on the propagation of waves and the equations of hydrodynamics, Hermann and Co., Paris, 1903, Part I, Part II.
H. Weyl, "The asymptotic distribution law for the eigen-oscillations of an elastic body of arbitrary form," Rend. Circ. Mat. Palermo 34 (1914), 1-49.
G. Petiau, "On a nonlinear generalization of wave mechanics and the properties of the corresponding wave functions," Supp. Nuov. Cim.
9 (1958), 542-568.
J. van Mieghem, Contribution to the theory of Huygens's enveloping-wave principle, Palais des Academies, Brussels, 1936.
G. Boillat, “The propagation of waves,” Traité de physique théorique et de physique mathématique , t. XXIII, Gauthier-Villars, Paris, 1965.
G. Petiau, "On the quantum theory of fields that are associated with some simple models of nonlinear field equations," Nuov. Cim. 40 (1965), 84-101.
E. Hölder, “Historical overview of the mathematical theory of discontinuity waves since Riemann and Christoffel,” from E. B. Christoffell: the Influence of his Work on Mathematics and the Physical Sciences, ed. P. L. Butzer and F. Feher, Birkhäuser, Basel, 1981.
Geometrical optics
E. Malus, “Optics," J. Ec. poly. 7 (1808), 1-44.
Ch. Dupin, "On the paths that are followed by light and elastic bodies in general under the phenomena of reflection and refraction," from Applications de geometrie et de mecanique..., Courcier, Paris, 1822.
M. Gergonne, "Optics. Purely geometric proof of a fundamental principle in the theory of caustics," Ann. de Math. pures et appl. 16 (1825-1826), 307-314.
J. Plucker, "Discussion of the general form for light waves," J. f. reine u. angew. Math. 19 (1839), 1-44.
J. Bertrand, "Memoir on the theory of surfaces," J. Math. pures et appl. 9 (1844), 133-154.
J. Bertrand, "Note on the wave surface," Comptes rendus 47 (1858), 817-819.
F. Brioschi, "On the line of curvature of the wave surface," Ann. mat. pura ed appl. 2 (1859), 135-136.
E. Combescure, "On the lines of curvature of the wave surface," Ann. di. mat. pura ed. appl. 2 (1859), 278-285.
E. Kummer, "General theory of rectilinear ray systems," J. f. reine u. angew. Math. 57 (1860), 189-230.
E. Kummer, (no title: rectilinear systems of light rays), Monatsber. d. konig.-preuss. Akad. d. Wiss. zu Berlin, phys-math. Klasse, 30 Juli 1860, pp. 469-474.
E. E. Kummer, "Three models of general, infinitely-thin, rectilinear ray sheaves that are composed of filaments, Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1860), 469-474.
E. E. Kummer, "On atmospheric ray refraction," J. f. d. reine u. angew. Math. 61 (1863), 263-275.
E. E. Kummer, "On the fourth-degree surfaces with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 246-270.
E. E. Kummer, "On the ray systems whose focal surfaces are surfaces of degree four with sixteen singular points," Monats. d. Konig. Preuss. Akad. Wiss. Berlin (1864), 495-499.
R. Meibauer, "Theory of rectilinear systems of light rays," C. Luderitz, Berlin, 1864.
E. E. Kummer, "On algebraic ray systems; in particular, on those of orders one and two," Abh. d. Kon. Akad. Wiss. zu Berlin (1866), 1-120.
A. Levistal, "Research in geometrical optics," Ann. de l'E.N.S. (1) 4 (1867), 195-254.
M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes," J. f. reine u. angew. Math. 76 (1873), 156-169.
M. Blasendorff, "On the relations between two general ray systems...," Inaugural Dissertation, Berlin, 1883.
M. Blasendorff, "On optical ray systems," J. f. d. reine u. angew. Math. 97 (1884), 172-176.
S. Lie and F. Klein, "On the principal tangent curves of the fourth-degree Kummer surface with 16 nodes," Math. Ann. 23 (1884); F. Klein, Ges. math. Werke, art. VI.
G. Darboux, "On the wave surface," Ann.sci. de l'E.N.S (3) 6 (1889), 379-388.
G. Darboux, Chapter XIII: "Normal lines to a surface," from Lecons sur la theorie des surfaces, etc., Book IV, Gauthier-Villars, Paris, 1889.
F. Klein, “On the recent English papers on mechanics,” Jber. d. Deutscher Mathematiker-Vereinigung 1 (1891/92).
H. Bruns, The Eikonal, S. Hirzel, Leipzig, 1895.
G. Darboux, Note VIII: "On the asymptotic lines and lines of curvature of the Fresnel wave surface," from Lecons sur la theorie des surfaces, Part IV, Gauthier-Villars, Paris, 1896, pp. 466-488.
F. Haussdorff, "Infinitesimal maps in optics," Ber. Verh. Kön. Sächs. Ges. Wiss. zu Leipzig, math.-phys. Klasse 48 (1896), 79-130.
T. Levi-Civita, "Complements to the Malus-Dupin theorem. I," Rend. R. Accad. dei Lincei 9 (1900), 185-189.
F. Klein, “On the Bruns Eikonal,” Zeit. f. Math. u. Phys. 46 (1901); Ges. math. Abh., v. 2, art. LXXI, pp. 601.
F. Klein, "Spatial collineations in optical instruments," Zeit. Math. Phys. 46 (1901), 376-382.
L. Bianchi: Chapter X: "Systems of (infinity-squared) rays or rectilinear congruences," from Lezioni di geometria differenziale, 2nd ed., v. I, Enrico Spoerri, Pisa, 1902.
R. Straubel, "On a general theorem of geometrical optics and some applications," Phys. Zeit. 4 (1902-03), 114-117.
E. Study, "On Hamilton's geometrical optics and its relationship to the theory of contact transformations," Jber. d. Deutschen Math.-Verein. 14 (1905), 424-438.
K. Zindler, "The development and present state of differential line geometry," Jahres. der deutschen Math.-Verein. 15 (1906), 185-213.
W. Voigt, "Remarks on the theory of conical refraction," Ann. Phys. (Leipzig) (4) 19 (1906), 14-21.
A. Garbasso and G. Fubini, "On the most general problem in optics," Atti della R. accad. sci. Torino 44 (1908/1909), 159-170.
A. Sommerfeld and J. Runge, "The application of vector calculus to the foundations of geometrical optics," Ann. Phys. (Leipzig) 35 (1911), 277-298.
H. Schlusser, The focal theory of linear ray congruences, Hoffmann, Leipzig, 1912.
B. Caldonazzo, "Trajectories of light rays and material points in a gravitational field," Nuovo Cim. (5) 5 (1913), 267-300.
R. Dontot, "On integral invariants and some points of geometrical optics," Bull. Soc. math. France 42 (1914), 53-91.
T. De Donder, "On the integral invariants of optics," Bull. Soc. math. France 42 (1914), 91-95.
T. Levi-Civita, "Einstein's theory and Fermat's principle," Nuovo Cim. 16 (1918), 105-114.
M. v. Laue, "Theoretical remarks on some recent optical observations about the theory of relativity," Phys. Zeit. 21 (1920), 659-662.
G. Prange, "W. R. Hamilton's significance to geometrical optics," Jber. d. Deutschen Math.-Verein. 30 (1921), 69-82.
J. Le Roux, "Special relativity and the geometry of wave systems," J. math. pures appl. 1 (1922), 205-253.
J. Hadamard, "Huygens's principle," Bull. Soc. math. France 52 (1924), 610-640.
C. Caratheodory, "On the connection between the theory of absolute optical instruments and a theorem in the calculus of variations," Sitz. der math.-phys. der Abt. Bay. Akademie de Wiss. zu Munchen 1 (1926), 1-18.
C. Caratheodory, Geometrical Optics, Springer, Berlin, 1937.
P. M. Quan, "Electromagnetic inductions in general relativity and Fermat's principle," J. Rat. Mech. Anal. 1 (1957), 54-80.
J. Ehlers, "On the transition from wave optics to geometrical optics in the general theory of relativity," Z. Naturforschg. 22a (1967), 1328-1333.
V. C. Bouslaev, "Generating integral and Maslov's canonical operator for the W. K. B. method," Complement 2 in V. P. Maslov, Theorie des perturbations et methodes asymptotiques, Dunod, Gauthier-Villars, Paris, 1972.
H. Knorrer, "The Fresnel wave surface," Lecture four of Mathematische Miniaturen 3: Arithmetik und Geometrie. Vier Vorlesungen, Birkhauser, Basel, 1986.
Quantum waves
P. Ehrenfest, "Does the angle of aberration measure the wave velocity in the case of a dispersive ether?" Ann. Phys. (Leipzig) 338 (1910), 1571-1576.
E. Schrodinger, "Hertz's mechanics and Einstein's theory of gravitation," Unpublished notes from around 1918.
L. de Broglie, Waves and Motions, Gauthier-Villars, Paris, 1926.
G. Wentzel, "A generalization of the quantization conditions for the purpose of wave mechanics," Zeit. Phys. 38 (1926), 518-529.
H. A. Kramers, "Wave mechanics and half-integer quantization," Zeit. Phys. 39 (1926), 828-840.
L. Brillouin, "Schrodinger's wave mechanics: a general method of solution by successive approximations," C. R. Acad. Sci. 183 (1926), 24-26.
W. Gordon, "The Compton effect according to Schrodinger's theory," Zeit. Phys. 40 (1926), 117-133.
V. Fock, "On the invariant form of the wave equation and the equations of motion for a charged mass-point," Zeit. Phys. 39 (1926), 226-232.
L. Flamm, "The foundations of wave mechanics," Phys. Zeit. 27 (1926), 600-617.
P. Ehrenfest and G. E. Uhlenbeck, "Graphical illustration of De Broglie's phase waves in O. Klein's five-dimensional universe," Zeit. Phys. 39 (1926), 495-498.
F. London, "Quantum-mechanical interpretation of Weyl’s theory," Zeit. Phys. 42 (1927), 375-389.
L. de Broglie, "The five-dimensional universe and wave mechanics," J. Phys. Radium (2) 8 (1927), 65-73.
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