The electrodynamic 2-body problem and the origin of quantum mechanics

Foundations of Physics

Volumn 34, Issue 6, 2004, Pages 937-962

  Raju, C K ^a  

^a Centre for Computer Science   (India)

Author keywords

Functional differential equations; Interpretation of quantum mechanics; Many body problem; Protein dynamics; Relativistic many body problem

Indexed keywords

EID: 3142780129     PISSN: 00159018     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:FOOP.0000034223.58332.d4     Document Type: Article

References (49)

1. C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994). (Fundamental Theories of Physics, Vol. 65.)
2. Preliminary versions of aspects of this paper have been circulating for some time, having been presented and discussed at various conferences and lectures over the past several years, e.g., "Simulating a tilt in the arrow of time: preliminary results," Seminar on Some Aspects of Theoretical Physics, Indian Statistical Institute, Calcutta, 14-15 May 1996, "The electrodynamic 2-body problem and the origin of quantum mechanics." Paper presented at the International Symposium on Uncertain Reality, New Delhi, 5-9 Jan 1998, "Relativity: history and history dependence." Paper presented at the On Time Seminar, British Society for History of Science, and Royal Society for History of Science, Liverpool, August 1999. "Time travel," invited talk at the International Seminar, Retrocausality Day, University of Groningen, September 1999, and in talks at the Universities of Southampton, Utrecht, Pittsburgh, etc.
3. Preliminary versions of aspects of this paper have been circulating for some time, having been presented and discussed at various conferences and lectures over the past several years, e.g., "Simulating a tilt in the arrow of time: preliminary results," Seminar on Some Aspects of Theoretical Physics, Indian Statistical Institute, Calcutta, 14-15 May 1996, "The electrodynamic 2-body problem and the origin of quantum mechanics." Paper presented at the International Symposium on Uncertain Reality, New Delhi, 5-9 Jan 1998, "Relativity: history and history dependence." Paper presented at the On Time Seminar, British Society for History of Science, and Royal Society for History of Science, Liverpool, August 1999. "Time travel," invited talk at the International Seminar, Retrocausality Day, University of Groningen, September 1999, and in talks at the Universities of Southampton, Utrecht, Pittsburgh, etc.
4. Preliminary versions of aspects of this paper have been circulating for some time, having been presented and discussed at various conferences and lectures over the past several years, e.g., "Simulating a tilt in the arrow of time: preliminary results," Seminar on Some Aspects of Theoretical Physics, Indian Statistical Institute, Calcutta, 14-15 May 1996, "The electrodynamic 2-body problem and the origin of quantum mechanics." Paper presented at the International Symposium on Uncertain Reality, New Delhi, 5-9 Jan 1998, "Relativity: history and history dependence." Paper presented at the On Time Seminar, British Society for History of Science, and Royal Society for History of Science, Liverpool, August 1999. "Time travel," invited talk at the International Seminar, Retrocausality Day, University of Groningen, September 1999, and in talks at the Universities of Southampton, Utrecht, Pittsburgh, etc.
5. Preliminary versions of aspects of this paper have been circulating for some time, having been presented and discussed at various conferences and lectures over the past several years, e.g., "Simulating a tilt in the arrow of time: preliminary results," Seminar on Some Aspects of Theoretical Physics, Indian Statistical Institute, Calcutta, 14-15 May 1996, "The electrodynamic 2-body problem and the origin of quantum mechanics." Paper presented at the International Symposium on Uncertain Reality, New Delhi, 5-9 Jan 1998, "Relativity: history and history dependence." Paper presented at the On Time Seminar, British Society for History of Science, and Royal Society for History of Science, Liverpool, August 1999. "Time travel," invited talk at the International Seminar, Retrocausality Day, University of Groningen, September 1999, and in talks at the Universities of Southampton, Utrecht, Pittsburgh, etc.
6. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
7. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
8. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
9. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
10. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
11. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
12. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
13. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
14. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
15. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
16. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
17. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
18. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
19. Some attempts have been made to study the 2-body problem in 1 and 2-dimensions, and some approximation procedures have been suggested for three dimensions, but none of these are critically relevant to the question at hand. For a quick review see C. K. Raju, chap. Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994) chap. 56. For the 1-body problem, see C. F. Eliezer, Rev. Mod. Phys. 19 147(1947): G. N. Plass, Rev. Mod. Phys. 33, 37-62. (1961) For the 2-body problem, see J. L. Synge, Proc. R. Soc. A177 118-139: (1940) R. D. Driver, Phys. Rev. 178 2051-2057, (1969) D. K. Hsing, Phys. Rev. D16 974-82; (1977) A. Schild, Phys. Rev. 131, 2762; (1963) C. M Anderssen and H. C. von Baeyer, Phys. Rev. D5, 802; (1972) Phys. Rev. D5, 2470; (1972). R. D. Driver, Phys. Rev. D19, 1098; (1979) L. S. Schulman, J. Math. Phys. 15, 205-208, (1974); K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl. 24, 372-387; (1968). H. Van Dam and E. P. Wigner, Phys. Rev. 138B, 1576; (1965). 142, 838 (1966).
20. J. Andrew McCammon and Stephen C. Harvey, Dynamics of Proteins and Nucleic Acids (Cambridge University Press), 1987, p. 61.
21. David J. Griffiths, Introduction to Electrodynamics (Prentice Hall, India, 3rd edn. 1999), p. 435, Eq. (10.46). Cf. J. D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, 2001). Griffiths' book is more convenient for our purpose.
22. David J. Griffiths, Introduction to Electrodynamics (Prentice Hall, India, 3rd edn. 1999), p. 435, Eq. (10.46). Cf. J. D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, 2001). Griffiths' book is more convenient for our purpose.
23. Cf. Griffiths, Electrodynamics, 3rd edn., Eqs. (10.65), (10.66) and (10.67), pp. 438-439.
24. Usually called just the Lorentz force law.
25. Griffiths, Electrodynamics, 3rd edn., 439, Eq. (10.67).
26. Griffiths, Electrodynamics, 3rd edn., p. 421. O. L. Brill and B. Goodman, Am. J. Phys. 35, 832, (1967).
27. Griffiths, Electrodynamics, 3rd edn., p. 421. O. L. Brill and B. Goodman, Am. J. Phys. 35, 832, (1967).
28. C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994). chapp. 5b.
29. L. E. El'sgol'tz, Introduction to the Theory of Differential Equations with Deviating Arguments, trans. R. J. McLaughlin (Holden-Day, San Francisco, 1966), pp. 13-19. R. D. Driver, Introduction to Differential and Delay Equations (Springer, Berlin, 1977).
30. L. E. El'sgol'tz, Introduction to the Theory of Differential Equations with Deviating Arguments, trans. R. J. McLaughlin (Holden-Day, San Francisco, 1966), pp. 13-19. R. D. Driver, Introduction to Differential and Delay Equations (Springer, Berlin, 1977).
31. C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994). chapp. 5b.
32. C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994).
33. An earlier method suggested by Synge also assumes instantaneity, but it has the advantage that it can actually be implemented. This implementation will be considered in subsequent articles. See, J. L. Synge, J. R. Soc. A177 118-139, (1940).
34. El'sgol'tz, Differential Equations with Deviating Arguments, pp. 13-19. Driver, Differential and Delay Equations.
35. El'sgol'tz, Differential Equations with Deviating Arguments, pp. 13-19. Driver, Differential and Delay Equations.
36. E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations (Springer Series in Computational Mathematics, Vol. 8) (Springer, Berlin, 1987; Revised edn. 1991).
37. El'sgol'ts, Differential Equations with Deviating Arguments, p. 9.
38. R. D. Driver, Differential and Delay Equations.
39. C. A. H. Paul, Numerical Analysis Report No. 283, Department of Mathematics, University of Manchester, 1995.
40. D. R. Wille and C. T. H. Baker, Appl. Num. Math. 9, 209-222, (1992).
41. Griffiths, Electrodynamics, 3rd edn., Eqs. (7.40), and (11.80), pp. 326 and 467.
42. E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations, cited above. (The above figures used the newer version of retard.)
43. P. A. M. Dirac, Proc. R. Soc. A 167, 148 (1938). Also F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, MA, 1985) p. 142.
44. P. A. M. Dirac, Proc. R. Soc. A 167, 148 (1938). Also F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, MA, 1985) p. 142.
45. Griffiths, Electrodynamics, 3rd edn., § 11.80, p. 467.
46. El'sgol'tz, Differential Equations with Deviating Arguments, pp. 32-33.
47. El'sgol'tz, Differential Equations with Deviating Arguments, pp. 28-41.
48. C. K. Raju, "Simulating a tilt in the arrow of time," cited earlier. Such an approximation is a priori plausible, though there is, at present, no formal proof of its validity.
49. C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic, Dordrecht, 1994), Chap. 6.