The Invariance Principle and Income-Wealth Conservation Laws

  • Ryuzo Sato
  • Rama V. Ramachandran
Part of the Advances in Japanese Business and Economics book series (AJBE, volume 1)


In the early part of the nineteenth century William Rowan Hamilton discovered a principle which can be generalized to encompass many areas of physics, engineering and applied mathematics. Hamilton’s principle roughly states that the evolution in time of a dynamic system takes place in such a manner that integral of the difference between the kinetic and potential energies for the system is stationary. If the “action” integral is free of the time variable, the sum of the kinetic and potential energies, the Hamiltonian, is constant—the conservation law of the total energy.


Discount Rate Optimal Path Invariance Principle Japanese Economy Infinitesimal Transformation 


  1. Bessel-Hagen, E. (1921). Über die Erhaltungssätze der Elektrodynamik. Mathematische Annalen, 84, 258–276.CrossRefGoogle Scholar
  2. Caton, C., & Shell, K. (1971). An exercise in the theory of heterogeneous capital accumulation. Review of Economic Studies, 32, 233–240.Google Scholar
  3. Gelfand, I.M., & Fomin, S.V. (1963). Calculus of variations (translated from the Russian by Silverman, R.A.). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  4. Kataoka, H. (1983). On the local conservation laws in the von Neumann model. In R. Sato, & M.J. Beckmann (Eds.), Technology, organization and economic structure: lecture notes in economics and mathematical systems, vol. 210 (pp. 253–260).Google Scholar
  5. Kemp, M.C., & Long, N.V. (1982). On the evaluation of social income in a dynamic economy. In G.R. Feiwel (Ed.), Samuelson and neoclassical economics. Boston: Kluwer-Nijhoff.Google Scholar
  6. Klein, F. (1918). Über die Differentialgesetze für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie. Nachr. Akad. Wiss. Göttingen, Math-Phys. Kl. II, 171–189.Google Scholar
  7. Lie, S. (1891). In G. Scheffers (Ed.), Vorlesungen liber Differentialgleichungen, mit bekannten infinitesimalen Transformationen. Leipzig: Teubner. Reprinted (1967) New York: Chelsea.Google Scholar
  8. Liviatan, N., & Samuelson, P.A. (1969). Notes on turnpikes: stable and unstable. Journal of Economic Theory, 1, 454–475.CrossRefGoogle Scholar
  9. Logan, J.D. (1977). Invariant variational principles. Mathematics in science and engineering, vol. 138. New York: Academic Press.Google Scholar
  10. Lovelock, D., & Rund, H. (1975). Tensors, differential forms and variational principles. New York: Wiley.Google Scholar
  11. Moser, J. (1979). Hidden symmetries in dynamical systems. American Scientists, 67, 689–695.Google Scholar
  12. Noether, E. (1918). Invariante Variationsprobleme. Nachr. Akad. Wiss. Göttingen, Math-Phys. KL II, 235–257. Translated by Tavel, M.A. (1971). Invariant variation problems. Transport Theory and Statistical Physics, 1, 186–207.Google Scholar
  13. Nôno, T. (1968). On the symmetry groups of simple materials: applications of the theory of Lie groups. Journal of Mathematical Analysis and Applications, 24, 110–135.CrossRefGoogle Scholar
  14. Nôno, T., & Mimura, F. (1975/1976/1977/1978). Dynamic symmetries I/III/IV/V. Bulletin of Fukuoka University of Education, 125/126/127/128.Google Scholar
  15. Ramsey, F. (1928). A mathematical theory of saving. Economic Journal, 38, 543–559.CrossRefGoogle Scholar
  16. Rund, H. (1966). The Hamilton–Jacobi theory in the calculus of variations. Princeton, NJ: Van Nostrand-Reinhold.Google Scholar
  17. Sagan, H. (1969). Introduction to the calculus of variations. New York: McGraw-Hill.Google Scholar
  18. Samuelson, P.A. (1970a). Law of conservation of the capital-output ratio: Proceedings of the National Academy of Sciences. Applied Mathematical Science, 67, 1477–1479.Google Scholar
  19. Samuelson, P.A. (1970b). Two conservation laws in theoretical economics. Cambridge, MA: MIT Department of Economics mimeo. Reprint (1990) In R. Sato, & R. Ramchandran (Eds.), Conservation laws and symmetry: applications to economics and finance (pp.57–70). Norwell, MA: Kluwer.Google Scholar
  20. Samuelson, P.A. (1976). Speeding up of time with age in recognition of life as fleeting. In A.M. Tang et al. (Eds.), Evolution, welfare, and time in economics: essays in honor of Nicholas Georgescu-Roegen. Lexington, MA: Lexington/Heath Books.Google Scholar
  21. Samuelson, P.A. (1982). Variations on capital-output conservation laws. Cambrige, MA: MIT Mimeo.Google Scholar
  22. Sato, R. (1981). Theory of technical change and economic invariance: application of Lie groups. New York: Academic Press. Updated edition (1999) Cheltenham: Edward Elgar.Google Scholar
  23. Sato, R. (1982). Invariant principle and capital-output conservation laws. Providence, RI: Brown University working paper No. 82-8.Google Scholar
  24. Sato, R. (1985). The invariance principle and income-wealth conservation laws: application of Lie groups and related transformations. Journal of Econometrics, 30, 365–389.CrossRefGoogle Scholar
  25. Sato, R. (2002). Optimal economic growth: test of income/wealth conservation laws. Macroeconomic Dynamics, 6, 548–572.CrossRefGoogle Scholar
  26. Sato, R. (2006). Biased technical change and economic conservation laws. Springer.Google Scholar
  27. Sato, R., & Fujii, M. (2006). Evaluating corporate performance: empirical tests of a conservation law. Japan and the World Economy, 18, 158–168.CrossRefGoogle Scholar
  28. Sato, R., Nôno, T., & Mimura, F. (1984). Hidden symmetries: Lie groups and economic conservation laws, essay in honor of Martin Beckmann. In H. Hauptman, W. Krelle, & K.C. Mosler (Eds.), Operations research and economic theory. Springer.Google Scholar
  29. Weitzman, M.L. (1976). On the welfare significance of national product in a dynamic economy. Quarterly Journal of Economics, 90, 156–162.CrossRefGoogle Scholar
  30. Whittaker, E.T. (1937). A treatise on the analytical dynamics of particles and rigid bodies. Cambridge: Cambridge University Press, 4th edn. Reprinted (1944), New York: Dover.Google Scholar
  31. Young, E.C. (1978), Vector and tensor analysis. New York:Decker.Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Ryuzo Sato
    • 1
  • Rama V. Ramachandran
    • 2
  1. 1.Stern School of BusinessNew York UniversityNew YorkUSA
  2. 2.Pebble Brook LanePlanoUSA

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