Lanczos Approach to Noether’s Theorem

Lam-Estrada, P.; Lopez-Bonilla, José Luis; López-Vázquez, R.

doi:10.18052/www.scipress.com/BSMaSS.11.1

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Lanczos Approach to Noether’s Theorem

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Abstract:

If the action A=∫_t₁^t2L(q,q,t)dt is invariant under the infinitesimal transformation t˜=t+ετ(q,t), q˜=q_r+εζ_r(q,t), r-1,...,n with ε=constant≤1, then the Noether’s theorem permits to construct the corresponding conserved quantity. The Lanczos method accepts that ε=q_n+1is a new degree of freedom, thus the Euler-Lagrange equation for this new variable gives the Noether’s constant of motion.

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Periodical:

The Bulletin of Society for Mathematical Services and Standards (Volume 11)

Pages:

1-3

DOI:

https://doi.org/10.18052/www.scipress.com/BSMaSS.11.1

Citation:

P. Lam-Estrada et al., "Lanczos Approach to Noether’s Theorem", The Bulletin of Society for Mathematical Services and Standards, Vol. 11, pp. 1-3, 2014

Online since:

September 2014

Authors:

P. Lam-Estrada, José Luis Lopez-Bonilla, R. López-Vázquez

Keywords:

Invariance of the Action, Noether’s Theorem, Variational Lanczos Technique

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References:

G. Leibnitz, Dynamica de potentia et leqibus nature corporae (1669) (published in 1890).

E. Noether, Invariante variationsprobleme, Nachr. Ges. Wiss. Göttingen 2 (1918) 235-257.

Y. Kosmann-Schwarzbach, The Noether theorems, Springer, New York (2011).

P. Havas, The connection between conservation laws and invariance groups: folklore, fiction and fact, Acta Physica Austriaca 38 (1973) 145-167.

E. Bessel-Hagen, Über die erhaltungssätze der elektrodynamik, Mathematische Annalen 84 (1921) 258-276.

D. E. Neuenschwander, Symmetries, conservation laws, and Noether's theorem, Radiations, fall (1998) 12-15.

D. E. Neuenschwander, Emmy Noether's wonderful theorem, The Johns Hopkins University Press, Baltimore (2011).

M. Havelková, Symmetries of a dynamical system represented by singular Lagrangians, Comm. in Maths 20, No. 1 (2012) 23-32.

A. Trautman, Noether's equations and conservation laws, Commun. Math. Phys. 6 (1967) 248261.

H. Rund, A direct approach to Noether's theorem in the calculus of variations, Utilitas Math. 2 (1972) 205-214.

M. A. Sinaceur, Dedekind et le programme de Riemann, Rev. Hist. Sci. 43 (1990) 221-294.

D. Laugwitz, Bernhard Riemann 1826-1866. Turning points in the conception of mathematics, Birkhäuser, Boston MA (2008).

N. Byers, Emmy Noether's discovery of the deep connection between symmetries and conservation laws, Proc. Symp. Heritage E. Noether, Bar Ilan University, Tel Aviv, Israel, 2-3 Dec. (1996).

K. A. Brading, H. R. Brown, Symmetries and Noether's theorems, in Symmetries in Physics, Philosophical Reflections, Eds. Katherine A. Brading, Elena Castellani, Cambridge University Press (2003) 89-109.

L. M. Lederman, Ch. T. Hill, Symmetry and the beautiful Universe, Prometheus Books, Amherst, New York (2004) Chaps. 3 and 5.

C. Lanczos, The variational principles of mechanics, University of Toronto Press (1970) Chap. 11.

C. Lanczos, Emmy Noether and the calculus of variations, Bull. Inst. Math. and Appl. 9, No. 8 (1973) 253-258.

P. Lam-Estrada, J. López-Bonilla, R. López-Vázquez, G. Ovando, Lagrangians: Symmetries, gauge identities and first integrals, The SciTech, J. of Sci. & Tech. 3, No. 1 (2014) 54-66.

M. Henneaux, C. Teitelboim, J. Zanelli, Gauge invariance and degree of freedom count, Nucl. Phys. B332, No. 1 (1990) 169-188.

H. J. Rothe, K. D. Rothe, Classical and quantum dynamics of constrained Hamiltonian systems, World Scientific Lecture Notes in Physics 81, Singapore (2010).

G. F. Torres del Castillo, Point symmetries of the Euler-Lagrange equations, Rev. Mex. Fís. 60 (2014) 129-135.

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