Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y) = 0 with respect to the standard Lagrangian L = xny′2/2 — xn ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αyr , where r ≠ 0,1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

[1]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[2]  Abdul-Majid Wazwaz,et al.  A new algorithm for solving differential equations of Lane-Emden type , 2001, Appl. Math. Comput..

[3]  R. Fowler.  FURTHER STUDIES OF EMDEN'S AND SIMILAR DIFFERENTIAL EQUATIONS , 1931 .

[4]  Fazal M. Mahomed,et al.  A note on the solutions of the Emden-Fowler equation , 1993 .

[5]  P. Burt.  Nonperturbative solution of nonlinear field equations , 1987 .

[6]  A. Kara,et al.  Lie–Bäcklund and Noether Symmetries with Applications , 1998 .

[7]  Pedro M. Lima.  Numerical methods and asymptotic error expansions for the Emden-Fowler equations , 1996 .

[8]  W. Mccrea.  An Introduction to the Study of Stellar Structure , 1939, Nature.

[9]  Shijun Liao,et al.  A new analytic algorithm of Lane-Emden type equations , 2003, Appl. Math. Comput..

[10]  R. Emden,et al.  Gaskugeln : Anwendungen der mechanischen Wärmetheorie auf Kosmologische und meteorologische Probleme , 1907 .

[11]  Pedro M. Lima,et al.  Numerical solution of a singular boundary-value problem in non-Newtonian fluid mechanics , 2000 .

[12]  O. Richardson.  The Emission of Electricity from Hot Bodies , 2007, Nature.

[13]  G. Adomian,et al.  On the analytic solution of the lane-emden equation , 1995 .

[14]  I. Roxburgh,et al.  Power series solutions of the polytrope equations , 1999 .

[15]  Fazal M. Mahomed,et al.  Lie and Noether Counting Theorems for One-Dimensional Systems , 1993 .

[16]  C. Bender,et al.  A new perturbative approach to nonlinear problems , 1989 .

[17]  H. Goenner,et al.  Exact solutions of the generalized Lane–Emden equation , 2000 .

[18]  Y. Bozhkov,et al.  LIE POINT SYMMETRIES AND EXACT SOLUTIONS OF QUASILINEAR DIFFERENTIAL EQUATIONS WITH CRITICAL EXPONENTS , 2004 .

[19]  Fazal M. Mahomed,et al.  Equivalent lagrangians and the solution of some classes of non-linear equations q̈ + p(t)q̊ + r(t)q = μq̈2q−1 + f(t)qn , 1992 .

[20]  Fazal M. Mahomed,et al.  Solution of generalized emden-fowler equations with two symmetries , 1994 .

[21]  G. P. Horedt,et al.  Seven-digit tables of Lane-Emden functions , 1986 .

[22]  F. Mahomed,et al.  NOETHER SYMMETRIES OF Y' = F(X)YN WITH APPLICATIONS TO NON-STATIC SPHERICALLY SYMMETRIC PERFECT FLUID SOLUTIONS , 1999 .

[23]  L. M. Berkovich.  The Generalized Emden-Fowler Equation , 1997 .

[24]  Petrovich Vasilij Ermakov,et al.  Second-order differential equations: Conditions of complete integrability , 2008 .

[25]  Yuri Bozhkov,et al.  Lie point symmetries of the Lane–Emden systems , 2004 .

[26]  N. Shawagfeh.  Nonperturbative approximate solution for Lane–Emden equation , 1993 .

[27]  P. Leach.  First integrals for the modified Emden equation q̈+α(t) q̇+qn =0 , 1985 .

[28]  Frans Cantrijn,et al.  GENERALIZATIONS OF NOETHER'S THEOREM IN CLASSICAL MECHANICS* , 1981 .

[29]  N. Bahcall.  Core radii and central densities of 15 rich clusters of galaxies , 1975 .

[30]  N. Bahcall.  The galaxy distribution in the cluster Abell 2199 , 1973 .

[31]  Lawrence Dresner,et al.  Similarity solutions of nonlinear partial differential equations , 1983 .

[32]  N. Euler.  Transformation Properties of , 1997 .

[33]  H. Goenner,et al.  Symmetry Transformations for the Generalized Lane–Emden Equation , 2001 .

[34]  Integration of Ordinary Differential Equations via Nonlocal Symmetries , 2002 .

[35]  H. Davis.  Introduction to Nonlinear Differential and Integral Equations , 1964 .