Partial Noether operators and first integrals via partial Lagrangians

The notions of partial Lagrangians, partial Noether operators and partial Euler–Lagrange equations are used in the construction of first integrals for ordinary differential equations that need not be derivable from variational principles. We obtain a Noether-like theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler–Lagrange equations. Applications given include those that admit a standard Lagrangian such as the harmonic oscillator, modified Emden and Ermakov–Pinney equations and systems of two second-order equations that do not have standard Lagrangians. Copyright © 2007 John Wiley & Sons, Ltd.

[1]  Fazal M. Mahomed,et al.  Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians , 2006 .

[2]  G. Bluman,et al.  Erratum: Integrating factors and first integrals for ordinary differential equations , 1999, European Journal of Applied Mathematics.

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

[4]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[5]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

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

[7]  H. Steudel.  Über die Zuordnung zwischen lnvarianzeigenschaften und Erhaltungssätzen , 1962 .

[8]  Geoff Prince,et al.  Symmetries of the time-dependent N-dimensional oscillator , 1980 .

[9]  Sergio A. Hojman,et al.  On the inverse problem of the calculus of variations , 1981 .

[10]  E. Bessel-Hagen.  Über die Erhaltungssätze der Elektrodynamik , 1921 .

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

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

[13]  P. Leach.  The first integrals and orbit equation for the Kepler problem with drag , 1987 .

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

[15]  P. Leach.  Applications of the Lie theory of extended groups in Hamiltonian mechanics: the oscillator and the Kepler problem , 1981, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[16]  T. Duchamp,et al.  Variational principles for second-order quasi-linear scalar equations , 1984 .

[17]  Jesse Douglas.  Solution of the inverse problem of the calculus of variations , 1941 .

[18]  M. Lutzky.  Symmetry groups and conserved quantities for the harmonic oscillator , 1978 .

[19]  Jarmo Hietarinta,et al.  Direct methods for the search of the second invariant , 1987 .

[20]  Symmetry Properties of Autonomous Integrating Factors , 2005, nlin/0512012.

[21]  W. Sarlet,et al.  First integrals versus configurational invariants and a weak form of complete integrability , 1985 .

[22]  H. Lewis,et al.  A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians , 1982 .

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

[24]  Jean-Marc Lévy-Leblond,et al.  Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics , 1971 .

[25]  Fazal M. Mahomed,et al.  Relationship between Symmetries andConservation Laws , 2000 .