Partial Noether operators and first integrals via partial Lagrangians

Abstract

SUMMARYThe 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.

A partial Hamiltonian approach for current value Hamiltonian systems

“…We introduce the definition of what we call the partial Hamiltonian operator below. This is motivated by the analogous definition of the partial Noether operator given in [24,25].…”

“…An approach in proving Theorem 3 is by invoking the Legendre transformation (5) on the partial Noether operators and partial Noether theorem given in [24].…”

Exact solution of fin problem with linear temperature-dependent thermal conductivity

“…Theorem [15,16]. If the Lie operator (11) is a partial Noether operator corresponding to a partial Lagrangian ‫ܮ‬ of Eq.…”

Noether, Partial Noether Operators and First Integrals for the Coupled Lane-Emden System

“…The following theorems are taken from [8] and [29] respectively. Theorem 1 (Noether [8]) If X as given in (3) is a Noether point symmetry generator corresponding to a Lagrangian…”

“…See, e.g., [18,31]. Theorem 2 (Partial Noether [29]) If X is a partial Noether operator corresponding to a partial Lagrangian (10) is a first integral of (5) associated with X. Proof.…”