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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
(ODEs) that need not be derivable from variational principles. We obtain a Noetherlike
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. In order to investigate the effectiveness of the
partial Lagrangian approach, some models such as the oscillator systems both linear and
nonlinear, Emden and Ermakov-pinnery equations and the Hamiltonian system with two
degrees of freedom are considered in this work. We study a general linear system of
two second-order ODEs with variable coefficients. Note that, a Lagrangian exists for the
special case only but, in general, the system under consideration does not have a standard
Lagrangian. However, partial Lagrangians do exist for all such equations in the absence
of Lagrangians. Firstly, we classify all the Noether and partial Noether operators for the
case when the system admits a standard Lagrangian. We show that the first integrals
that result due to the partial Noether approach is the same as for the Noether approach.
First integrals are then constructed by the partial Noether approach for the general case
when there is in general no Lagrangian for the system of two second-order ODEs with variable coefficients. We give an easy way of constructing first integrals for such systems
by utilization of a partial Noether’s theorem with the help of partial Noether operators
associated with a partial Lagrangian.
Furthermore, we classify all the potential functions for which we construct first integrals
for a system with two degrees of freedom. Moreover, the comparison of Lagrangian and
partial Lagrangian approaches for the two degrees of freedom Lagrangian system is also
given.
In addition, we extend the idea of a partial Lagrangian for the perturbed ordinary differential
equations. Several examples are constructed to illustrate the definition of a partial Lagrangian in the approximate situation. An approximate Noether-like theorem which
gives the approximate first integrals for the perturbed ordinary differential equations
without regard to a Lagrangian is deduced.
We study the approximate partial Noether operators for a system of two coupled
nonlinear oscillators and the approximate first integrals are obtained for both resonant
and non-resonant cases. Finally, we construct the approximate first integrals for a system
of two coupled van der Pol oscillators with linear diffusive coupling. Since the system
mentioned above does not satisfy a standard Lagrangian, the approximate first integrals
are still constructed by invoking an approximate Noether-like theorem with the help of
approximate partial Noether operators. This approach can give rise to further studies
in the construction of approximate first integrals for perturbed equations without a
variational principle. |
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