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On the inverse problem of calculus of variations for fourth-order equations. (English) Zbl 1200.35054

The necessary and sufficient conditions under which a fourth-order equation \[ u^{(\mathrm{iv})}=F(t,u,u',u'',u''')\tag{1} \] admits a unique Lagrangian, namely \[ \frac{\partial^3F}{\partial(u''')^3}=0\tag{2} \] and \[ \frac{\partial F}{\partial u'}+\frac12\frac{d^2}{dt^2}\left(\frac{\partial F}{\partial u'''}\right)- \frac{d}{dt}\left(\frac{\partial F}{\partial u''}\right)- \frac34\frac{\partial F}{\partial u'''}\frac{d}{dt}\left(\frac{\partial F}{\partial u'''}\right)+ \frac12\left(\frac{\partial F}{\partial u''}\right)^2+ \frac18\left(\frac{\partial F}{\partial u'''}\right)^3=0\tag{3} \] were derived by M. E. Fels [Trans. Am. Math. Soc. 348, No.12, 5007–5029 (1996; Zbl 0879.34016)]. The authors propose to extend the use of the Jacobi last multiplier in order to find the Lagrangian for ordinary differential equations of fourths order satisfying Fels’ conditions (1) and (3). It is shown that if a Lagrangian exists for an equation of any even order, then it can be derived from the Jacobi last multiplier. The known connection between Jacobi last multiplier and Lie symmetries is also exploited. The Lagrangians of two fourth-order equations drawn from physics are determined with the same method.

MSC:

35G20 Nonlinear higher-order PDEs
34C14 Symmetries, invariants of ordinary differential equations
34A55 Inverse problems involving ordinary differential equations

Citations:

Zbl 0879.34016
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References:

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