1 Introduction

The generalized Emden–Fowler differential equation (GEFE) \( \ddot{q} + {{n\dot{q}} \mathord{\left/ {\vphantom {{n\dot{q}} t}} \right. \kern-0pt} t} + f(q) = 0,n \in \Re \) was the focus of many studies over the last years mainly in the study of stellar structures, pattern formation and chemical reactions [2,3,4,5,6]. Numerical and perturbative approaches were used to solve GEFE for different forms of f(q) (see [8] and references therein). The so-called GEFE of the first-kind \( t\ddot{q} + \alpha \dot{q} + \beta t^{\gamma } q^{\beta } = 0,(\alpha ,\beta ,\gamma ) \in \Re \) has been discussed in [8, 9, 11] and more recently in [1] where exact analytical solutions are given. A Lagrangian derivation of GEFE was done in [8], however in this communication we introduce a different approach based on a generalized second-order derivative operator (GDO) which will lead to interesting nonlinear differential equations for different forms of f(q) mainly a GEFF starting from basic tools of the calculus of variations. The present work is organized as follows: In Sect. 2, we introduce the basic setups of our approach and in Sect. 3 we extend our arguments for the case of higher-order Lagrangians derivatives; Conclusions are given in Sect. 4.

2 Generalized derivative operator and Lagrangian description

To start, we let tbe an independent variable and q = (q 1, q 2, …, q m) be an independent variable with coordinate q s(s = 1, 2, …, m) with respect to t. We define the GDO of second-order by:

$$ \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} = \frac{d}{dt} + \alpha \left( {\dot{q}^{s} + \beta \frac{{q^{s} }}{t}} \right)\frac{\partial }{{\partial q^{s} }} + \gamma \left( {\ddot{q}^{s} + \delta \frac{{\dot{q}^{s} }}{t}} \right)\frac{\partial }{{\partial \dot{q}^{s} }}, $$
(1)

where α, β, γ, δ ∊ ℜ, \( \dot{q}^{s} = {{dq^{s} } \mathord{\left/ {\vphantom {{dq^{s} } {dt}}} \right. \kern-0pt} {dt}} \) and \( \ddot{q}^{s} = {{d^{2} q^{s} } \mathord{\left/ {\vphantom {{d^{2} q^{s} } {dt^{2} }}} \right. \kern-0pt} {dt^{2} }} \). Noticeably we have:

$$ \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q^{s} = (1 + \alpha )\dot{q}^{s} + \alpha \beta \frac{{q^{s} }}{t}, $$
(2)
$$ \text{D}_{t}^{\alpha ,\beta } \dot{q}^{s} = (1 + \gamma )\ddot{q}^{s} + \gamma \delta \frac{{\dot{q}^{s} }}{t}, $$
(3)
$$ \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q^{s} = (1 + \alpha )(1 + \gamma )\ddot{q}^{s} + (\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\frac{{\dot{q}^{s} }}{t} + \alpha \beta (\alpha \beta - 1)\frac{{q^{s} }}{{t^{2} }}. $$
(4)

By letting:

$$ {\mathcal{I}} = \left\{ {q^{s} \in C^{1} [t_{1} ,t_{2} ]\left| {q^{s} (t_{1} ) = q_{1}^{s} ,} \right.q^{s} (t_{2} ) = q_{2}^{s} } \right\}, $$

and

$$ S:{\mathcal{I}} \to \Re , $$

be a function of the form:

$$ S = \int\limits_{{t_{1} }}^{{t_{2} }} {\text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q^{s} ,q^{s} ,t)dt} $$

where

$$ \text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q^{s} ,q^{s} ,t) = \text{L}\left( {(1 + \alpha )\dot{q}^{s} + \alpha \beta \frac{{q^{s} }}{t},q^{s} ,t} \right) \equiv \text{L}(\dot{q}^{s} ,q^{s} ,t), $$

be a function with continuous first and second partial derivatives with respect to \( (\dot{q}^{s} ,q^{s} ,t), \) then classically if \( S \) has an extremum at q s0  ≡ Q ∊ S, Q satisfies the following Euler–Lagrange equation:

$$ \frac{{\partial \text{L}(\dot{Q},Q,t)}}{\partial Q} - \frac{d}{dt}\left( {\frac{{\partial \text{L}(\dot{Q},Q,t)}}{{\partial \dot{Q}}}} \right) = 0. $$
(5)

As an exemplification, let:

$$ \text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} Q,Q,t) = \frac{1}{2}\left( {\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} Q} \right)^{2} - V(Q) = \frac{1}{2}\left( {(1 + \alpha )\dot{Q} + \alpha \beta \frac{Q}{t}} \right)^{2} - \frac{1}{2}Q^{2} = \text{L}(\dot{Q},Q,t). $$

Here the potential is given by \( V(Q) = \tfrac{1}{2}Q^{2} \). Then the corresponding equation of motion is:

$$ (1 + \alpha )^{2} \ddot{Q} + \left( {\frac{{(1 + \alpha )\alpha \beta - \alpha^{2} \beta {}^{2}}}{{t^{2} }} + 1} \right)Q = 0. $$
(6)

For illustration, we set α = β = 1 and consequently Eq. (6) is reduced to \( 4\ddot{Q} + (t^{ - 2} + 1)Q = 0 \) where the solution in term of q s0 is given by:

$$ q_{0}^{s} (t) = \sqrt t \left( {c_{1} J_{0} \left( {\frac{t}{2}} \right) + c_{2} Y_{0} \left( {\frac{t}{2}} \right)} \right) , $$
(7)

J n (z) and Y n (x) are respectively Bessel functions of first and second kinds. c 1 and c 2are constants of integrations. We plot in Figs. 1 and 2 the solutions of Eq. (6) for two different initial conditions:

Fig. 1
figure 1

The numerical solution of Eq. (6) for Q(1) = 1 and \( \dot{Q}(1) = 0 \)

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Fig. 2
figure 2

The numerical solutions of Eq. (6) for Q(1) = 0 and \( \dot{Q}(1) = 1 \)

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This corresponds to a quasi-linear oscillatory periodic dynamics which is represented by a closed curve as it is noticeable from the shape of the ellipse in the phase-space. It is noteworthy that Eq. (6) may be viewed as harmonic oscillators with time-dependent frequency which have attracted considerable interest in the last years since they occur in different areas of physics like quantum optics and plasma physics (see [12] and references therein).

3 Higher-order Lagrangians derivatives description

The previous argument may be extended for the case of higher-order Lagrangians derivatives, i.e. functionals depending on higher-order derivatives. We focus on functionals holding second-order Lagrangians derivatives of GDO-type, i.e.:

$$ \text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} ,\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} q_{0,k}^{s} ,t) \equiv \text{L}(\ddot{q}_{0,k}^{s} ,\dot{q}_{0,k}^{s} ,q_{0,k}^{s} ,t). $$

We let again:

$$ {\mathcal{I}} = \left\{ {q^{s} \in C^{1} [t_{1} ,t_{2} ]\left| {q^{s} (t_{1} ) = q_{1}^{s} ,} \right.q^{s} (t_{2} ) = q_{2}^{s} } \right\}, $$

and

$$ \text{S}:{\mathcal{I}} \to \Re , $$

be a function of the form:

$$ \text{S} = \int\limits_{{t_{1} }}^{{t_{2} }} {\text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} ,\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} ,q_{0,k}^{s} ,t)dt} \equiv \int\limits_{{t_{1} }}^{{t_{2} }} {\text{L}(\ddot{q}_{0,k}^{s} ,\dot{q}_{0,k}^{s} ,q_{0,k}^{s} ,t)dt} . $$

Then classically if \( \text{S} \) has an extremum at \( q_{0}^{s} \equiv Q \in \text{S,} \) then \( Q \) satisfies the following Euler–Lagrange equation:

$$ \frac{{\partial \text{L}(\ddot{Q},\dot{Q},Q,t)}}{\partial Q} - \frac{d}{dt}\left( {\frac{{\partial \text{L}(\ddot{Q},\dot{Q},Q,t)}}{{\partial \dot{Q}}}} \right) + \frac{{d^{2} }}{{dt^{2} }}\left( {\frac{{\partial \text{L}(\ddot{Q},\dot{Q},Q,t)}}{{\partial \ddot{Q}}}} \right) = 0. $$
(8)

To illustrate, we consider the higher-order derivative Lagrangian:

$$ \begin{aligned} \text{L}(\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} ,\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} q_{0,k}^{s} ,t) = \frac{1}{2}\left( {\text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} \text{D}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} Q} \right)^{2} - \frac{1}{2}Q^{2} \equiv \text{L}(\ddot{Q},\dot{Q},Q,t) \hfill \\ = \frac{1}{2}(1 + \alpha )^{2} (1 + \gamma )^{2} \ddot{Q}^{2} + \frac{1}{2}(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )^{2} \frac{{\dot{Q}^{2} }}{{t^{2} }} \hfill \\ + \frac{1}{2}\alpha^{2} \beta^{2} (\alpha \beta - 1)^{2} \frac{{Q^{2} }}{{t^{4} }} + (1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\frac{{\ddot{Q}\dot{Q}}}{t} + (1 + \alpha )(1 + \gamma )\alpha \beta (\alpha \beta - 1)\frac{{Q\ddot{Q}}}{{t^{2} }} \hfill \\ + (\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\alpha \beta (\alpha \beta - 1)\frac{{Q\dot{Q}}}{{t^{3} }} - \frac{1}{2}Q^{2} . \hfill \\ \end{aligned} $$

The corresponding equation of motion is derived from Eq. (8) and takes the form:

$$ \begin{aligned} \alpha^{2} \beta^{2} (\alpha \beta - 1)^{2} t^{ - 4} Q + 2(1 + \alpha )(1 + \gamma )\alpha \beta (\alpha \beta - 1)t^{ - 2} \ddot{Q} + (\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\alpha \beta (\alpha \beta - 1)t^{ - 3} \dot{Q} - Q \hfill \\ - (\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )^{2} t^{ - 2} \ddot{Q} + 2(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )^{2} t^{ - 3} \dot{Q} + (1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )t^{ - 2} \ddot{Q} \hfill \\ - (1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )t^{ - 1} \dddot Q - (\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\alpha \beta (\alpha \beta - 1)t^{ - 3} \dot{Q} \hfill \\ + 3(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )\alpha \beta (\alpha \beta - 1)t^{ - 4} Q + (1 + \alpha )^{2} (1 + \gamma )^{2} Q^{(4)} + (1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )t^{ - 1} \dddot Q \hfill \\ - 2(1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )t^{ - 2} \ddot{Q} + 2(1 + \alpha )(1 + \gamma )(\alpha \beta + \gamma \delta (1 + \alpha ) + \alpha^{2} \beta )t^{ - 3} \dot{Q} \hfill \\ - 4(1 + \alpha )(1 + \gamma )\alpha \beta (\alpha \beta - 1)t^{ - 3} \dot{Q} + 6(1 + \alpha )(1 + \gamma )\alpha \beta (\alpha \beta - 1)t^{ - 4} Q = 0. \hfill \\ \end{aligned} $$
(9)

Obviously the choices \( \alpha \beta = 1 \) and \( \gamma = - 1 \) with δ ≠ 1 reduces Eq. (9) to:

$$ (( - \delta + 1)(1 + \alpha ))^{2} t\ddot{Q} - 2(( - \delta + 1)(1 + \alpha ))^{2} \dot{Q} + t^{3} Q = 0 , $$
(10)

which is the GEFE of the first-kind. The solution is given by:

$$ Q(t) = \frac{{t^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}{{(( - \delta + 1)(1 + \alpha ))^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}} }}\left( {c_{3} \sqrt 2 \varGamma \left( {\frac{5}{4}} \right)J_{{ - \tfrac{3}{4}}} \left( {\frac{{t^{2} }}{2( - \delta + 1)(1 + \alpha )}} \right) + c_{4} \frac{{\varGamma \left( {\frac{7}{4}} \right)}}{\sqrt 2 }J_{{\tfrac{3}{4}}} \left( {\frac{{t^{2} }}{2( - \delta + 1)(1 + \alpha )}} \right)} \right), $$
(11)

and the GDO takes the special form:

$$ {\text{D}}_{t}^{{(\alpha ,{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-0pt} \alpha }),( - 1,\delta )}} = \frac{d}{dt} + \left( {\alpha \dot{q}^{s} + \frac{{q^{s} }}{t}} \right)\frac{\partial }{{\partial q^{s} }} - \left( {\ddot{q}^{s} + \delta \frac{{\dot{q}^{s} }}{t}} \right)\frac{\partial }{{\partial \dot{q}^{s} }}. $$
(12)

The Lagrangian of the theory which in principle holds second-order derivative is reduced to a function of first-order derivative and takes the form:

$$ {\text{L}}({\text{D}}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} {\text{D}}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} ,{\text{D}}_{t}^{(\alpha ,\beta ),(\gamma ,\delta )} q_{0,k}^{s} q_{0,k}^{s} ,t) \equiv {\text{L}}(\dot{Q},Q,t) = \frac{1}{2}((\alpha + 1)(1 - \delta ))^{2} \frac{{\dot{Q}^{2} }}{{t^{2} }} - \frac{1}{2}Q^{2} , $$

which coincides with the standard Lagrangian [9]. As a numerical illustration, we set: α = 1, δ = 1/12.2 and we plot in Figs. 3 and 4 the solutions of Eq. (10) for two different initial conditions:

Fig. 3
figure 3

The numerical solutions of Eq. (10) for Q(1) = 1 and \( \dot{Q}(1) = 0 \)

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Fig. 4
figure 4

The numerical solutions of Eq. (10) for Q(1) = 0 and \( \dot{Q}(1) = 1 \)

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Figures 3 and 4 prove the existence of oscillatory behavior for \( V(Q) = \tfrac{1}{2}Q^{2} \) in agreement with [10], although the frequency of the system produce a kind of weird phase plot trajectories that bear a resemblance to a disordered motion. However, it should be pointed out that Eq. (10) may be written for \( t \ne 0 \) as:

$$ \ddot{Q} + \xi (t)\dot{Q} + \omega^{2} (t)Q = 0, $$
(13)

where

$$ \xi (t) = - \frac{{2(( - \delta + 1)(1 + \alpha ))^{2} }}{t}, $$
(14)

and

$$ \omega^{2} (t) = \frac{{t^{2} }}{{(( - \delta + 1)(1 + \alpha ))^{2} }}. $$
(15)

Equation (13) describes a harmonic oscillator with simultaneously a time-dependent frequency ω(t) and a time-dependent mass M(t)such that:

$$ \xi (t) = \frac{d}{dt}\left( {\ln M(t)} \right), $$
(16)

with \( M(t) \propto t^{{ - 2(( - \delta + 1)(1 + \alpha ))^{2} }} \) [12]. The oscillator is hence characterized by a decaying mass and an increasing frequency.

If we choose V(Q) = e nQ, n ∊ ℜ, then we obtain for αβ = 1, γ = − 1 and δ ≠ 1the nonlinear differential equation:

$$ (( - \delta + 1)(1 + \alpha ))^{2} t\ddot{Q} - 2(( - \delta + 1)(1 + \alpha ))^{2} \dot{Q} - nt^{3} e^{nQ} = 0, $$
(17)

which is the GEFE of the second-kind studied in [7, 13] and its integrability was discussed in [9]. More generally, we can write Eq. (17) as:

$$ \ddot{Q} - \frac{2}{t}\dot{Q} - \frac{{t^{2} }}{{(( - \delta + 1)(1 + \alpha ))^{2} }}\frac{dV}{dQ} = 0, $$
(18)

which in principle may be derived from the Lagrangian:

$$ L_{s} = \frac{1}{2}(( - \delta + 1)(1 + \alpha ))^{2} \frac{{\dot{Q}^{2} }}{{t^{2} }} + V(Q), $$

as it is expected.

4 Conclusions

To conclude, we have introduced in this communication a new method to derive the GEFE starting from a GDO and based on Lagrangian analysis. Lie symmetry analysis and Noether symmetry classification based on Eq. (12) will the subject of a forthcoming paper yet we believe that the GDO represented by Eq. (12) will smooth the progress toward finding the corresponding Noether operators. It is noteworthy that Eqs. (1) and (12) may be generalized as well for higher-order derivatives larger than 2 as follows:

$$ {\text{D}}_{t}^{{\left( {\alpha_{1} , \ldots ,\alpha_{n} } \right),\left( {\beta_{1} , \ldots ,\beta_{n} } \right)}} = \frac{d}{dt} + \sum\limits_{i = 1}^{n} {\alpha_{i} \left( {\left( {q^{s} } \right)^{(i)} + \beta_{i} \frac{{\left( {q^{s} } \right)^{(i - 1)} }}{t}} \right)\frac{\partial }{{\partial \left( {q^{s} } \right)^{(i - 1)} }}} . $$

It will be of interest to explore in the future a variety of higher-order dynamical models derived from a Lagrangian description holding such types of GDO. Their symmetries and stabilities/instabilities deserve to be studied further. We expect that these models will have interesting properties both at classical and quantum levels.