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Lagrangian Representations and Solutions of Modified Emden-Type Equations Aparna Saha1 and Benoy Talukdar1
1
Department of Physics, Visva-Bharati University, Santiniketan 731235, India DOI 10.13140/RG.2.1.2683.0323
Keywords: variation of parameters, factorization method, modified Emden-type equations, solutions, Lagrangian representation.
ABSTRACT. We derive two novel methods to construct solutions of the physically important modified Emden-type ethod of variation of parameters, we make use of a trivial particular solution of the problem to construct expressions for the nontrivial general solutions. Secondly, we judiciously adapt the factorization method of differential operators to present solutions of certain oscillator equations obtained by adding a linear term to the MEEs. We provide Lagrangian and Hamiltonian formulations of these equations in order to look for another useful theoretical model for solving MEEs.
PACS numbers: 02.30.Hq, 02.30.Ik, 05.45.-a
1. Introduction. The modified Emden-type equation [1] is given by:
,
where .
and
,
(1)
are arbitrary parameters, and overdots denote differentiation with respect to time
This equation plays a role in several applicative contexts including the study of equilibrium configurations of a spherical gas cloud with mutual interaction between the molecules subject to laws of thermodynamics [2]. An important mathematical observation on (1) is that only for it possess eight parameter Lie point symmetries [3] and is exactly solvable. In the recent past Chandrasekar, Senthilvelan and Lakshmanan [4] made use of an extension of the semi-decision algorithmic method [5] supplemented by a suitable Hamiltonian theory to demonstrate that (1) is integrable for arbitrary values of and . Iacono [6] proved the integrability of (1) by mapping it nonlinear differential equations once their forms are found by a specific method [7]. However, we note that, as opposed to the theory of linear differential equations, the nonlinear theory is highly fragmented. Thus the identification of a new mathematical procedure and demonstration of its usefulness in solving nonlinear equations will always remain interesting provided the method of solution is uncomplicated enough to be appreciated by a wide variety of audience. The present work is an effort in this direction. In particular, we shall present two independent methods to solve MEEs one b suitable factorization of the differential operators [1] that give the equations. To the best of our knowledge these methods have not hitherto been discussed in the literature. In addition, we provide MMSE Journal. Open Access www.mmse.xyz
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for these equations a suitable Lagrangian and Hamiltonian formulation which paves the path to introduce still another uncomplicated method for their solution. In section 2 we apply the first method to solve (1) for and with a view to compare our solutions with other available results [3, 4]. In section 3 we provide a solution of the nonlinear oscillator equation
,
where
(2)
by the second method. -type differential equation of the form:
.
(3)
With
and
.
(4)
We shall see in the course of our study that form in (3) is very suitable to develop the second method. It is interesting to note that the second method is also applicable to (1). The system (3) has been found to exhibit unusual nonlinear dynamical properties [8] in that for the frequency of oscillation is completely independent of its amplitude as found in the case of a linear harmonic oscillator. We devote section 4 to study the Lagrangian and Hamiltonian representations of our equations and demonstrate that the Jacobi integrals [9] can be used to solve the equations for only. The equation with needs altogether a separate consideration. Here we also introduce a family of Emden-type equations and study their Hamiltonian structure. Finally, in section 5 we make some concluding remarks with a view to summarize our outlook on the present work. 2. Modified Emden-type equations (MEEs). For simplicity of presentation we write two different forms of (1) as follows:
.
(5)
0.
(6)
And
x 4kxx 2k 2 x 3
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Comparing (5) and (6) with (1) it is evident that these equations refer to the cases
and
respectively. It is interesting to note that
,
(7)
where c an arbitrary constant represents a particular solution of both (5) and (6). We postulate that the general solutions of these equations are given by:
,
(8)
where the constant c in (7) has been replaced by a function of , namely, The replacement sought by us forms the basic philosophy of parameters [1] for finding general solution of second-order linear differential equations from the particular solutions. We now use the following steps to solve the above MEEs. (i) We substitute (8) either in (5) or (6) to obtain an ordinary differential equation for
and then
(ii) Use the solution of it in (8) to find a general solution of the chosen MEE. (a) Equation (5). Substituting (8) in (5) we obtain the differential equation for
.
as
(9)
Albeit nonlinear, (9) can be solved analytically to get
,
where For
and
(10)
are constants of integration. we have
(11)
Which in conjunction with (8) leads to the well-known general solution
,
(12)
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as given in [3]. (b) Equation 6. In this case an equation analogous to that of (9) reads
.
(13)
Unlike (9), (13) cannot be integrated analytically. We thus try to find a solution of it by introducing a change of variable written as
.
(14)
From (13) and (14) we get
,
which is again almost of the same form as that in (13). However, if we choose to work with (15) can be solved to get
1 4
g (t )
where, as before,
and
c1 k2
(15)
,
c1 2
4 erf
1
2i e 2 k (t c 2 )
,
e
(16)
are arbitrary constants of integration;
stands for the inverse error function. Since implies, that is not the most general one.
, the reciprocal of the result in (16) refers to a solution of (6) which
In [4] Chandrasekar et. al solved (6) by treating it as a Hamiltonian system and then taking recourse to the use of a canonical transformation. Interestingly, our solution of (6) for c1 1 and is in exact agreement with that of (16) in their work where it was also taken not as a general solution. The general solution was found by substituting the result in the equation for the canonical transformation of . We follow a different viewpoint to find the general solution. The function
represents the solution of (6) only when the first derivative of the variable
parameter is fixed at . The question is how can one relax this condition and find a general solution of the equation. In this context we note, that the transformation:
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,
(17)
reduces (5) to the linear form
,
(18)
and the solution of (18) when substituted in (17) gives the exact solution of (5). Unfortunately, no such transformation can be found to reduce (6) to the linear form. In this situation we suggest that the choice:
(19)
together with (17) will at least lead to an approximate general solution of (6). Following this viewpoint we obtain: (20)
where:
.
(21)
The solution (21) of (6) as obtained by us is in exact agreement with that found in [4] where the authors claimed that their result is exact. 3. Nonlinear oscillator equation. As given in (7) we could write a simple form for the particular solution of (5) or (6) only by inspection. For the oscillator equation (2) written as
(22)
it is, however, not possible to write a similar particular solution. As a result we take recourse to the -type equations to construct the general solution of (22). Equation (3) can be written in the factored form
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.
(23)
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It is rather straightforward to deduce that
.
(24)
and
.
(25)
Since (3) has been factored as (23), a particular solution of the equation can be found from the firstorder equation ,
.
(26)
In applying the factorization method to problems of physical interest one proceeds by writing and with some pre-factors so as to satisfy (25). Understandably, if the pre-factor for is taken as , that for should be .The chosen results for and when substituted in (24) leads to an equation for . One can assume that is a c- number. Alternatively, one can regard as a function of say .It is straightforward to see that (i) satisfies an algebraic equation while (ii) satisfies a first-order differential equation. In two interesting publications Rosu and Cornejoas a c number and found that the above factorization method provides an efficient tool to obtain particular postulate that the solution of the differential equation for in conjunction with (26) gives the general solution of (3). We first verify the conjecture with attention to (5) which is also of the and . Thus we introduce:
and
.
(27)
.
(28)
to write (24) as
The solution of (28) can be found as
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.
It is now straightforward to substitute the expression for to get the general solution of (5) as given in (12). For the nonlinear equation in (22) the value of
(29)
in (26) and solve the resulting equation
is same as that in (5) but
Thus we take as given in (27). Here in (28) is only slightly different and is given by
.
. In this case an equation similar to that
.
(30)
We obtained the solution of (30) as:
.
(31)
For , (30) coincides with (28). But on the same limit (31) does not go over to (29). Thus before proceeding further to use (31) to deal with the oscillator equation one would like to check if
(32)
When used in (26) leads to the solution of (5). We found that this is indeed the case. The factorization method also works for (6) for which . In this case we, however, need to invoke the set of approximations as used in deriving the result in (20). Coming back to the oscillator equation we now substitute (31) in (26) and solve the resulting linear first-order differential equation to get
(33)
Where:
n1 n2
2
(1 3 ) sin(2 2 3
2
sin(
t) , t) .
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(34a) (34b)
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And
.
It is of interest to note, that the real part of
(34c)
corresponds to the solution of (22) given in [8].
4. Lagrangian and Hamiltonian formulations and MEEs. Traditionally, a Lagrangian function L is called natural (mechanical type) when it is of the form where is a quadratic kinetic term and is a potential function. For the equation of motion of a free particle moving in the
x direction the natural or standard Lagrangian is given by and
. However, one can verify that
via the Euler-Lagrange equations reproduce the equation of motion of the
free particle. Thus, these two also equations represent admissible Lagrangians for the motion of a free particle. Following this viewpoint we introduce:
(35)
And
,
(36)
and verify that (35) and (36) when substituted in the appropriate Euler-Lagrange equation lead to (5) and (6). A Lagrangian for (22) similar to that in (35) reads
.
(37)
One can check that
(38)
Is associated with the oscillator equation:
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(39)
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From the expressions for
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presented here it is clear that Emden-type equations for
follow from the inverse type of Lagrangians while those for follow from a logarithmic class. Both inverse and logarithmic class of Lagrangians do involve neither the kinetic energy term nor the potential function. As a result they were qualified as nonstandard [12]. The Jacobi integral for the one-dimensional motion can be written as [9]
.
For the equations in (5) and (6) the expressions for written as
(40)
obtained from (35), (36) and (40) can be
(41)
And .
(42)
For (41) we can express as a function of and and solve the resulting equation to get the solution of (5), which agrees with that given in (12). On the other hand in (42) occurs in essentially non-algebraic way and does not permit one to write as a function of and . This is precisely the reason why the MEE for
poses inordinate complications to find its solution.
Let us generalize (35) and (36) to write
(43)
And
.
(44)
The equations of motion corresponding to (43) and (44) are:
(45)
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and
(46)
respectively. Equations (45) and (46) give the equations of motion for the over-damped and critically damped harmonic oscillators for with appropriate non-standard Lagrangians obtained from (43) and (45) respectively. This provides an example of the non-standard Lagrangian representation for a system which follows from an explicitly time-dependent standard Lagrangian [13]. The non-standard Lagrangians are explicitly time independent by choice although the associated equations of motion involve velocity terms. Thus the Hamiltonians corresponding to both will refer to the total energies of the systems represented by (45) and (46). Using the Legendre transformation between the Hamiltonian and Lagrangian
,
(47)
we obtain the Hamiltonian functions
(48)
And
(49)
For (45) and (46). Here
and
are given by
(50)
And
.
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(51)
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If we substitute back the values of respectively the Jacobi integrals explicitly
and and
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in (50) and (51) in (48) and (49) we shall get for the equations of motion in (45) and (46). Written
(52)
And
.
For
(53)
(52) and (53) reduce to the results in (41) and (42).
Summary. We introduced, in this paper, two methods to solve modified Emden-type equations. The merit of the approaches developed by us lies in their simplicity and effectiveness to deal with the problem. The first method, where we obtained the general solutions from a particular solution satisfying the asymptotic boundary condition only has a deep root in the scientific literature. For example, Sommerfeld [14] used the exact particular solution to construct a beautiful well behaved analytic solution for the famous Thomas-Fermi equation (TF).The method of Majorana [15] for constructing semi-analytical solution of the TF equation also provides another example in respect of this. In the second method we modified a factorization technique of differential operators -type equations for finding particular solutions. The modification used by us -type oscillator. References [1] E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956). [2] S. Chandrasekher, An Introduction to the study of Stellar Structure (Dover Pub., New York, 1957). [3] P. G. Leach, J. Math. Phys. Vol. 26, 2310 (1985). [4] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the general solution for the modified Emden-type equation, J. Phys. A : Math. Theor. 40, 4717 (2007). [5] M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279, 215 (1983). [6] R. Iacono 2008, Comment on 'On the general solution for the modified Emden-type equation, J. Phys. A : Math. Theor. 41, 068001(2008). [7] V. K. general solution for the modified Emden type equation, J. Phys. A : Math. Theor. 41, 0680002(2008). -type nonlinear oscillator, Phys. Rev. E 72, 066203(2005). MMSE Journal. Open Access www.mmse.xyz
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[9] H. Goldstein Classical Mechanics, 7th reprint (Narosa Pub. House, New Delhi, India, 1998). [10] H. C. Rosu and O. Cornejononlinearities, Phys. Rev. E 71, 046607(2005). [11] O. Cornejosolutions, Prog. Theor. Phys. 114, 533 (2005). ar secondorder Riccati systems: one-dimensional integrability and two-dimensional superintegrability, J. Math. Phys. 46, 062703 (2005). [13] P. Caldirola Nuovo. Cim. 18. 393 (1941); E. Kanai Prog. Theor. Phys. 20,440(1948). [14] A. Sommerfeld, 1932 Z. Phys. 78,19 (1932). [15] S. Esposito, Majorana solution physics/0111167v1(2001) [physics. atom-ph]
of
the
Thomas-Fermi
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equation,
arXiv
: