Appl. Math. Mech. -Engl. Ed. 31(7), 911–916 (2010) DOI 10.1007/s10483-010-1325-x c Shanghai University and Springer-Verlag Berlin Heidelberg 2010
Applied Mathematics and Mechanics (English Edition)
Group classification for path equation describing minimum drag work and symmetry reductions∗ M. PAKDEMIRLI,
Y. AKSOY
(Department of Mechanical Engineering, Celal Bayar University, 45140, Muradiye, Manisa, Turkey) (Communicated by Xing-ming GUO)
Abstract The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered (Pakdemirli, M. The drag work minimization path for a flying object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223(5), 1113– 1116 (2009)). The Lie group theory is applied to the general equation. The group classification with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates. Key words
minimum drag work, Lie group theory, group classification
Chinese Library Classification O175.1, O35 2000 Mathematics Subject Classification 34A05, 34A34, 34C14, 76M60
1
Introduction
Drag forces are the major source of energy loss for objects moving in a fluid medium. The minimization of work due to the drag force may reduce fuel consumption. Several assumptions can be made for this purpose. One way to minimize the drag work may be to search for an optimum path. The drag force depends on the density of fluid, the drag coefficient, the crosssectional area, and the velocity. If all parameters are assumed to be constant, then the minimum drag work path would be a linear path. However, these parameters change during motion. A general case in which all parameters are the functions of altitude is treated. Using variational calculus, the differential equation describing the optimum path was derived recently[1] . A special case in which there was the exponential dependence on the altitude coordinate was treated, and an exact solution was presented for the problem. The same problem was also solved by the homotopy analysis method[2] . A more theoretical approach is followed in this study. The combination of the altitudedependent parameters, such as the density, the drag coefficient, the cross sectional area, and the velocity, is expressed as a single arbitrary function. The path equation corresponding to the minimum drag work given in [1] is then treated using the Lie group theory[3–5] . The symmetries of the equation are calculated. The group classification is performed by the calculation of the principal Lie algebra and its extensions for special forms of the arbitrary function. The symmetries are used in two different ways: (i) Some group-invariant solutions are constructed. ∗ Received Oct. 13, 2009 / Revised Apr. 9, 2010 Corresponding author M. PAKDEMIRLI, Professor, Ph. D., E-mail:
[email protected]
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(ii) The reduction of order by the canonical coordinates is performed. The literatures on group classifications of ordinary differential equations are extensive, and a review of all results is beyond the scope of this study. The reader is referred to the survey paper of Mahomed[6] for details.
2
Equation of motion and Lie group theory
The differential equation describing the path of the minimum drag work has been derived previously[1], i.e., y −
f (y) (1 + y 2 ) = 0, f (y)
(1)
where y = y(x) is the altitude function, and f (y) = ρ(y)Cd (y)A(y)U (y)2 . The density ρ, the drag coefficient Cd , the cross sectional area A, and the velocity of the object U are the functions of the y coordinate. The Lie group theory[3–5] is applied to the equation to determine symmetries. First, the higher order variables are defined as y1 = y ,
y2 = y .
(2)
In terms of the new variables, the equation is expressed as follows: y2 −
f (y) (1 + y12 ) = 0, f (y)
(3)
where the prime in f indicates the differentiation with respect to y. The prolonged generator is X = ξ(x, y)
∂ ∂ ∂ ∂ + η(x, y) + η1 (x, y, y1 ) + η2 (x, y, y1 , y2 ) , ∂x ∂y ∂y1 ∂y2
(4)
where η1 and η2 are given as[3–5] η1 = ηx + (ηy − ξx )y1 − ξy y12 , η2 = ηxx + (2ηxy − ξxx )y1 + (ηyy −
(5) 2ξxy )y12
−
ξyy y13
+ (ηy − 2ξx )y2 − 3ξy y1 y2 .
(6)
The application of (4) to (3) yields the invariance condition η2 −
f (y) f (y) 2y1 η1 − (1 + y12 )η = 0. f (y) f (y)
(7)
y2 is inserted from (3) to (6), and (5) and (6) are substituted into (7). This yields a block of equations that can be separated with respect to y1 . The resulting equations are f (y) f (y) −η = 0, f (y) f (y) f (y) = 0, 2ηxy − ξxx − (3ξy + 2ηx ) f (y) f (y) f (y) ηyy − 2ξxy − ηy −η = 0, f (y) f (y) f (y) ξyy + ξy = 0. f (y) ηxx + (ηy − 2ξx )
(8) (9) (10) (11)
Group classification for path equation describing minimum drag work and symmetry reductions 913
3
Group classification
To find the infinitesimals, (8)–(11) should be solved. The principal Lie algebra corresponding to arbitrary f (y) and specific forms of this function, for which there are extensions of the algebra, are determined. (11) is integrated first as follows: dy + b(x). (12) ξ = a(x) f (y) Integrating (10) with respect to y, i.e., ηy − 2ξx − η
f (y) = c (x), f (y)
and solving for η yield 1 dy dy + 2a (x)f (y) dy + d(x)f (y). η = c (x) + 2b (x) f (y) f (y) f (y) f (y) The infinitesimals ξ and η are inserted into (9). Then, f (y) dy 2c (x) + 3b (x) + 3a (x) − 3a(x) 2 =0 f (y) f (y)
(13)
(14)
(15)
is obtained. Differentiating the above equation with respect to y gives the following classifying relationship: f (y) a (x) − a(x) 2 = 0. (16) f (y) f (y) Several cases should be treated separately. 3.1 Arbitrary f (y) For arbitrary f (y), a(x)=0 from (16). The solution satisfying (8)–(11) yields ξ = a,
η = 0.
(17)
The above result corresponds to the principal Lie algebra. The principal Lie algebra is a one parameter finite Lie group of transformations. 3.2 f (y)=1/(k1 y+k2 ) This specific choice corresponds to f /f 2 = 0. From (16), a(x)=a1 x+a2 . After some manipulations, (8)–(11) yield ⎧ 1 1 1 ⎪ 2 ⎪ ξ = (ax + b) k y + k y + k1 ax3 + cx2 + dx + e, 1 2 ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ 1 2 4 1 1 k y2 + k y ⎨ k y + k1 k2 y 3 + k22 y 2 1 2 η= (c − 3k1 b) x + g + 2d 2 +a4 1 (18) 2 k1 y + k2 k1 y + k2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 1 k 2 ax4 − 14 k1 (c + k1 b) x3 + 12 k1 gx2 + hx + ⎪ ⎪ ⎩ + 4 1 . k1 y + k2 The algebra for this specific case can be extended to an eight parameter finite Lie group of transformations. 3.3 f (y)=k Another special form is f (y)=k, where k is a constant. For this case, (8)–(11) finally yield ξ = axy + by + cx2 + dx + e, (19) η = cxy + gy + ay 2 + hx + . The algebra consists of an eight parameter finite Lie group of transformations.
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3.4 f (y)=k exp(αy) The solution of (8)–(11) yields an eight parameter Lie group of transformations
ξ = (a cos αx + b sin αx)e−αy + c cos 2αx + d sin 2αx + e, η = (a sin αx − b cos αx)e−αy + c sin 2αx − d cos 2αx + g + (h cos αx + sin αx)eαy .
(20)
The results of all cases are summarized in Table 1 for convenience. Table 1 Function
Group classification summary
Symmetry ξ =a
Arbitrary f (y)
η=0
“1 ” 1 1 ξ = (ax + b) k1 y 2 + k2 y + k1 ax3 + cx2 + dx + e 2 2 2 1 2 4 “1` ” 1 k y2 + k y ´ k y + k1 k2 y 3 + k22 y 2 1 2 η= +a 4 1 c − 3k1 b x + g + 2d 2 2 k1 y + k2 k1 y + k2 ` ´ − 14 k12 ax4 − 14 k1 c + k1 b x3 + 12 k1 gx2 + hx + + k1 y + k2
f (y)=1/(k1 y+k2 )
ξ = axy + by + cx2 + dx + e
f (y)=k
η = cxy + gy + ay 2 + hx + ξ = (a cos αx + b sin αx)e−αy + c cos 2αx + d sin 2αx + e η = (a sin αx − b cos αx)e−αy + c sin 2αx − d cos 2αx + g
f (y)=k exp(αy)
+(h cos αx + sin αx)eαy
Note that for a second-order equation, the symmetry can be 0, 1, 2, 3, or 8[6] . In all special cases, the principal Lie algebra can be extended from 1 to 8. The above symmetries can be used in two different ways: (i) The group-invariant solutions can be constructed. (ii) The reduction of order by the canonical coordinates can be performed.
4
Group-invariant solutions
Some group-invariant solutions are constructed from the symmetries. Two of the special forms are used. 4.1 f (y)=k exp(αy) Since several parameters (8 parameters) exist, a number of solutions can be constructed from the symmetries. (i) Parameters a and b If parameters a and b are selected and the remaining ones are set to be zero, the determining equation is dy dx = . (a cos αx + b sin αx)e−αy (a sin αx − b cos αx)e−αy
(21)
Integrating (21) yields y=−
1 ln(a cos αx + b sin αx) + c1 , α
(22)
Group classification for path equation describing minimum drag work and symmetry reductions 915
which satisfies the original equation. A similar solution is presented in [1–2]. (ii) Parameters g and e For a selection of parameters g and e, the determining equation is dy dx = . e g
(23)
y = mx + n,
(24)
The integration yields
where m=g/e. Substituting it into the original equation yields m=±i. Hence, the groupinvariant solution is y = ±ix + n, which is a trivial solution. (iii) Parameters e, h, and For parameters e, h, and , the determining equation is dx dy = . e (h cos αx + sin αx)eαy
(25)
(26)
Selecting e=1 and solving the determining equation yield 1 y = − ln( cos αx − h sin αx − c1 ), (27) α which satisfies the original equation for the exponential function form. This group-invariant solution is similar to (22). 4.2 f (y)=1/(k1 y+k2 ) Two different selections are considered. (i) Parameter d If parameter d is selected and the remaining ones are set to be zero, the determining equation is dy dx = k y2 +2k y . (28) 1 2 x k1 y+k2
Integrating and substituting it into the original equation yield the group-invariant solution k1 y 2 + 2k2 y + k1 x2 = 0,
(29)
k22 − k12 x2 . k1
(30)
from which y can be solved as y=
−k2 ∓
(ii) Parameters e, h, and For a selection of parameters e, h, and , the determining equation is dx = e
dy hx+ k1 y+k2
.
(31)
Without loss of generality, e=1 is chosen. Integrating and substituting (31) into the original equation yield k1 y 2 + 2k2 y + k1 x2 − 2x + c1 = 0, which is similar to the previous solution −k2 ∓ k22 − k1 (k1 x2 − 2x + c1 ) y= . k1
(32)
(33)
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Reduction by canonical coordinates
For ordinary differential equations, one of the strategies is to reduce the order of the equation by defining the canonical coordinates[3] . One example for case f (y)=1/(k1 y+k2 ) is considered. Parameter is selected. The corresponding generator is X=
∂ 1 . k1 y + k2 ∂y
(34)
The canonical coordinates r and s are determined from the equations[3] Xr = 0,
Xs = 1.
(35)
The solutions are 1 k1 y 2 + k2 y. 2 In terms of the canonical coordinates, the original equation reduces to r = x,
s=
d2 s + k1 = 0. dr2
(36)
(37)
Solving (37) and returning back to the original equation yield k1 y 2 + 2k2 y + k1 x2 − 2c1 x + c2 = 0.
(38)
This solution is the same solution given in (32).
6
Concluding remarks
The path equation describing the minimum drag work first proposed in [1] is reconsidered. The group classification of the equation is performed with respect to an altitude-dependent arbitrary function. The function is a combination of the density, the drag coefficient, the cross sectional area, and the velocity. The principal Lie algebra and its extensions for the specific forms are found. The symmetries are used in two different ways: (i) The group-invariant solutions are constructed. (ii) The order of the equation is reduced by defining the canonical coordinates, and a solution is presented. Specific boundary value problems are not treated in this study. See examples of boundary value problems in [1–2] for the exponential form of the function.
References [1] Pakdemirli, M. The drag work minimization path for a flying object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223(5), 1113–1116 (2009) [2] Abbasbandy, S., Pakdemirli, M., and Shivanian, E. Optimum path of a flying object with exponentially decaying density medium. Zeitschrift f¨ ur Naturforschung A 64a(7-8), 431–438 (2009) [3] Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer-Verlag, New York (1989) [4] Stephani, H. Differential Equations: Their Solution Using Symmetries, Cambridge University Press, New York (1989) [5] Ibragimov, N. H. CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, CRC Press, Boca Raton (1994) [6] Mahomed, F. M. Symmetry group classification of ordinary differential equations: survey of some results. Mathematical Methods in the Applied Sciences 30(16), 1995–2012 (2007)
