SOME INVARIANTS CONNECTED WITH EULER-LAGRANGE EQUATIONS

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1 U.P.B. Sci. Bull., Series A, Vol. 71, Iss.2, 29 ISSN: SOME INVARIANTS CONNECTED WITH EULER-LAGRANGE EQUATIONS Irena ČOMIĆ Lucrarea descrie mai multe tipuri de omogenitate definite în spaţiul Osc k M. Sunt prezentate ecuaţiile Euler-Lagrange pentru diverse tipuri de variabile, iar legătura dintre Lagrangienii spaţiilor L (kn şi Hamiltonienii spaţiilor H (kn este precizată. Ecuaţiile diferă semnificativ, funcţie de paritatea ordinului k. Se arată că în Osc 3 M, doar E i şi E 3 i sunt covectori. Se demonstrează că ecuaţiile Euler-Lagrange au caracter invariant, şi se determină un nou câmp scalar. In Osc k M, different kinds of homogeneity are defined, and properties of homogeneous functions are pointed out. The Euler-Lagrange equations for different types of variables in Osc k M are given; they are coordinate invariant, and have different forms for k = 2l and for k = 2l+1. The connection between Lagrangians in the space L (kn and Hamiltonians in H (kn, is outlined. It is shown that in Osc 3 M, only Ei and Ei 3 are covectors, and one new scalar field is determined. Keywords: Homogeneity, Euler-Lagrange equations, Lagrangians and Hamiltonians of higher order, Legendre transformation. AMS Subject Classification: 53B4, 53C6. 1. Different kinds of homogeneity. Euler s equations Let us denote by Osc k M the k-th order tangent space of M. In some local chart the point u Osc k M has coordinates (x i = y i, y 1i,..., y ki, i = 1, n, y Ai = da y i, A =, k. The transformation group is well known ([1], [2], [3], dt A [4], etc. Some properties of Osc 3 M are given by (41-(43. Definition 1. The scalar function F (y i, y 1i,..., y ki is homogeneous of first kind and order r (r Z if F (y i, λy 1i, λ 2 y 2i,..., λ k y ki = λ r F (y i, y 1i,..., y ki, λ >. (1 Professor, Faculty of Technical Sciences, University of Novi Sad, Serbia

2 4 Irena Čomić It is homogeneous of second kind and order r if F (y i, λy 1i, λy 2i,..., λy ki = λ r F (y i, y 1i,..., y ki. (2 It is homogeneous of third kind and order r with respect to variable y Ai, if F (y i, y 1i,..., y (A 1i, λy Ai, y (A+1i,..., y ki = λ r F (y i, y 1i,..., y ki. (3 It is homogeneous of fourth kind and order r with respect to variable y Ai if F (y i, y 1i,..., y (A 1i, λ A y Ai, y (A+1i,..., y ki = (λ A r F (y i, y 1i,..., y ki. (4 Theorem 1. For the homogeneous function F (y i, y 1i,..., y ki of first kind and order r the following Euler type equations are valid: y 1i y1i + y 2i 2y2i + + y ki kyki = rf, r (5 y 1i y 1j y1i y 1j F y 2i y 2j 2 2y2i y 2j + + [ y 1i y 1 2j 2y1i y 2j + 2 F y 1i y 1 3j 3y1i y 3j + 2 F y ki y kj k kyki y kj + (6 + 2 F y 1i y 1 kj ky1i y kj 2 ] F + + y (k 1i y (k kj 1ky(k 1i y kj + y 2 2i 1y2i + y 3 3i 2y3i + + y k(k ki 1yki = r(r 1F, r 1. Proof. If we take the first derivative of (1 with respect to λ, we get A=1 (λ A y Ai AλA 1 y Ai = rλ r 1 F. (7 If we put λ = 1 in (7, we get (5. If we take the derivative of (7 with respect to λ, we get B=1 A=1 (λ A y Ai (λ B y Bi AλA 1 y Ai Bλ B 1 y Bi + (8 A=2 If we put λ = 1 in (8, we obtain (6. (λ A y Ai A(A 1λA 2 y Ai = r(r 1λ r 2 F.

3 Some invariants connected with Euler-Lagrange equations 5 Theorem 2. For the homogeneous function F (y i, y 1i,..., y ki of second kind and order r the following equations are valid: y 1i y1i + y 2i y2i + + y ki yki = rf, r (9 y 1i y 1j y1i y 1j + 2 [ y 1i y 2j y1i y 2j + + r(r 1F, r 1. 2 F y 2i y 2j y2i y 2j F y 1i y kj y1i y kj F y ki y kj yki y kj + (1 2 ] F y (k 1i y kj y(k 1i y kj = Proof. The first and second derivative of (2 with respect to λ give: A=1 (λy Ai yai = rλ r 1 F (11 B=1 A=1 (λy Ai (λy Bj yai y Bj = r(r 1F. (12 For λ = 1 the explicit form of (11 and (12 are (9 and (1, respectively. The higher order derivatives give more complicated formulae. Theorem 3. If the function F (y i, y 1i,..., y Ai,..., y ki is homogeneous of third kind and order r with respect to the variable y Ai, then y Ai yai = rf, y Ai y Aj yai y Aj = r(r 1F,..., (13 r F y Ai 1 y Ai 2... y Ai r y Ai 1 y Ai 2... y Air = r!f. Proof. The proof is similar to the proof of previous Theorems. Theorem 4. If the function F (y i, y 1i,..., y Ai,..., y ki is homogeneous of fourth kind and order r with respect to variable y Ai, then y Ai AyAi = raf, A = 1, k (14 y Ai y Aj A2 y Ai y Aj + y Ai A(A 1yAi = ra(ra 1F. (15

4 6 Irena Čomić Proof. The first and second derivatives of (4 with respect to λ are (λ A y Ai AλA 1 y Ai = raλ ra 1 F (16 (λ A y Ai (λ A y Aj (AλA 1 2 y Ai y Aj + (17 (λ A y Ai A(A 1λA 2 y Ai = ra(ra 1λ ra 2 F. For λ = 1, (16 and (17 give (14 and (15 respectively. Remark 1. For the scalar function on Osc 1 M, i.e. F = F (x, y = F (y i, y 1i the definitions of first, second and third kinds of homogeneity coincide. Example 1. The function F (y i, y 1i, y 2i = a i(xy 2i b j (xy 1j (y i = x i is homogeneous of first kind and order 1, or of second kind and order, namely F (y i, λy 1i, λ 2 y 2i = a i(xλ 2 y 2i b j (xλy 1j = λf (y i, y 1i, y 2i F (y i, λ 1i, λy 2i = F (y i, y 1i, y 2i. For this function, the relations (5 and (9 give Example 2. The function a i(xy 2i b j (x(y 1j 2 y1j + a i(x b j (xy 1j 2y2i = a i(xy 2i b j (xy 1j a i(xy 2i b j (x(y 1j 2 y1j + a i(x b j (xy 1j y2i =. F (y i, y 1i, y 2i = a i (xy 2i + b ij (xy 1i y 1j + c i(xy 3i d j (xy 1j

5 Some invariants connected with Euler-Lagrange equations 7 is homogeneous of first kind and order 2. For this example (5 and (6 reduce to: y 1i y1i + y 2i 2y2i + y 3i 3y3i = 2F y 1i y 1j y1i y 1j + 2 y 1i y 3j 3y1i y 3j + y 2i 2y2i + y 3i 6y3i = 2F. Straightforward calculations lead to these equations, Example 3. The function F (x, y (1, y (2, y (3 = a ijk (xy 1i (y 2j 3 (y 3k 2 is homogeneous of third kind and order 3 with respect to y (2 homogeneous of third kind and order 2 with respect to y (3, namely and F (x, y (1, λy (2, y (3 = λ 3 F (x, y (1, y (2, y (3 F (x, y (1, y (2, λy (3 = λ 2 F (x, y (1, y (2, y (3. For this example, (13 has the form y 2i y2i = 3F, y 2i y 2j y2i y 2j = 3 2F y 3i y3i = 2F, Example 4. The vector field [5] y 3i y 3j y3i y 3j = 2 1F. z 2i = y 2i γ i j ky 1j y 1k in L (2n is homogeneous of first kind and order 2. The scalar functions: α(x, y 1, y 2 = [a ij (xz 2i z 2j ] 1/2, β(x, y 1, y 2 = b i (xz 2i are homogeneous of first kind and order 2. The fundamental functions F (x, y (1, y (2 = α + β F (x, y (1, y (2 = α2 β F (x, y (1, y (2 = α2 α β (Randers metric (Kropina metric (Matsumoto metric

6 8 Irena Čomić are homogeneous of first kind and order 2. In the Lagrange space L (2n = (M, L(x, y (1, y (2 with the metric tensor: L(x, y (1, y (2 = F 2 (x, y (1, y (2, c ij (x, y (1, y (2 = 1 2 L(x, y (1, y (2 2 y 2i y 2j is homogeneous of first kind and order, i.e. c ij (x, λy (1, λ 2 y (2 = λ c ij (x, y (1, y (2. 2. The Euler-Lagrange equations for functions of several higher order variables To obtain the general Euler-Lagrange equations for functions of several higher order variables we shall restrict our attention to one special case. Let us consider I = F (t, x, ẋ, ẍ, x (3, p, ṗ, p, p (3 dt, (18 where x = x(t, p = p(t, ẋ = ẋ(t, ṗ = ṗ(t,... and δx(t, δẋ(t, δẍ(t, δp(t, δṗ(t, δ p(t at t = and t = 1 have value zero. Using the partial integration d and the property: δx = δẋ,..., d δẍ = dt dt δx(3 (similarly for p, from (18 we get the first variation of I [6]: δi = = ( δx + δẋ + δẍ + ẋ ẍ (3 δx(3 + + δp + δṗ + ṗ p + ( δx + δp dt + ( d δp + dt ṗ p δ p + δp(3 dt (3 ( d δx + dt ẋ ẍ δṗ + δ p dt (3 δẋ + δẍ dt + (3

7 Some invariants connected with Euler-Lagrange equations 9 1 ( d dt ẋ δx + d dt ( d dt ṗ δp + d dt ẍ δẋ + d dt p δṗ + d dt (3 δẍ (3 δ p dt dt. We have because ( d δx + dt ẋ ẍ δẋ + δẍ dt = (3 ẋ δx 1 + ẍ δẋ δẍ (3 δx(1 = δẋ(1 = δẍ(1 =, δx( = δẋ( = δẍ( =. The same is true if in the above equation x is changed with p. Now we have = δi = where A = = ( d δxdt + dt ẋ ( d d dt dt ẍ δx + d dt 1 ( d d dt dt p δp + d dt ( d δpdt dt ṗ δẋ (3 δṗ (3 dt + dt + ( d dt ẋ + d2 dt 2 δxdt + ẍ ( d 2 dt 2 ( d d 2 dt dt 2 d2 δẋ + (3 dt 2 d2 δx + (3 dt 2 1 δṗ dt = (3 ( d 3 δp dt (3 dt 3 ( d 2 d2 dt 2 δx + ẍ dt 2 δẋ dt (3 ( d 2 d2 dt 2 δp + p dt 2 δṗ dt = (3 ( d dt + d2 dt 2 δpdt + A, p d3 δx + (3 dt 3 δp dt. (3

8 1 Irena Čomić Using the same procedure as before we have δi = 1 ( d dt ẋ + d2 dt 2 ẍ d3 δxdt + (19 dt 3 (3 ( d dt ṗ + d2 dt 2 p d3 δpdt. dt 3 (3 We shall use the notations x = x (, ẋ = x (1, ẍ = x (2,..., d dt G = G. From (19, using similar calculations, we obtain Theorem 5. The extreme value of functional I = F (t, x (, x (1, x (2,..., x (k, p (, p (1, p (2,..., p (k dt, where x (l = x (l (t, p (l = p (l (t, l =, k and δx (l (1 =, δx (l ( =, δp (l (1 =, δp (l ( =, l =, k, can be obtained, when ( 1 l dl =, l= dt l (l l= ( 1 l dl =, dt l (l i.e. d dt d dt d2 + (1 dt 2 d2 + (1 dt 2 dk + ( 1k (2 dt k dk + ( 1k (2 dt k F = (2 (k =. (21 (k The above equations are Euler-Lagrange equations. In the case when x denotes (x 1, x 2,..., x n = (x i and p = (p 1, p 2,..., p n = (p i, i.e. when x and p are points in n-dimensional spaces, then the Euler-Lagrange equations have the form: d (i dt d2 + (1i dt 2 d + d2 (i dt (1i dt 2 dk + ( 1k (2i dt k (2i + ( 1 k dk dt k = (22 (ki (ki =.

9 Some invariants connected with Euler-Lagrange equations 11 In the recent literature it is more usual to use the notations: x (li = y (li = y li, p (li = p li, x i = x i, p i = p i, l =, k, i = 1, n. (23 3. The connection between Lagrangians and Hamiltonians of higher order Let L(t, y ( (t, y (1 (t,..., y (k (t be a regular Lagrangian in L (kn and H(t, p ( (t, p (1 (t,..., p (k (t the Hamiltonian in the H (kn space. We shall examine the variation of F (t, y (, y (1,..., y (k, p (, p (1,..., p (k = (24 H(t, p (, p (1,..., p (k + L(t, y (, y (1,..., y (k y i p ki y 1i p (k 1i y (k 1i p 1i y ki p i. To obtain the general formula we shall similar to (18 first consider the special form of F : F = F (t, x, ẋ, ẍ, x (3, p, ṗ, p, p (3 = (25 H(t, p i, ṗ i, p i, p (3 i + L(t, x i, ẋ i, ẍ i, x (3i x i p (3 i ẋ i p i ẍ i ṗ i x (3i p i. From (18 and (22 it follows that δi = 1 F dt = if From the above we get i p(3 i d dt d i dt ẋ + d2 i dt 2 ẍ d3 =, i = 1, n, i dt 3 (3i d + d2 d3 i dt ṗ i dt 2 p i dt 3 ( ẋ i p i (3 i =, i = 1, n. ( ( + d2 dt 2 ẍ ṗ i i d3 dt 3 p (3i i = x (3i d ( ( ( ẍ i + d2 ẋ i d3 x i =. i dt ṗ i dt 2 p i dt 3 (3 i

10 12 Irena Čomić As ṗ i = d dt p i, ẋ i = d dt xi,..., we have d i dt ẋ + d2 i dt 2 ẍ d3 = (26 i dt 3 (3i d + d2 d3 i dt ṗ i dt 2 p i dt 3 (3 i =. (27 The Euler-Lagrange equation of F given by (25 are (26 and (27. From (25 we get ( df = t + ( ( dt + t i p(3 i dx i + ẋ p i i dẋ i + (28 ( ( ẍ ṗ i i dẍ i + p (3i i dx (3i + ( ( x (3i dp i + ẍ i dṗ i + i ṗ i ( ( ẋ i d p i + x dp (3 p i (3 i =. i As the variables are independent, from (28 it follows, that df = is equivalent to the following relations t = t = i p(3 i i = x (3i ẋ i = p i ṗ i = ẍ i ẍ i = ṗ i p i = ẋ i (3i = p i (29 (3 i = x i. If we replace equations (29 into (26 and (27 we obtain =. For the function F (t, x, ẋ, ẍ, p, ṗ, p = H(t, p, ṗ, p + L(t, x, ẋ, ẍ x i p i ẋ i ṗ i ẍ i p i, the Euler-Lagrange equations (26 and (27 have the form: d i dt ẋ + d2 i dt 2 ẍ p i i = (3

11 Some invariants connected with Euler-Lagrange equations 13 and equation (29 have the form d + d2 ẍ i = (31 i dt ṗ i dt 2 p i t = t i = p i ẋ i = ṗ i ẍ i = p i (32 i = ẍ i ṗ i = ẋ i p i = x i. If we substitute (32 into (3 and (31 we obtain =, i.e. (32 satisfy (3 and (31. From the above it follows: Theorem 6. The Euler-Lagrange equations for the function F determined by (24 are: (a for k = 2l + 1 A = y d d2 + i dt y1i dt 2 y 2i d3 dk + + dt 3 ( 1k = (33 y3i dt k yki B = d + d2 d3 + + ( 1 k dk = (34 i dt 1i dt 2 2i dt 3 3i dt k ki (b for k = 2l the Euler-Lagrange equations have the form: A p ki =, B y ki = (35 where p ki = dk p i, y ki = dk x i. The equations which give relations between L dt k dt and H for both cases (a and k (b have the form t = t, y i = p ki, y 2i = p (k 2i,..., y ki = p i = p i i = y ki, 1i = y (k 1i,..., y 1i = p (k 1i, (36 (k 1i = y 1i, ki = y i. Theorem 7. If L(t, y (, y (1,..., y (k and H(t, p (, p (1,..., p (k are homogeneous functions of first kind and order k, i.e. L(t, y (, λy (1,..., λ k y k = λ k L(t, y (, y (1,..., y (k, (37

12 14 Irena Čomić H(t, p (, λp (1,..., λ k p (k = λ k H(t, p (, p (1,..., p (k, (38 then the function F (t, y (, y (1,..., y (k, p (, p (1,..., p (k defined by (24 is also homogeneous of first kind and order k, i.e. F (t, y (, λy (1,..., λ k y (k, p (, λp (1,..., λ k p (k = λ k F (t, y (, y (1,..., y (k, p (, p (1,..., p (k. If we suppose that L(t, y (,..., y (k with the property (37 is the fundamental function in the Finsler space of order k(f (kn, then the metric tensor in this space can be defined by g ij (t, y (,..., y (k = 2 L 2 (t, y (,..., y (k y ki y kj. (39 If H(t, p (, p (1,..., p (k with the property (38 is the fundamental function in the Cartan space of order k (C (kn, then the metric tensor in this space can be determined by c ij (t, p (, p (1,..., p (k = 2 H 2 (t, p (, p (1,..., p (k ki kj. (4 Both metric tensors g ij and c ij given by (39 and (4 respectively are homogeneous of first kind and order zero. The former statements can be very useful in the investigation of F (kn and C (kn spaces recently introduced. The connection between the function F in (24 and energies of higher order can be investigated. 4. Invariant form of Euler-Lagrange equations We want to show that Euler-Lagrange equations of type (2 are coordinate invariant. For that reason we restrict k to k = 3, and let F = L(t, y i, y 1i, y 2i, y 3i be a regular Lagrangian in Osc 3 M. We shall use the notations li = y, li dl t = dl, l =, 3. dtl In Osc 3 M the allowable transformations are: ([1], [7], [2], [3], [4], etc y i = y i (y i, y i = x i (41

13 Some invariants connected with Euler-Lagrange equations 15 y 1i = ( i y i y 1i = B i i y 1i y 2i = (d 1 t B i i y 1i + B i i y 2i y 3i = (d 2 t B i i y 1i + 2(d 1 t B i i y 2i + B i i y 3i. The transformations of the form (41 form a group. The natural basis of T (Osc 3 M is { i, 1i, 2i, 3i } and it is transforming as follows: i = ( i y i i + ( i y 1i 1i + ( i y 2i 2i + ( i y 3i 3i 1i = ( 1i y 1i 1i + ( 1i y 2i 2i + ( 1i y 3i 3i 2i = ( 2i y 2i 2i + ( 2i y 3i 3i (42 3i = ( 3i y 3i 3i. It is known that ([8], [1], [9], [7], [2], [3], [4] i y i = 1i y 1i = 2i y 2i = 3i y 3i, (43 i y 1i = 1 2 1iy 2i = 1 3 2iy 3i, i y 2i = 1 3 1iy 3i,... ( i y si = 1 ( r+s r ri y (r+si, d 1 t i y i = i y 1i, d 2 t i y i = i y 2i, d 1 t i y 1i = i y 2i, d 1 t i y 2i = i y 3i. Let us denote δx i = εv i (t = εv i, δẋ i = εd 1 t v i = εv 1i, δẍ = εd 2 t v i = εv 2i, δx (3 = εd 3 t v i = εv 3i. Using the similar procedure as in (19 (only the initial conditions are not substituted, we obtain for I = 1 Ldt: Using the notations δi ε = [v i ( i L + v 1i ( 1i L + v 2i ( 2i L + v 3i ( 3i L]dt. (44 E i = i d 1 t 1i + d 2 t 2i d 3 t 3i (45 E 1 i = 1i d 1 t 2i + d 2 t 3i E 2 i = 2i d 1 t 3i E 3 i = 3i

14 16 Irena Čomić (44 can be written in the form δi ε = {v i (Ei L + d 1 t [v i (Ei 1 L + v 1i (Ei 2 L + v 2i (Ei 3 L]}dt. (46 The comparison of (44 and (46 gives From (45, it follows that (v i i + v 1i 1i + v 2i 2i + v 3i 3i L = (47 v i (E i L + d 1 t [v i (E 1 i L + v 1i (E 2 i L + v 2i (E 3 i L. E i + d 1 t E 1 i = i, E 1 i + d 1 t E 2 i = 1i, (48 E 2 i + d 1 t E 3 i = 2i, E 3 i = 3i. If we use the condition that v i (t, v 1i (t and v 2i (t vanish for t = and t = 1, then from (46, we get δi ε = v i (Ei Ldt. (49 Lemma 8. Under the transformation (41, E1 is covariant vector field, i.e. ([8], [3], [4] Ei = ( i y i Ei = Bi i Ei. (5 Proof. Using (42 and (43, we have Ei = ( i y i i + ( i y 1i 1i + ( i y 2i 2i + ( i y 3i 3i d 1 t [( i y i 1i + 2( i y 1i 2i + 3( i y 2i 3i ] +d 2 t [( i y i 2i + 3( i y 1i 3i ] d 3 t [( i y i 3i ] = ( i y i i + [( i y 1i ( i y 1i ] 1i +[( i y 2i 2( i y 2i + ( i y 2i ] 2i +[( i y 3i 3( i y 3i + 3( i y 3i ( i y 3i ] 3i ( i y i d 1 t 1i + ( i y i d 2 t 2i ( i y i d 3 t 3i

15 Some invariants connected with Euler-Lagrange equations 17 +( i y 1i [ 2d 1 t 2i + 2d 1 t 2i ] +( i y 2i [ 3d 1 t 3i + 6d 1 t 3i 3d 1 t 3i ] +( i y 1i [ 3d 2 t 3i + 3d 2 t 3i ] = ( i y i [ i d 1 t 1i + d 2 t 2i d 3 t 3i ] = B i i E i. In the above the Leibniz rule of differentiation was used. Theorem 9. The Euler-Lagrange equation (v i E i L = is coordinate invariant if v i = B i i vi, i.e. if v i = v i is a contravariant vector field. Proof. From (5, we get i.e., v j = v i B j i. The fields Ei 1, Ei 2, Ei 3 the following way: v i E i = v i B j i E j = vj E j (51 defined by (45 are transforming in E 1 i = ( i y i E 1 i + ( iy 1i E 2 i + ( iy 2i E 3 i (52 E 2 i = ( i y i E 2 i + 2( iy 1i E 3 i, E3 i = ( i y i E 3 i. From (5 and (52 it is clear, that only E i and E 3 i are covector fields. Theorem 1. The expression v i E i + d 1 t B, which appears in (46 is coordinate invariant, where B = v i E 1 i + v 1i E 2 i + v 2i E 3 i. (53 Proof. From (51, we have v i Ei = v i Ei. Further: B = ( j y i v j [( i y i Ei 1 + ( iy 1i Ei 2 + ( iy 2i Ei 3 ] + (54 Using the relations [( j y 1i v j + ( j y i v 1j ][( i y i E 2 i + 2( iy 1i E 3 i ] + [( j y 2i v j + 2( j y 1i v 1j + ( j y i v 2j ]( i y i E 3 i. y 1i = B i h y 1h, y 1i = B i h y1h, y 2i = B i h ky 1h y 1k + B i h y 2h, y 2i = B i h k y1h y 1k + B i h y2h, i y 1h =, i y 2h =, i y 1h =, i y 2h =

16 18 Irena Čomić obtained from (41 and (42, and Bj i k h Bi i ( j y i ( i y i = Bj i Bi i = δj i, B j i k Bi i + Bj i i Bi kbk k =, + Bj i i k Bi hbh h + B i j i h Bi kbk k + Bi j ihkb h Bi h Bk k + Bi i j Bi kbk k h = and similar type relations obtained from Bj i Bj j = δj, i after longer calculations, we get B = v i Ei 1 + v1i Ei 2 + v2i Ei 3. (55 The other coefficients besides v j Ei 2, vj Ei 3, v1j Ei 3 Using (55, (51 and (53, we get in (54 are equal to zero. [v i E i + d 1 t (v i E 1 i + v 1i E 2 i + v 2i E 3 i ]L = (56 [v i E i + d1 t (v i E 1 i + v1i E 2 i + v3i E 3 i ]L. From (53 and (55, we obtain that B defined by (53 is a scalar field. R E F E R E N C E S 1. I. Čomić, Some invariants in the higher order geometry, Tagungsband, Geometrie-Tagung in Vorau 17 Drehfluchtprinzip, pp , R. Miron: Spaces with higher order metric structures, Tensor N.S. 53, pp. 1 23, R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces. Theory and Applications, Kluwer Acad. Publishers, R. Miron, The Geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publishers, I. Maşca, V.S. Sabău, H. Shimada, The L-dual of an (α, β-finsler space of order two, Balkan J. Geom. Appl. 12, No. 1, pp , B. Vujanović, D. Spasić, Metodi optimizacije, Univerzitet u Novom Sadu Ed. 34, I. Čomić, The Einstein-Yang-Mills equations in Gauge spaces of order k, (Beograd Proceedings of the Conference Contemporary Geometry and Related Topics, pp , I. Bucataru, R. Miron, Finsler and Lagrange Geometry, Applications to dynamical systems, Editura Academiei Romane, I. Čomić, On Miron s Geometry in Osc 3 M, II Studia Scientiarum Mathematicarum Hungarica 35, pp , 1999.