APPLICATION OF CANONICAL TRANSFORMATION TO GENERATE INVARIANTS OF NON-CONSERVATIVE SYSTEM

Transcription

1 Indian J pure appl Math, 39(4: , August 2008 c Printed in India APPLICATION OF CANONICAL TRANSFORMATION TO GENERATE INVARIANTS OF NON-CONSERVATIVE SYSTEM ASHWINI SAKALKAR AND SARITA THAKAR Department of Mathematics, North Maharashtra University, Jalgaon (MS , India Department of Mathematics, Shivaji University, Kolhapur (MS , India (Received 16 January 2006; after final revision 18 December 2007; accepted 1 May 2008 In this paper, a method based upon Hamilton s integral variational principle is developed to determine invariants for non-conservative dynamical system d dt ( L L ( q q = N t, q T, q T denoted by (L, N using Hamiltonian approach In the Hamiltonian form the system is denoted by (H, N where N = N (t, ( q T, q T t, q T, p T A family of canonical transformations is generated that preserves the form of Hamilton s equations of motion for some Hamiltonian like function These functions consist of the Hamiltonian function K in the transformed coordinates and an additional function R dependent on the non-conservative force N In the new co-ordinates the non-conservative system can be completely described by (K + R, N Noether s theorem is applied to the transformed system to determine invariant of the system in the original variables The method is illustrated for few examples Key Words: Non-conservative system; Hamiltonian approach; canonical transformations; conservation laws

2 354 ASHWINI SAKALKAR AND SARITA THAKAR 1 INTRODUCTION Conservation laws play a very important role in analyzing the dynamics of any physical or mechanical system Solutions to generalized Emden equations are determined in 13, 14, 15 by formation of first integrals using extended Prelle-Singer method These methods are applicable for the differential equations involving rational functions of time, displacement and generalized velocity Noether s identity is one of the tools for generating invariants of motion for any dynamical system Noether s theorem using variational principle has been presented in 3 Vujanovic and Jones 7 have extended the Noether s theorem for non-conservative dynamical system using D Alembert s principle Velocity dependent form of infinitesimal transformations generates higher order conservation laws 7, 10, 11, 12 Various approaches to the generalization of Noether s theorem are revived in 4 The logical equivalence between Hamiltonian approach and Lagrangian approach is established through Legender transformation The transformations provide Hamilton s equations of motion not only for conservative dynamical system but also for non-conservative dynamical system 6 Crespo-da-silva 1 has used the time dependent canonical transformations, linear in generalized co-ordinates and moments to generate the conservation laws for the Hamiltonian system For nonconservative system Hamilton s modified equations of motion in the canonical variables are derived in 8 Since canonical transformations alter the form of Hamilton s equations of motion, Hamiltonian form of Noether s theorem 7 is not applicable for the non-conservative system We consider those canonical transformations that preserve the form of modified Hamilton s equations of motion for non-conservative system We define the Hamiltonian like function, dependent on the non-conservative force The corresponding non-conservative force is the nonconservative force in the transformed co-ordinates Application of Noether s theorem for the Hamiltonian like function and the corresponding non-conservative force provides new conservation laws 2 PRELIMINARIES The position and momentum co-ordinates of the dynamical system are described by the vector q T p T = q 1, q 2, q 3,, q n, p 1, p 2, p 3,, p n If q T (t, p T (t and q T (t, p T (t are co-ordinates of the point on the varied path and actual path respectively then a Lagrangian variation is δq T = q T (t q T (t δp T = p T (t p T (t The infinitesimal transformations are of the form t = t + ετ(t, q T

3 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 355 q( t = qt + εξ(t, q T where ξ T = ξ 1, ξ 2, ξ 3,, ξ n and τ are of the class C 2 and are called space and time generators of the infinitesimal transformations respectively The parameter ε is a positive constant 0 ε 1 For the Hamiltonian H (t, q T, p T = ( q i p i L t, q T, q T q = H p, ṗ = H q + N, where NT = N 1, N 2, N 3,, N n are Hamilton s equations of motion for non-conservative system Thus we denote this non-conservative dynamical system as (H, N where N = N (t, ( q T, q T t, q T, p T Noether s theorem in Hamiltonian form: If the Hamiltonian H (t, q T, p T and the non-conservative force N satisfy (21 H ( t τ + dξ 1 p i dt + ξ i dq i dt τ H ξ i H dτ q i dt dφ dt = 0 (22 for some function φ known as gauge function, then p i ξ i Hτ φ = constant (23 holds along the motion (21 Equation (22 is called as Noether s identity Noethers identity (22 is a differential equation in space, time generators ξ, τ and a gauge function φ Sinee q = H and ξ, τ, φ are functions of generalized coordinates and time, equation p (22 can be partitioned into a system of partial differential equations by equating equal powers of p i s If this system of partial differential equations has a non trivial solution then the expression (23 becomes the first integral of the motion represented by (H, N Since q = H, by inverse p function theorem 2 the generalized moments ( p i s can be expressed as functions of q 1, q 2, q 3,, q n and expression (23 treated as function of t, q T, q T is an integral surface of the system denoted by (L, N 3 GENERATION OF NEW TRANSFORMATIONS Here we define the Hamiltonian like function H and the canonical transformations that preserve the form of Hamilton s equations of motion for non-conservative dynamical system Thus in the new set of canonical variables (Q T, P T equations of motion become Q = H P and Ṗ = H Q + N

4 356 ASHWINI SAKALKAR AND SARITA THAKAR where ( ( ( ( N t, Q T, P T = N t, q T t, Q T, P T, p T t, Q T, P T The dynamical system in new co-ordinates is denoted by (H, N Lemma For the vector N T = N 1, N 2, N 3,, N n if the canonical transformation ( ( q = q t, Q T, P T, p = p t, Q T, P T (31 where Q T P T = Q 1, Q 2, Q 3,, Q n, P 1, P 2, P 3,, P n satisfies {Q r, Q j {P r, P j (N,q = N j Q r N r (N,q = 0 (32 {P r, Q j (N,q = N j P r j, r = 1, 2, 3,, n where { denotes the Lagrange s bracket with respect to ( N, q, then the system of equations has a solution R Q = q i R + N, Q P = q i P (33 PROOF : The system of equations (33 is integrable if (i (ii (iii 2 R Q r = 2 R P r P j = 2 R P r = 2 R Q r, j, r = 1, 2, 3,, n 2 R P j P r, j, r = 1, 2, 3,, n 2 R P r, j, r = 1, 2, 3,, n The first equation in the system (33 is R = q i + N j, j = 1, 2, 3,, n

5 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 357 On differentiating above equation with respect to Q r, we get, 2 R Q r = Q r q i 2 q i + N j Q r Q r Similarly 2 R Q r = q i Q r 2 q i + N r Q r The first integrability condition gives Q r q i q i Q r = N j Q r N r But Ni q i N i q i = {Q r, Q j Q r Q r (N,q The term on the right denotes Lagrange bracket of Q r, Q j with respeet to (N, q Thus {Q r, Q j The second integrability condition gives Similarly the third integrability condition gives (N,q = N j Q r N r Ni q i N i q i = {P r, P j P r P j P j P = 0 r (N,q Ni q i N i q i = {P r, Q j P r P = N j r (N,q P r Theorem For non conservative dynamical system (H, N, the equations of motion (21 remain invariant under the transformation (31, where the Hamiltonian is replaced by H given by ( ( ( H t, Q T, P T = K t, Q T, P T + R t, Q T, P T The function K (t, Q T, P T is the Hamiltonian in the transformed co-ordinates, R (t, Q T, P T is the solution of the system of equations (33 and ( ( ( ( N t, Q T, P T = N t, q T t, Q T, P T, p T t, Q T, P T

6 358 ASHWINI SAKALKAR AND SARITA THAKAR PROOF : The Hamiltonian function for conservative dynamical system in the new set of canonical variables 5 is K = H + F (t, q T, Q T t In the transformed state (t, Q T, P T the function K is not modified to accommodate the nonconservative part We define the function H the new co-ordinates in such a way that the absence of non-conservative forces will reduce the function H into K (t, Q T, P T Therefore, it is natural to define ( ( ( H t, Q T, P T = K t, Q T, P T + R t, Q T, P T (34 We call this function H as modified Hamiltonian Hamilton s principle for the non-conservative system 7 is t1 t 0 δ ( ( q i p i H t, q T, p T + ( t, q T, p T δq i dt = 0 (35 Since the transformation (31 is canonical, t1 ( t1 δ q i p i H t, q T, p T dt = δ t 0 t 0 j=1 Q j P j K (t, Q T, P T dt For the conservative system δ and integral are commutative 7 Hence the Hamiltonian principle (35 in the new co-ordinates can be put in the form t1 t 0 δ ( ( Q j P j K t, Q T, P T + ( t, Q T, P T j=1 ( qi δq j + q i δp j dt = 0 P j Since the Lagrangian variation δ and the differentiation d are commutative 9 on integrating the above equation with respect to t we get P j δq j j=1 t 1 t1 + t 0 t 0 j=1 ( P j δq j + Q j δp j δk + j=1 ( qi δq j + q i δp j dt = 0 P j Since the variation δq is zero at end points t 0 and t 1, above equation becomes

7 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 359 t1 t 0 ( ( P j K q i + δq j + j=1 Q j K P j + q i P j δp j dt = 0 Since the variations δq and δp are arbitrary, we have From (33 we get Ṗ = K Q + q i Q and Q = K P q i P Q = H P and Ṗ = H Q + N 4 INVARIANTS OF NON-CONSERVATIVE SYSTEM In the study of non conservative systems we come across many situations For certain non conservative systems and apparently non conservative systems explicit Lagrangian and Hamiltonian functions may exist In some situations it may happen that with the proper choice of the coordinate system, the non conservative system can be explained completely with the help of Lagrangian or a Hamiltonian function For example, the system with n degrees of freedom q + 2k q + Aq = 0 (41 where A has all real roots, can be completely explained with the help of Lagrangian or Hamiltonian function by proper choice of generalized coordinates 7 If the roots of matrix A are not all real then the system is purely non-conservative For Lagrangian systems Noether s theorem 3 may provide first integral of the system under consideration The non-zero solution ξ, τ, φ of Noethers identity (22 with N = 0 integrates the given system Equation (22 may have many independent solutions Corresponding to each solution we get one conservation law The existence of non-zero solution depends upon the form of Hamiltonian function Similarly for non-conservative system the existence of non-zero solution of equation (22 depends upon the Hamiltonian function and the corresponding non-conservative force For equation (41 Noether s identity (22 may have only a trivial solution Example 2 in Section 5 Therefore, the application of Noether s theorem to equation (41 is insufficient to derive the invariant surface Some times the equation representing a physical situation contains sufficient number of parameters In such situations more information about the system can be obtained by studying equivalent forms For systems that are completely described with the help of Hamiltonian function, equivalent forms of Noethers identity can be developed by applying a family of canonical

8 360 ASHWINI SAKALKAR AND SARITA THAKAR transformations Since the form of equations of motion remain invariant under the canonical transformation, Noether s theorem equation (22 with N = 0 and equation (23 is applicable for the system in the new set of canonical variables If the system is non-conservative application of canonical transformation alters the form of Hamilton s canonical equations of motion 8 As such Noether s theorem equation (22 and (23 is not applicable for the system in the set of new canonical variables To overcome this situation in Section 3 Hamiltonian like function is defined and the family of canonical transformations is identified that preserve the form of Hamilton like equations (21 for the non-conservative system in the new canonical variables This choice of Hamiltonian like function is now appropriate to apply Noether s theorem in the new set of canonical variables If the Hamiltonian like function H and the non conservative force N (t, Q T, P T satisfy H ( t τ + dξ i P i dt + ξ i dq i dt τ H ξ i H dτ Q i dt dφ dt = 0 then the expression denoted by ϕ ϕ = P i ξ i (t, Q T ( Hτ t, Q T ( φ t, Q T = constant holds along the motion in the new set of canonical variables (Q T, P T Since the canonical transformations are invertible the expression FI can be expressed in terms of old co-ordinates (q T, p T and ϕ = ( t, q T, p T ( = ϕ t, Q T( t, q T, p T, P T( t, q T, p T = constant (42 becomes the solution of the system in the Hamiltonian form (H, N Equation q = H q states the relation between q and p j, j =1, 2, 3,, n By inverse function theorem 2 all the generalized moments p i s can be expressed as functions of q 1, q 2, q 3,, q n and the expression ( ( F I t, q T, q T = ϕ t, q T, q T = constant holds along the system in the Lagrangian form (L, N

9 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM ILLUSTRATIONS Example 1 Consider the dynamical system (41, where k is constant A is n n constant matrix and q is n 1 vector Every matrix A can be uniquely represented as the addition of a symmetric and upper triangular matrices The symmetric part corresponds to the conservative potential function where as the triangular part becomes the non-conservative potential Thus the system (41 can be represented by (H, N with H = 1 2 q T p T e 2kt S 0 0 e 2kt I N = Ue 2kt q 0 p where A = S + U, S T = S and U is an upper triangular matrix q p For the canonical transformation q = p T 11 T 12 e 2kt T 21 e 2kt T 22 Q P, equation (32 holds if T 11 and T 12 has the following properties (i T T 12 UT 12 is a diagonal matrix (ii T T 11 UT 11 UT 11 is a symmetric matrix (iii T T 11 (U-U T T 12 = UT 12 (51 In the transformed co-ordinates the modified Hamiltonian H = 1 2 Q T P T e 2kt H 1 H 2 H 3 e 2kt H 4 Q P and the non-conservative force where N = N 1 e 2kt N 2 Q P ( H 1 = T T 11(S + UT 11 + T T kT T 11 T 21 UT 11 H 2 = H T 3 = T T 11ST 12 + T T 12UT 11 + T T 21 (T kT 12

10 362 ASHWINI SAKALKAR AND SARITA THAKAR H 4 = T T 12(S+UT 12 + T T 22 (T kT 12 N 1 = UT 11 N 2 = UT 12 The Noether s identity (22 for (H, N becomes Q T P T E 11 E 12 E 21 E 22 Q P = 0 (52 where E 11 = {e 2kt H 1 (kiτ + B + T T 11U T (B H 3 τ + ϕ + H T 3 ϕ + ϕ T H 3, E 12 = T T 11U T τ e 2kt ϕ T H 4, E 21 = B H 3 + T T 12U T B + B + T T 12U T τ H 3 e 2kt H 4 ϕ, E 22 = e 2kt kiτ + B + T T 12U T τ H 4, τ = constant, ξ = BQ and φ = Q T ϕq Equation (52 is equivalent to E 11 + E T 11 = 0 E 21 + E T 12 = 0 E 22 + E T 22 = 0 Above equations take the following form

11 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 363 ( (i e 2kt 2kH 1 T T 11 UT H 3 H T 3 UT 11 τ + (H 1 + T T 11 B UT + B (H T 1 + UT 11 +ϕ + ϕ T + 2(H T 3 ϕ + ϕ T H 3 = 0 (ii B H 3 B + BH 3 T T 11 UT B + (T T 12 UT H 3 + H 4 UT 11 τ 2H 4 ϕe 2kt = 0 (iii (2kH 4 + T T 12 UT H 4 + H 4 UT 12 τ + BH 4 + H 4 B = 0 (53 Equation (53-iii can be written as ( ki + T T 12U T τ + B H 4 + H 4 (ki + UT 12 τ + B = 0 ( Observe that B = ki + T T 12 UT + H 4 G τ identically satisfy equation (53-iii for any skew symmetric matrix G From equation (53-ii we get ϕ = e2kt ( 2 H 1 4 T T 12U T ki + T T 12U T + H 4 G + H 4 (UT 11 GH 3 + H 3 (T T 12U T + H 4 G τ We choose the transformation such that the above matrix ϕ becomes symmetric and equations (51 and (53-i are true for some skew symmetric matrix G The inverse canonical transformation Q P = T T 22 T T 12 e 2kt T T 12 e2kt T T 11 q p and the relation between the generalized velocities and the generalized moments makes equation (42 as e 2kt q T q T T 22 T 21 T 12 T 11 ( where B = ki + T T 12 UT + H 4 G τ H 1 + 2ϕe 2kt H 2 H 3 2B H 4 T T 22 T T 12 T T 21 T T 11 q q = constant (54

12 364 ASHWINI SAKALKAR AND SARITA THAKAR Equation (54 is the first integral of equation (41 provided that T 11, T 12, T 21, T 22 satisfy equation (51 and equation (53-i Example 2 Consider the dynamical system q 1 + 2k q 1 + aq 1 + 4bq 2 = 0 q 2 + 2k q 2 + bq 1 + aq 2 = 0 (55 where k, a and b are constant The system (55 can be represented by (H, N where H = e 2kt ( p p e2kt (aq aq bq 1 q N 1 = 3be 2kt q 2 N 2 = 0 Applying the Noether s identity (22 for above system (H, N and comparing the coefficients of equal powers of P i we get the following system of Killing s equations (i τ qi = 0, i = 1, 2 (ii τ t 2kτ 2ξ 1q1 = 0 (iii τ t 2kτ 2ξ 2q2 = 0 (iv ξ 1q2 + ξ 2q1 = 0 (v φ q1 = e 2kt (ξ 1t + 3bq 2 τ (vi φ q2 = e 2kt ξ 2t (vii φ t = e 2kt ( aq aq bq 1q 2 ( τ12 + kτ + (aq 1 + 4bq 2 ξ 1 + (bq 1 + aq 2 ξ 2 (56 From equations (56-i, ii, iii, we get the form of the time and space generators as

13 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 365 τ(t = e (2 2kt f 1 (te 2kt dt + c ξ 1 (t, q 1, q 2 = f 1 (tq 1 + f 2 (t, q 2 (57 ξ 2 (t, q 1, q 2 = f 3 (t, q 1 + f 1 (tq 2 Substituting the above forms of the time and space generators in equation (56-iv, we get the relation between the functions f 2 q 2 + f 3 q 1 = 0 Thus, ξ 1 and ξ 2 are linear in generalized co-ordinates q 1, q 2 respectively Then ξ 1 = f 1 (tq 1 + f 4 (tq 2 + f 6 (t ξ 2 = f 5 (tq 1 + f 1 (tq 2 + f 7 (t Since φ q1 t = φ tq1, f 4 (t = f 5 (t = f 6 (t = f 7 (t = 0 and we get τ = constant, ξ 1 = kτq 1, ξ 2 = kτq 2 Since ξ 1 and ξ 2 are functions of τ, the integrability condition φ q2 t = φ tq2 therefore ξ 1 = ξ 2 = 0 gives τ = 0 and Hence equations (56-v, vi, vii provides φ = constant Thus we get the trivial solution To obtain the non-trivial solution to the system 5 in Example 1 with A = a b 4b a, S = a b b a, U = 0 3b 0 0 The transformation define by T 11 = 1 k 1 0 nk 1, T 12 = 0 a 14 0 na 14 T 21 = k(1 + n kn(1 + n 2 1 n (na 14 1, T 22 = 1 ka 14 n 1 kna 14 where n and a 14 are non-zero arbitrary constants This transformation is canonical and satisfy all the conditions in equation (51 for the existence of Hamiltonian like function

14 366 ASHWINI SAKALKAR AND SARITA THAKAR From equation (53-iii, we get B = k 3nba 14 + g((a k 2 (1 + n 2 + 5nba 2 14 g k τ Since φ is symmetric matrix from equation (53-ii, we get ϕ = φ 11 φ 12 φ 12 φ 22 where φ 11 = 3n 2(a k2 2 4b 2 + 6nb(a k 2 + n 2 (2(a k b 2 + n 3 b(a k 2 4b + 10n(a k n 2 b + 10n 3 (a k 2 + 4bn 4 φ 12 = 3n k2 ((a k 2 (1 + n 2 + 5nb + (1 + n 2 2 ((a k 2 2 4b 2 a 14 k4b + 10n(a k n 2 b + 10n 3 (a k 2 + 4bn 4 a 14 φ 22 = 32(a k 2 + 9nb + 2(a k 2 n 2 + 4n 3 b n4b + 10n(a k n 2 b + 10n 3 (a k 2 + 4bn 4 a 2 14 From equation (53-i, we get g = 3n(a k 2 + 5nb + (a k 2 n 2 4b + 10n(a k n 2 b + 10n 3 (a k 2 + 4bn 4 a 14 For n = 1 equation (54 becomes e 2kt{ 8(a k b ( q 1 + kq ( q 2 + kq b( q 1 + kq 1 ( q 2 + kq 2 ( + 8(a k b b(a k 2 q q b b(a k 2 q 1 q 2 = constant For n = -1 equation (54 becomes { e 2kt 8(a k 2 20b ( q 1 + kq ( q 2 + kq b( q 1 + kq 1 ( q 2 + kq 2 ( + 8(a k b 2 20b(a k 2 q q b b(a k 2 q 1 q 2 = constant For every value of n we get an invariant Thus we get a family of invariants where n is any real number This family of invariants need not be independent but will be useful to analyze the system

15 CANONICAL TRANSFORMATION TO GENERATE NON-CONSERVATIVE SYSTEM 367 Vujanovic and Jones 7 have suggested a method to calculate the invariants of the system (41 This method is applicable only if all the characteristic roots of the matrix A are real If the roots are complex, the method is inadequate to generate invariants Example 3 For the system in Example 1 with A = the characteristic roots of A are complex and we do not get invariant by the method describe in 7 Here S = and U = The procedure described in example (1 and the transformation defined in example (2 provide ( e 2kt 8k q q 1 q 2 + (8k 2 11 q ( 16k kq 1 q 1 4kq 1 q 2 4kq 2 q 1 +(16k 3 22kq q 2 + ( 8k q ( 16k q 1 q 2 + (16k 2 24q 2 2 = constant and ( e 2kt 8k 2 15 q q 1 q 2 + ( 8k q (16k 3 30kq 1 q kq 1 q kq 2 q 1 +( 16k kq 2 q 2 + (8k q (16k 2 6q 1 q 2 + ( 16k q 2 2 = constant 5 CONCLUSION Every dynamical system is completely described by (H, N, though the representation may not be unique A family of (H, s generated with the help of canonical transformation satisfying certain conditions Noether s identity has Lagrangian as well as Hamiltonian form The Hamiltonian form has been selected to develop invariance for the dual set (H, N under specified conditions The method is illustrated for few examples

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