Elementary work, Newton law and Euler-Lagrange equations

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1 Elementary work, Newton law and Euler-Lagrange equatons Constantn Udrşte, Oltn Dogaru, Ionel Ţevy, Dumtru Bala Abstract. The am of ths paper s to show a geometrcal connecton between elementary mechancal work, Newton law and Euler-Lagrange ODEs or PDEs. The sngle-tme case s wellknown, but the multtme case s analyzed here for the frst tme. Secton 1 ntroduces the Newton law va a covarant vector or va a tensoral 1-form. Secton 2 shows that the untemporal Euler-Lagrange ODEs can be obtaned from mechancal work and sngle-tme Newton law. Secton 3 descrbes the Noether Frst Integrals n the untemporal Lagrangan dynamcs. Secton 4 shows that the multtemporal Euler-Lagrange PDEs can be obtaned from the mechancal work and multtme Newton law. Secton 5 descrbes the Frst Integrals n multtemporal ant-trace Lagrangan dynamcs. M.S.C. 2010: 70H03, 70H05, 53C40. Key words: mechancal work; Newton law; Euler-Lagrange ODEs or PDEs; submanfold. 1 Elementary work and Newton law Let y = (y I, I = 1,..., N, be an arbtrary pont n R N. In case of forces defned on R N, the elementary mechancal work can be wrtten as an 1-form ω = f I (ydy I. On a submanfold M of dmenson n n R N, descrbed by the equatons y I = y I (x, x = (x, = 1,..., n, we have dy I = yi dx. Consequently, t appears the pull-back ω = F (xdx, F (x = f I (y(x yi (x. Sngle-tme Newton law. Introducng the tme t, we can wrte the untemporal Newton law on R N as equalty of 1-forms f I = mδ IJ dẏ J dt. Balkan Journal of Geometry and Its Applcatons, Vol.15, No.2, 2010, pp (electronc verson; pp (prnted verson. c Balkan Socety of Geometers, Geometry Balkan Press 2010.

2 Elementary work, Newton law and Euler-Lagrange equatons 101 The representaton of untemporal Newton law on the submanfold M s (1.1 F = mδ IJ dẏ I dt y J. Multtme Newton law. Introducng the multtme t = (t α, α = 1,..., m, we can wrte the multtemporal (tensoral Newton law as equalty of 1-forms 2 y J f I = mδ IJ δ α t α t. The representaton multtemporal Newton law on the submanfold M s F = mδ IJ δ α 2 y I y J t α t. An ant-trace of the force F s the Newton tensoral 1-form (1.2 F σ α = mδ IJ δ σ 2 y I t α t y J, wth F = F α α. 2 Sngle-tme Euler-Lagrange ODEs obtaned from mechancal work Lookng at the Newton law (1.1 and usng the operator d dt, we observe the dentty δ IJ dẏ I dt y J = d dt (δ IJ ẏ I yj δ IJ ẏ I d y J dt or otherwse Consequently δ IJ dẏ I dt y J = d dt F m = d dt (δ IJ ẏ I yj δ IJ ẏ I dy J dt. (δ IJ ẏ I yj δ IJ ẏ I ẏj. Snce ẏ I = yi ẋ, the Jacoban matrx satsfes yi or F = d dt F m = d dt = ẏi ẋ. Hence (δ IJ ẏ I ẏj ẋ δ IJ ẏ I ẏj ( ( m ẋ 2 δ IJẏ I ẏ J ( m 2 δ IJẏ I ẏ J. If we use the knetc energy T = m 2 δ IJẏ I ẏ J, we can wrte F = d T dt ẋ T. Now, we suppose that the pullback ω = F (xdx s a completely ntegrable (closed 1-form,.e., t s assocated to a conservatve force. Settng ω = dv = V dx,.e.,

3 102 Constantn Udrşte, Oltn Dogaru, Ionel Ţevy, Dumtru Bala F = V and ntroducng the Lagrangan L = T V, t follows the Euler-Lagrange ODEs d dt ẋ = 0, whose solutons are the curves x(t. Partcularly, the prevous theory survve for any changng of coordnates. Summng up, for sngle-tme case, t appears the followng Theorem. 1 A constraned conservatve movement s descrbed by the Euler- Lagrange ODEs. 2 For conservatve systems, the Euler-Lagrange ODEs represents the nvarant form of Newton law, wth or wthout constrants. 3 Frst ntegrals n sngle-tme Lagrangan dynamcs If L(x(t, ( ẋ(t s an autonomous Lagrangan, satsfyng the regularty condton det 2 L ẋ ẋ 0 (see the Legendran dualty, then the Hamltonan j or shortly H(x, p = ẋ (x, p (x, ẋ(x, p L(x, ẋ(x, p ẋ H(x, p = p ẋ (x, p L(x, p s a frst ntegral both for Euler-Lagrange and Hamlton equatons. Whch chances we have to fnd new frst ntegrals? Noether Theorem Let T (t, x be the flow generated by the C 1 vector feld X(x = (X (x. If the autonomous Lagrangan L s nvarant under ths flow, then the functon I(x, ẋ = ẋ (x, ẋx (x s a frst ntegral of the movement generated by the Lagrangan L. Proof. We denote x s (t = T (s, x(t. The nvarance of L means 0 = dl ds (x s(t, ẋ s (t s=0 = (x(t, ẋ(t X ẋ j (x(tẋj (t + (x(t, ẋ(tx (x(t. Consequently, by the dervaton formulas and by the Euler-Lagrange equatons, we fnd ( di d (x(t, ẋ(t = (x, ẋ X (x + (x, ẋ X dt dt ẋ ẋ j (xẋj ( d = (x, ẋ (x, ẋ X (x = 0. dt ẋ In ths way, the functon I(x, ẋ s a frst ntegral.

4 Elementary work, Newton law and Euler-Lagrange equatons Multtme Euler-Lagrange PDEs obtaned from the mechancal work We start from the Newton law (1.2. Now we use the dentty δ IJ δ σ 2 y I y J t α t = D σ yi y α (δ J ( σ yi y J IJ δ t δ IJ δ t D α or otherwse Snce 1 m F α σ σ yi y = D α (δ J σ yi IJ δ t δ IJ δ t the Jacoban matrx satsfes It follows or or y I t γ y I γ λ = yi t γ, = yi δλ γ. ( y J t α ( Fαδ σ γ λ σ yi y J ( = D α mδ IJ δ t δλ σ yi y J γ mδ IJ δ t t α δλ γ F σ αδ λ γ = D α ( σ yi yγ J mδ IJ δ t δγ λ λ. ( m 2 δ σ yi y J t t α δλ γ ( ( m Fαδ σ γ λ = D α λ 2 δ σ yi y J t t γ ( m 2 δ σ yi y J t t α δλ γ. Contractng λ wth α and σ wth γ, we fnd ( ( m Fα α = F = D α α 2 δ γ yi y J t t γ If we use the multtemporal knetc energy then we can wrte T = m 2 δ γ yi y J t t γ, T F = D α T α. ( m 2 δ γ yi y J t t γ. Now, we suppose that the pullback ω = F (xdx s a completely ntegrable (closed 1-form,.e., t s assocated to a conservatve force. Settng ω = dv = V dx,.e., F = V and ntroducng the Lagrangan L = T V, t follows the multtemporal Euler-Lagrange PDEs D α α = 0,

5 104 Constantn Udrşte, Oltn Dogaru, Ionel Ţevy, Dumtru Bala whose solutons are m-sheets x(t. Partcularly, the prevous theory survve for any changng of coordnates. Summng up, for multtme case, t appears the followng Theorem. 1 A constraned conservatve movement s descrbed by the Euler- Lagrange PDEs. 2 For conservatve systems, the Euler-Lagrange PDEs represents the nvarant form of Newton law, wth or wthout constrants. 5 Frst ntegrals n multtme Lagrangan dynamcs An autonomous multtme Lagrangan s a functon of the form L(x, x γ. We reconsder the multtme ant-trace Euler-Lagrange PDEs ([4], [5] δγ D γ = 0, (At E L n order to ntroduce multtemporal ant-trace Hamlton PDEs. Lagrangan L(x, x γ (x, p, satsfyng the regularty condton ( 2 L det α j 0 Startng from the (see the Legendran dualty, defne the Hamltonan H(x, p = x α(x, p (x, x γ (x, p L(x, x γ (x, p α or shortly H(x, p = p α x α(x, p L(x, p. Theorem (multtme ant-trace Hamlton PDEs Let x( be a soluton of multtemporal ant-trace Euler-Lagrange PDEs (At E L. Defne p( = (p α ( va Legendran dualty. Then the par x(, p( s a soluton of multtemporal ant-trace Hamlton PDEs H (t = t p (x(t, p(t, p α t (t = H δα (x(t, p(t. (At H Moreover, f the Lagrangan L(x, x γ (x, p s autonomous, then the Hamltonan H(x, p s a frst ntegral of the system (At-H. Here we have a system of nm(m + 1 PDEs of frst order wth n(1 + m unknown functons x (, p α (. Proof.: We fnd H(x, p = L(x, x γ(x, p.

6 Elementary work, Newton law and Euler-Lagrange equatons 105 By hypothess p α (t = (x(t, x γ (t f and only f α t (t = x α α (x(t, p(t. Consequently, multtemporal ant-trace Euler-Lagrange PDEs (At E L mply p α t (t = δ α (x(t, x γ (t = δ α (x(t, x γ (x(t, p(t = δ α H (x(t, p(t,.e., we fnd the multtemporal ant-trace Hamlton PDEs on the second place, Moreover, the equalty H p α On the other hand, p α (t = p α t (t = H δα (x(t, p(t. (x, p = x α(x, p produces H (x(t, p(t = x α(x(t, p(t. (x(t, x γ (t and so x α (t = x α (x(t, p(t. In ths α way, t appears the multtemporal ant-trace Hamlton PDEs on the frst place, p α H (t = t p (x(t, p(t. Snce the Hamltonan s autonomous, usng multtemporal ant-trace Hamlton PDEs, we fnd D γ H = H t γ + H p λ p λ t γ = 0. If the Lagrangan s autonomous, then the Hamltonan s a frst ntegral both for multtemporal ant-trace Euler-Lagrange PDEs and multtemporal ant-trace Hamlton PDEs. Whch chances we have to fnd new frst ntegrals? Theorem Let T (t, x be the m-flow generated by the C 1 vector felds X α (x = (Xα(x. If the autonomous Lagrangan L s nvarant under ths flow, then the functon I(x, x γ = (x, x γ X(x s a frst ntegral of the movement generated by the Lagrangan L va multtemporal ant-trace Euler-Lagrange PDEs. Proof. We denote x s (t = T (s, x(t. The nvarance of L means 0 = D α L(x s (t, x sγ (t s=0 = (x(t, x γ (t X j (x(txj α(t+ (x(t, x γ(txα(x(t. Consequently, by dervaton formulas and by multtemporal ant-trace Euler-Lagrange PDEs, we fnd ( D α I(x(t, x γ (t = D α (x, x γ X(x + (x, x γ X j (xxj α ( = D α ẋ (x, x γ (x, x γδα X(x = 0. In ths way, the functon I(x, x γ s a frst ntegral.

7 106 Constantn Udrşte, Oltn Dogaru, Ionel Ţevy, Dumtru Bala 6 Concluson The results explaned n the prevous sectons show that the Euler-Lagrange ODEs or PDEs, for the Lagrangan L = T V, can be obtaned usng the elementary mechancal work, Newton law and technques from dfferental geometry. On the other hand, the Euler-Lagrange ODEs or PDEs are usually ntroduced va varatonal calculus [16]. It follows that the conservatve Newton law s nvarant representable as Euler-Lagrange equatons. Other results regardng the multtemporal Euler-Lagrange or Hamlton PDEs can be found n our papers [2]-[15]. Acknowledgements. Partally supported by Unversty Poltehnca of Bucharest, and by Academy of Romanan Scentsts. References [1] A. Ptea, Null Lagrangan forms on 2-nd order jet bundles, J. Adv. Math. Studes, 3, 1 (2010, [2] A. Ptea, C. Udrşte, Şt. Mttelu, P DI&P DE-constraned optmzaton problems wth curvlnear functonal quotents as objectve vectors, Balkan J. Geom. Appl. 14, 2 (2009, [3] C. Udrşte, Mult-tme maxmum prncple, Short Communcaton, Internatonal Congress of Mathematcans, Madrd, August 22-30, ICM Abstracts, 2006, p. 47, Plenary Lecture at 6-th WSEAS Internatonal Conference on Crcuts, Systems, Electroncs, Control&Sgnal Processng (CSECS 07, p and 12-th WSEAS Internatonal Conference on Appled Mathematcs, Caro, Egypt, December 29-31, 2007, p.. [4] C. Udrşte, I. Ţevy, Mult-tme Euler-Lagrange-Hamlton theory, WSEAS Transactons on Mathematcs, 6, 6 (2007, [5] C. Udrşte, I. Ţevy, Mult-tme Euler-Lagrange dynamcs, Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton (ISTASC 07, Voulagmen Beach, Athens, Greece, August 24-26, 2007, [6] C. Udrşte, Controllablty and observablty of multtme lnear PDE systems, Proceedngs of The Sxth Congress of Romanan Mathematcans, Bucharest, Romana, June 28 - July 4, 2007, vol. 1, [7] C. Udrşte, Mult-tme stochastc control theory, Selected Topcs on Crcuts, Systems, Electroncs, Control&Sgnal Processng, Proceedngs of the 6-th WSEAS Internatonal Conference on Crcuts, Systems, Electroncs, Control&Sgnal Processng (CSECS 07, Caro, Egypt, December 29-31, 2007, [8] C. Udrşte, Fnsler optmal control and Geometrc Dynamcs, Mathematcs and Computers n Scence and Engneerng, Proceedngs of the Amercan Conference on Appled Mathematcs, Cambrdge, Massachusetts, 2008, [9] C. Udrşte, Lagrangans constructed from Hamltonan systems, Mathematcs a Computers n Busness and Economcs, Proceedngs of the 9th WSEAS Internatonal Conference on Mathematcs a Computers n Busness and Economcs (MCBE-08, Bucharest, Romana, June 24-26, 2008,

8 Elementary work, Newton law and Euler-Lagrange equatons 107 [10] C. Udrşte, Multtme controllablty, observablty and bang-bang prncple, Journal of Optmzaton Theory and Applcatons 139, 1(2008, [11] C. Udrşte, L. Mate, Lagrange-Hamlton Theores (n Romanan, Monographs and Textbooks 8, Geometry Balkan Press, Bucharest, [12] C. Udrste, O. Dogaru, I. Tevy, Null Lagrangan forms and Euler-Lagrange PDEs, J. Adv. Math. Studes, 1, 1-2 (2008, [13] C. Udrşte, Smplfed multtme maxmum prncple, Balkan J. Geom. Appl. 14, 1 (2009, [14] C. Udrşte, Nonholonomc approach of multtme maxmum prncple, Balkan J. Geom. Appl. 14, 2 (2009, [15] C. Udrşte, I. Ţevy, Multtme Lnear-Quadratc Regulator Problem Based on Curvlnear Integral, Balkan J. Geom. Appl. 14, 2 (2009, [16] E. T. Whttaker, A Treatse on The Analytcal Dynamcs of Partcles & Rgd Bodes, Cambrdge Unversty Press, Authors addresses: Constantn Udrşte, Oltn Dogaru and Ionel Tevy Unversty Poltehnca of Bucharest, Faculty of Appled Scences, Department of Mathematcs-Informatcs I, 313 Splaul Independente, Bucharest, Romana. E-mal: udrste@mathem.pub.ro, anet.udr@yahoo.com; oltn.hora@yahoo.com; vascatevy@yahoo.fr Dumtru Bala Drobeta Turnu Severn, 4 Aleea Prvghetorlor, Bl. T3, Sc. 3, Ap. 14, Jud. Mehednţ, Romana. E-mal: dumtru bala@yahoo.com