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Deriving Lagrange's Equations Using Elementary Calculus

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Deriving Lagrange's Equations Using Elementary Calculus Deriving Lagrange’s equations using elementary calculus Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Edwin F. Taylorb) Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Slavomir Tulejac) Gymnazium arm. gen. L. Svobodu, Komenskeho 4, 066 51 Humenne, Slovakia ͑Received 30 December 2002; accepted 20 June 2003͒ We derive Lagrange’s equations of motion from the principle of least action using elementary calculus rather than the calculus of variations. We also demonstrate the conditions under which energy and momentum are constants of the motion. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1603270͔ I. INTRODUCTION The action S along a world line is defined as The equations of motion1 of a mechanical system can be Sϭ ͵ L͑x,v͒dt. ͑2͒ derived by two different mathematical methods—vectorial along the world line and analytical. Traditionally, introductory mechanics begins with Newton’s laws of motion which relate the force, mo- The principle of least action requires that between a fixed mentum, and acceleration vectors. But we frequently need to initial event and a fixed final event the particle follow a describe systems, for example, systems subject to constraints world line such that the action S is a minimum. without friction, for which the use of vector forces is cum- The action S is an additive scalar quantity, and is the sum bersome. Analytical mechanics in the form of the Lagrange of contributions L⌬t from each segment along the entire equations provides an alternative and very powerful tool for world line between two events fixed in space and time. Be- obtaining the equations of motion. Lagrange’s equations em- cause S is additive, it follows that the principle of least action ploy a single scalar function, and there are no annoying vec- must hold for each individual infinitesimal segment of the tor components or associated trigonometric manipulations. world line.6 This property allows us to pass from the integral Moreover, the analytical approach using Lagrange’s equa- equation for the principle of least action, Eq. ͑2͒,to tions provides other capabilities2 that allow us to analyze a Lagrange’s differential equation, valid anywhere along the wider range of systems than Newton’s second law. world line. It also allows us to use elementary calculus in The derivation of Lagrange’s equations in advanced me- this derivation. chanics texts3 typically applies the calculus of variations to We approximate a small section of the world line by two the principle of least action. The calculus of variation be- straight-line segments connected in the middle ͑Fig. 1͒ and longs to important branches of mathematics, but is not make the following approximations: The average position widely taught or used at the college level. Students often coordinate in the Lagrangian along a segment is at the mid- encounter the variational calculus first in an advanced me- point of the segment.7 The average velocity of the particle is chanics class, where they struggle to apply a new mathemati- equal to its displacement across the segment divided by the cal procedure to a new physical concept. This paper provides time interval of the segment. These approximations applied a derivation of Lagrange’s equations from the principle of to segment A in Fig. 1 yield the average Lagrangian LA and 4 least action using elementary calculus, which may be em- action SA contributed by this segment: ployed as an alternative to ͑or a preview of͒ the more ad- ϩ Ϫ vanced variational calculus derivation. x1 x2 x2 x1 L ϵLͩ , ͪ , ͑3a͒ In Sec. II we develop the mathematical background for A 2 ⌬t deriving Lagrange’s equations from elementary calculus. ϩ Ϫ Section III gives the derivation of the equations of motion x1 x2 x2 x1 S ϷL ⌬tϭLͩ , ͪ ⌬t, ͑3b͒ for a single particle. Section IV extends our approach to A A 2 ⌬t demonstrate that the energy and momentum are constants of the motion. The Appendix expands Lagrange’s equations to with similar expressions for LB and SB along segment B. multiparticle systems and adds angular momentum as an ex- ample of generalized momentum. III. DERIVATION OF LAGRANGE’S EQUATION II. DIFFERENTIAL APPROXIMATION TO THE We employ the approximations of Sec. II to derive PRINCIPLE OF LEAST ACTION Lagrange’s equations for the special case introduced there. A particle moves along the x axis with potential energy As shown in Fig. 2, we fix events 1 and 3 and vary the x coordinate of the intermediate event to minimize the action V(x) which is time independent. For this special case the 5 between the outer two events. Lagrange function or Lagrangian L has the form: For simplicity, but without loss of generality, we choose ͑ ͒ϭ Ϫ ϭ 1 2Ϫ ͑ ͒ ͑ ͒ ⌬ L x,v T V 2mv V x . 1 the time increment t to be the same for each segment, 510 Am. J. Phys. 72 ͑4͒, April 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 510 ϭ ⌬ ϩ ⌬ ͑ ͒ SAB LA t LB t. 6 The principle of least action requires that the coordinates of the middle event x be chosen to yield the smallest value of the action between the fixed events 1 and 3. If we set the 8 derivative of SAB with respect to x equal to zero and use the chain rule, we obtain L dxץ L dvץ L dxץ dS AB ϭ ϭ A A ⌬ ϩ A A ⌬ ϩ B B ⌬ t ץ t ץ t ץ 0 dx xA dx vA dx xB dx L dvץ ϩ B B ⌬ ͑ ͒ t. 7 ץ vB dx We substitute Eq. ͑5͒ into Eq. ͑7͒, divide through by ⌬t, and regroup the terms to obtain Lץ Lץ L 1ץ Lץ 1 Fig. 1. An infinitesimal section of the world line approximated by two A ϩ B Ϫ BϪ A ϭ ͑ ͒ 8 .0 ͪ ץ ץ ͩ ⌬ ͪ ץ ץ ͩ .straight line segments 2 xA xB t vB vA To first order, the first term in Eq. ͑8͒ is the average value x on the two segments A and B. In the limit ⌬tץ/Lץ which also equals the time between the midpoints of the two of segments. The average positions and velocities along seg- !0, this term approaches the value of the partial derivative ments A and B are at x. In the same limit, the second term in Eq. ͑8͒ becomes the time derivative of the partial derivative of the Lagrangian x ϩx xϪx v)/dt. Therefore in the limitץ/Lץ)ϭ 1 ϭ 1 ͑ ͒ with respect to velocity d xA , vA ⌬ , 4a 2 t ⌬t!0, Eq. ͑8͒ becomes the Lagrange equation in x: ץ ץ xϩx x Ϫx ϭ 3 ϭ 3 ͑ ͒ L d L xB , vB ⌬ . 4b Ϫ ͩ ͪ ϭ0. ͑9͒ vץ x dtץ t 2 ͑ ͒ The expressions in Eq. 4 are all functions of the single We did not specify the location of segments A and B along ͑ ͒ variable x. For later use we take the derivatives of Eq. 4 the world line. The additive property of the action implies with respect to x: that Eq. ͑9͒ is valid for every adjacent pair of segments. dx 1 dv 1 An essentially identical derivation applies to any particle A ϭ A ϭϩ ͑ ͒ , ⌬ , 5a with one degree of freedom in any potential. For example, dx 2 dx t the single angle ␸ tracks the motion of a simple pendulum, so its equation of motion follows from Eq. ͑9͒ by replacing x dxB 1 dvB 1 ϭ , ϭϪ . ͑5b͒ with ␸ without the need to take vector components. dx 2 dx ⌬t Let L and L be the values of the Lagrangian on segments A B IV. MOMENTUM AND ENERGY AS CONSTANTS A and B, respectively, using Eq. ͑4͒, and label the summed OF THE MOTION action across these two segments as SAB : A. Momentum We consider the case in which the Lagrangian does not depend explicitly on the x coordinate of the particle ͑for ex- ample, the potential is zero or independent of position͒. Be- cause it does not appear in the Lagrangian, the x coordinate is ‘‘ignorable’’ or ‘‘cyclic.’’ In this case a simple and well- known conclusion from Lagrange’s equation leads to the mo- mentum as a conserved quantity, that is, a constant of mo- tion. Here we provide an outline of the derivation. For a Lagrangian that is only a function of the velocity, LϭL(v), Lagrange’s equation ͑9͒ tells us that the time de- v is zero. From Eq. ͑1͒, we find thatץ/Lץ rivative of vϭmv, which implies that the x momentum, pϭmv,isץ/Lץ a constant of the motion. This usual consideration can be supplemented or replaced by our approach. If we repeat the derivation in Sec. III with LϭL(v) ͑perhaps as a student exercise to reinforce under- ͒ Fig. 2. Derivation of Lagrange’s equations from the principle of least action. standing of the previous derivation , we obtain from the prin- Points 1 and 3 are on the true world line. The world line between them is ciple of least action ͑ ͒ L dvץ L dvץ approximated by two straight line segments as in Fig. 1 . The arrows show dS that the x coordinate of the middle event is varied. All other coordinates are AB ϭ ϭ A A ⌬ ϩ B B ⌬ ͑ ͒ t. 10 ץ t ץ 0 fixed. dx vA dx vB dx 511 Am. J. Phys., Vol. 72, No.
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