An example of Lagrangian for a non-holonomic system

Transcription

1 Uniersit of North Georgia Nighthaks Open Institutional Repositor Facult Publications Department of Mathematics An eample of Lagrangian for a non-holonomic sstem Piotr W. Hebda Uniersit of North Georgia piotr.hebda@ung.edu Beata A. Hebda Uniersit of North Georgia beata.hebda@ung.edu Follo this and additional orks at: Part of the Mathematics Commons Recommended Citation Hebda Piotr W. and Hebda Beata A. "An eample of Lagrangian for a non-holonomic sstem" 05. Facult Publications. Paper. This Article is brought to ou for free and open access b the Department of Mathematics at Nighthaks Open Institutional Repositor. It has been accepted for inclusion in Facult Publications b an authorized administrator of Nighthaks Open Institutional Repositor.

2 An eample of a Lagrangian for a non-holonomic sstem Piotr W. Hebda a Beata A. Hebda Department of Mathematics Uniersit of North Georgia Oakood Georgia 0566 USA An adjustable to-mass-point Chaplgin Sleigh is used as an eample of a non-holonomic sstem. Netonian equations of motion based the assumption of zero irtual ork done b constraints are calculated. A Lagrangian that reproduces these equations as its unmodified Euler- Lagrange equations is then eplicitl gien. The Lagrangian uses ariables that are present in the Chaplgin Sleigh equations of motion as ell as some additional ariables. Some of the Euler- Lagrange equations of that Lagrangian are non-differential. These non-differential equations automaticall and completel reduce out all of these additional ariables so that onl the ariables that appear in the original equations of motion remain in the final dnamics of the sstem. I. INTRODUCTION While the Netonian equations of motion seem to be phsicall more fundamental than the Lagrangian that produces these equations as its Euler-Lagrange equations the Lagrangian is still of great interest since it proides a natural frameork for further stud of the sstem. For eample it is a starting point for calculating the Hamiltonian and the Poisson brackets structure. The problem of constructing a Lagrangian for gien equations of motion has been therefore etensiel studied but it is still not completel resoled. Quite often e stud a mechanical sstem for hich a Lagrangian is alread knon but hich is subsequentl modified b imposing additional constraints. The constraints a Author to hom correspondence should be addressed. Electronic mail: Piotr.Hebda@UNG.edu

3 modif the original equations of motion and the modifications then lead to the need of modifications of the Lagrangian. Modifing the Lagrangian is quite simple if the constraints are of holonomic tpe constraints that could be epressed b restricting the alloable positions of the sstem. In this case the ne Lagrangian is obtained b adding the constraints each constraint multiplied b its on so called Lagrange multiplier to the original Lagrangian. In the case of non-holonomic constraints these are constraints that inole elocities and cannot be reduced to restricting the positions onl the situation is not so simple. Adding constraints multiplied b the Lagrange multipliers to the original Lagrangian ill produce equations of motion that are acceptable from mathematical point of ie but the are different from actual phsical equations that result from such constraints. Specificall in the case of non-holonomic constraints the constraints forces resulting from the use of Lagrange multipliers do not satisf the condition of zero irtual ork hich is epected to be satisfied in real-orld mechanics. Because of that fact the use of Lagrange multipliers is generall rejected in the case of nonholonomic constraints. A commonl accepted approach for such a case is not to modif a Lagrangian at all but to obtain the Euler-Lagrange equations from the Lagrangian and then modif these equations to include forces resulting from the constraints. Hoeer as the resulting final equations of motion are not the usual Euler-Lagrange equations of a knon Lagrangian anmore some adantages of using the Lagrangian are lost. Another approach to the non-holonomic constraints can be done b adapting the Bateman- Morse-Feshbach approach. 5 In this approach a Lagrangian for an arbitrar sstem of equations of motion is constructed b Lagrangian being equal to sum of all equations of motion each equation multiplied b a ne ariable. The method is somehat analogous to Lagrange multipliers method etended not onl to equations representing constraints but to all equations

4 of the sstem. This approach results in getting correct equations of motion directl as the Euler- Lagrange equations but it also creates additional non-phsical ariables that ere not eisting in the original equations of motion. The additional ariables are then present in the Lagrange- Hamilton formalism that follos and it is not clear ho to interpret them. So this approach hile relatiel simple is not commonl accepted as a resolution of the Lagrangian construction problem. In this ork e sho a Lagrangian for one specific eample of a non-holonomic sstem using the eplicit solutions of the equations of motion to construct the Lagrangian rather than a combination of the kinetic and potential energies used in a tpical process of getting a Lagrangian. On some leel this is a quite satisfing approach since one ma claim that solutions are more fundamental objects than kinetic and potential energies - solutions of equations are directl obserable and the alas eists hile kinetic and potential energies are more abstract constructs and in some cases ma not eists at all. On the other hand generalization of our approach to other eamples ma be problematic because in man situations e do not hae eplicit solutions of the equations of motion and in these cases e ill not be able to get the Lagrangian eplicitl hich ma be a serious draback. Hoeer some preliminar results 6 suggest that een in such cases e can proe the eistence of a Lagrangian hich b itself is an interesting result. Our approach produces the correct equations of motion directl from the Lagrangian as the Euler-Lagrange equations ith no further modifications necessar. Similarl to the Bateman- Morse-Feshbach approach 5 e also use ariables that do not appear in the original equations of motion. We aoid the basic difficult of the Bateman-Morse-Feshbach approach though because some of the Euler-Lagrange equations obtained from our Lagrangian are constrains

5 rather than differential equations. The constraints then automaticall eliminate all ariables that do not appear in the original equations of motion hile leaing the original equations of motion intact. The organization of our presentation is as follos: In section II e define the adjustable to-mass-point Chaplgin Sleigh mechanical sstem e sho that it is a non-holonomic sstem and e derie its equations of motion. In section III e present the proposed Lagrangian and e derie and simplif its Euler- Lagrange equations. We sho that some of the equations are identical to the equations of motion for the adjustable to-mass-point Chaplgin Sleigh presented in section II. We also sho that other Euler-Lagrange equations are constraints or time deriaties of the constraints and that these constraints eliminate all the additional ariables used to create the Lagrangian leaing in the final dnamics onl the ariables that ere present in the original equations of motion. In section IV e comment on a possibilit to use Dirac s Theor of Constraints 78 to obtain the Hamiltonian formalism for our Lagrangian. II. THE ADJUSTABLE TWO-MASS-POINT CHAPLYGIN SLEIGH Phsicall our mechanical sstem ill be made of to particles each moing freel in to dimensions and each haing a unitar mass. Their position ill be gien b the usual ariables ith the ariables describing the first particle and ariables describing the second.

6 Using the time deriaties of the ariables produces the folloing equations of motion still unmodified b the constraints as: 0 i 0 ii 0 iii 0 i fi ii iii here a dot aboe a ariable means the time deriatie. To obtain an adjustable to-mass-point Chaplgin Sleigh from this free sstem e impose to constraints: 0 i 5

7 0 ii that are supposed to be satisfied b all solutions of the equations of motion. The constraint i is holonomic since it is obtained b taking a time deriatie of r. r ma ar ith different initial positions of the particles but the constraint i assures that it remains constant during the motion. The possibilit of haing different alues of r is the reason for calling our Chaplgin Sleigh adjustable. The constraint ii is non-holonomic. It can be interpreted as the elocit of the center of mass being parallel to the ector hich is starting at the first particle and ending at the second. The ector is parallel to since the constraint ii sas it is perpendicular to. 6 This constraint ii is non-holonomic since it allos a rotation of the particles around its center of mass and also it allos a translation along the ector starting at the first particle and ending at the second. Combinations of these rotations and translations allo to reach all possible positions of the particles once the distance beteen the particles is established b the constraint i. Therefore ii is not imposing an restrictions on the possible positions of the sstem hile imposing restrictions on possible elocities. k The constraints ma be ritten in standard form as a 0 jk k here j represents the constraint i and j represents the constraint ii. Direct comparison then gies:

8 a i a ii a iii a i a a i a ii a iii We assume that the forces of the constraint are such that the do zero ork during instantaneous irtual displacements. It can be shon that from this assumption e get the constraint s forces to be F i j a j ji These forces then modif the equations of motion giing 5i 5ii 5iii 5i 5 5i 7

9 8 5ii 5iii The constraints must be presered in time. So their time deriaties must be zero. This fact and the equations of motion 5 gie us after somehat tedious calculations the folloing epressions for and : Substituting that into the equations o motion 5 after some simplifications gies: 6i 6ii 6iii 6i 6 6i 6ii 6iii

10 9 0 6i 0 6 The last to equations are constraints. The equations 6 are the final equations of motion of the adjustable to-mass-point Chaplgin Sleigh. III. THE LAGRANGIAN AND EULER-LAGRANGE EQUATIONS Consider the Lagrangian. sin sin sin sin sin sin sin sin L 7 In the Lagrangian 7 all the 5 ariables namel 8 are treated on equal footing as independent ariables. The specific formula for this Lagrangian as obtained using the procedure described b authors in a separate ork. 6

11 0 To obtain the equations of motion from the Lagrangian 7 e use the standard Euler Lagrange equations in the form i i q L q L dt d here 5... i q i represents all 5 ariables 8. We obtain 5 equations: sin sin sin sin sin sin sin sin 9i 9ii 9iii 0 9i i 0 9ii 0 9iii 0 9i 0 9

12 0 9i sin sin sin sin 9ii sin sin sin sin 9iii sin sin sin sin 9i 9 9i 9ii sin 9iii sin 9i sin 9 sin 9i sin 9ii sin 9iii sin 9i

13 sin 9 Using equations 9i 9i to simplif equations 9i 9iii and 9ii 9i and rearranging the order of equations e get: 0 0i 0 0ii 0 0iii 0i 0 0i 0ii 0iii 0i sin 0 sin 0i sin 0ii 0iii sin sin 0i sin 0

14 sin 0i sin 0ii 0 0iii 0 0i i 0 0ii 0 0iii 0 0i 0 0 The equations 0i 0i gie us time deriaties of the ariables. The equations 0ii 0 are constraints. Constraints must hold as time progresses so for each constraint time deriaties of both sides must be equal. In general taking time deriaties of eisting constraints ma create ne constraints and/or gie time deriaties of the ariables that ere not included in the earlier equation. In case of equations 0 taking time deriaties of constraints creates no ne constraints. Instead it gies us time deriaties for all the ariables that had no time deriaties in equations 0 namel time deriaties of the ariables. This somehat tedious process produces the folloing formulas for the time deriaties of all the 5 ariables of the Lagrangian 7:

15 sin sin sin sin i ii iii i i ii iii i i 0 ii 0 iii 0 i 0 0 i 0 ii 0 iii

16 0 i 0 0 i 0 ii 0 iii 0 i 0 and the constraints sin i sin ii sin iii i sin sin sin i sin ii sin iii i 5

17 6 i 0 ii 0 iii 0 i 0 0 i 0 ii 0 iii 0 i B direct calculations using the constraints i - iii e get sin i sin ii sin iii

18 7 sin i 0 0 If e no use the equations i i to replace the right sides of the equations i i and include equations i and ith other equations e obtain the folloing sstem of equations of motion: i ii iii i i ii iii

19 ith constraints 0 i 0 sin i sin ii sin iii sin i sin sin i sin ii iii sin i i 0 ii 0 iii 0 i 0 8

20 0 i 0 ii 0 iii 0 i i ii iii 0 i 0 0 i 0 ii 0 iii 0 i 0 0 i 0 ii 0 iii 0 i 9

21 0 0 0 i 0 ii Let us stress that the entire sstem of equations and is completel equialent to the sstem 9 of the Euler-Lagrange equations of the Lagrangian 7. Let us no interpret the sstem of equations and. First the constraints ma be interpreted as implicit definition of all the ariables that do not appear in the adjustable tomass-point Chaplgin Sleigh 6 b the ariables that do appear there. Moreoer it can be shon that all time deriaties of the ariables not appearing in the Chaplgin Sleigh equations can be obtained directl b taking time deriaties of the constraints and then using equations. This means that the ariables are completel dependent of ariables. The former are defined b the latter and time deriaties of the former are the results of these definitions and the time deriaties of the latter. Therefore the ariables are just redundant ariables on the space described b the ariables. Please also notice that the equations are identical to equations of motion of the Chaplgin Sleigh 6 obtained in section II. Concluding the Lagrangian 7 gies the correct equations of motion for the Chaplgin Sleigh hile at the same time completel eliminating the additional ariables used for its construction from the final dnamics of the sstem.

22 V. A COMMENT ON A HAMILTONIAN Since our Lagrangian is degenerate to the etreme ith no elocities epressible b the canonical momenta the Dirac s Theor of Constraints 78 is a natural choice for creating the Hamiltonian formalism. Some preliminar results 6 suggest that it ill be possible to eplicitl calculate both the Hamiltonian and the Dirac s Brackets for the adjustable to-mass-point Chaplgin Sleigh shon in this ork and that Dirac s Brackets of all ariables appearing in the Lagrangian 7 but not appearing in the equations of motion 6 as ell as the canonical momenta of these ariables ill be equal to zero ith eer function using an ariable of the sstem. REFERENCES R. M. Santilli Foundations of Theoretical Mechanics Springer-Verlag Ne York 978 ol.. H. Goldstein C. P. Poole and J. L. Safko Classical Mechanics rd Ed. Addison-Wesle Boston 00 p. 6 A. M. Bloch Nonholonomic Mechanics and Control Springer Ne York 00 p. P. M. Morse and H. Feshbach Methods of Theoretical Phsics McGra-Hill Ne York 95 p H. Bateman Phs. Re P. W. Hebda and B. A. Hebda Spontaneous Dimension Reduction and the Eistence of a local Lagrange- Hamilton Formalism for Gien n-dimensional Netonian Equations of Motion Facult Publications paper P. A. M. Dirac Can. J. Math P. A. M. Dirac Lectures on Quantum Mechanics Belfer Graduate School of Science Ne York 96