# CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

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1 CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1

2 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O z P F 1 y m (x,y,0) f N The equaton of moton s mr F( r,, r t) p 2

3 In component form, the equaton of moton s mx my P mz P P F ( x, y, z, x, y, z, t) x F ( x, y, z, x, y, z, t) y F ( x, y, z, x, y, z, t) z Also, moton s restrcted to xy plane z = 0 - equaton of constrant It s a geometrc restrcton on where the partcle can go n the 3-D space. Clearly, there s a constrant reacton (force) that needs to be ncluded n the total force F. 3

4 Ex. 2: Consder a partcle movng on a surface. x O z y m Surface f(x,y,z)=c Now, the moton s confned to a prespecfed surface (e.g. a roller coaster). The surface s defned by the relaton: f (x, y, z) - c = 0 - equaton of constrant. The equaton of moton wll agan be the same. 4

5 The constrants n the two examples are geometrc or confguraton constrants. They could be ndependent of tme t, or could depend explctly on t. For an N partcle system, f the postons of partcles are gven by r, r, r, , the constrant can be wrtten as: f ( r, r,, r, t) Ths s an equaton of a fnte or geometrc or holonomc constrant. N 5

6 Ex. 3: Double pendulum: t conssts of two partcles and two massless rgd rods The masses are O m1:( x1, y1) l 1 m2:( x2, y2) (x 1,y 1 ) m ( z1 z2 0: planar moton) 1 l 2 Number of coordnates x m 2 requred s 4 - used to defne the confguraton There are certan constrants on moton: l ( x y ), l ( x x ) ( y y ) y (x 2,y 2 ) 6

7 2 equatons of constrant (they are holonomc, geometrc, fnte etc.) Degrees-of-freedom: the number of ndependent coordnates needed to completely specfy the confguraton of the system (4-2) = 2. One could perhaps fnd another set of two coordnates (varables) that are ndependent: e.g.,,, the two angles wth the vertcal. 1 2 Then, there are no constrants on 1, 2, 7

8 Ex. 4: A dumbbell movng n space Z C m 1 X O Y m 2 one possble specfcaton of poston s: m : x, y, z ; m : x, y, z these are 6 varables or coordnates, and there s one constrant ( x x ) ( y y ) ( z z )

9 degrees-of-freedom of the system 6-1 = 5 another possble specfcaton for the confguraton of the system: Z C m 1 O X Y m 2 Locaton of center of mass C: ( xc, yc, zc) ; and orentaton of the rod: (, ). These are ndependent no constrant relaton for these varables. 9

10 generalzed coordnates - any number of varables needed to completely specfy the confguraton of a system. e.g., for the dumbbell n space moton: (x,y,z,x,y,z ), (x,y,z,,θ) C C C there are two sets of generalzed coordnates Important: some sets consst of ndependent coordnates (no constrants) where as others are not ndependent. 10

11 Ex. 5: Ice skate Basc facts: Confguraton of the skate can be specfed by the coordnates (x, y) and the angle. The ce skates can only move along the plane of the skate,.e., n the tangent drecton specfed by angle. (a constrant) O Z v Y Path n x-y plane P n X 11 t

12 Let t - tangent to the path, n - normal to the path. Then v n 0 for the skate, or or ( x yj ) ( cos j sn ) 0 x sn y cos 0 In general a constrant whch depends both, on coordnates and ther tme dervatves. ( r,, r, r,, r, t) 1 N 1 N 0 Such a constrant s called a knematcal, dfferental, nonholonomc constrant. 12

13 We have seen then that, n general: Holonomc constrants are of the form j ( q,..., q, t) 0, j 1,2,3,..., g 1 N equalty constrants nvolvng only generalzed coordnates and tme Nonholonomc constrants are of the form ( q,..., q, q,..., q, t) 0, j 1,2,3,..., d j 1 N 1 N they depend on generalzed coordnates, veloctes, as well as tme. 13

14 Fundamental dfference: A geometrc constrant restrcts the confguratons that can be acheved durng moton. Certan regons (postons) are naccessble A knematc constrant only restrcts the veloctes that can be acqured at a gven poston. The system can, however, occupy any poston desred (e.g.: one can reach any pont n the skatng rnk - t s just that one cannot move n arbtrary drecton). 14

15 We can also wrte the constrants n the form: (n dfferental form) n 1 a ( q,..., q, t) dq a ( q,..., q, t) dt 0, j 1 n jt 1 n j = 1, 2,., d Whether a constrant s holonomc or nonholonomc depends on whether the dfferental form s ntegrable or nonntegrable. 15

16 Ex 6: Partcle model of a skate: two equal masses are connected by a massless rgd rod. They slde on the XY plane. G s the centrod of the system. Y t z1 z2 0 The other constrants G m on moton are: 1 length s constant n ( x x ) ( y y ) X (holonomc) O Skate cannot move vg n 0 along n drecton (nonholonomc) m 2 16

17 We now defne these constrants n terms of the physcal coordnates, and then the generalzed coordnates q : The CG has v G [( x 1 x 2) ( y 1 y 2) j]/ 2 Now n cos j sn cos ( x x ) / l, sn ( y y ) / l n [( y y ) ( x x ) j]/ l The nonholonomc constrant s ( x x )( y y ) ( y y )( x x )

18 The coordnates are: ( x1, y1, z1), ( x2, y2, z2) Generalzed coordnates: x q, y q, z q, x q, y q, z q Then, the constrants have to be wrtten n terms of q s: z 0, z constrants #1 and #2 ( x x ) ( y y ) ( z z ) constrant #3 ( x x )( y y ) ( y y )( x x ) constrant #4 18

19 Constrant #1: In dfferental form 0 z dz1 dt 0 dz1 0 or d( q3) 0 dt In general form, we have 6 a jdq a jtdt 0, j 1 1 or a dq a dq a dq a dt t z a 0, a 0, a 1, a 0, a 0, a 0, a t 0 19

20 Constrant #2: In dfferental form dz2 dt 0 dz dt 2 2 In general form, we have 6 z 0 z 0 0 or d( q ) a jdq a jtdt 0, j 2 1 or a dq a dq a dq a dt t a 0, a 0, a 0, a 0, a 0, a 1, a t 0 20

21 constrant #3: ( x x ) ( y y ) ( z z ) In dfferental form ( x x )( dx dx ) ( y y )( dy dy ) ( z z )( dz dz ) or ( x x ) dx ( x x ) dx ( y y ) dy ( y y ) dy or ( z z ) dz ( z z ) dz ( q q ) dq ( q q ) dq ( q q ) dq ( q q ) dq ( q q ) dq ( q q ) dq

22 constrant #4: ( x x )( y y ) ( y y )( x x ) In Dfferental form: ( q q )( dq dq ) ( q q )( dq dq ) 0 or ( q q ) dq ( q q ) dq ( q q ) dq ( q q ) dq (0) dq (0) dq

23 dfferental form of constrants (n n general): a dq a dt 0, j 1,2,3,... m A constrant (or dfferental form) s ntegrable f 1 j jt a q a q j k jk a t a q,, k 1,2,..., n j jt These are condtons for exactness (of a dfferental form) 23

24 Ex 7: Consder a constrant a x a x a t t In dfferental form, t s : a dx a dx a dt 0 Suppose that a, a, a Clearly, the constrant s ntegrable: The ntegrated form s: t are constants. a x a x a t c t Mathematcally, f ntegrable, there s a functon such that d dt 0 0 or dx dx dt x x t 1 2 a x, a x, a t t 24

25 Clearly, then or a x Smlarly a x a a x x x x x x a a a a 11 1t 12 1t 0, 0 t x t x These are suffcent condtons for the constrant to be ntegrable. 25

26 e.g.: consder the constrant #3 (j = 3) ( q1 q4 ) dq1 ( q4 q1 ) dq4 ( q2 q5 ) dq2 ( q q ) dq 0 Here Thus Smlarly and a / q 1 a / q, etc. j1 4 j a q q, a q q, a 0 j1 1 4 j2 2 5 j3 a / q 0 a / q j1 2 j2 1 a q q, a q q, a 0 j4 4 1 j5 5 2 j6 Ths constrant s ntegrable. 26

27 Also, a j1 / q3 0 a j3 / q1 a a q a q q q j5 / 0 ; j1 5 j5 5 2 a / q 0 a / q ; a / q a / t 0 j1 6 j6 1 jt 1 j1 1.. a / q 0 a / q j2 3 j3 2 27

28 Now, consder constrant #4: Then or, ( q5 q2 ) dq1 ( q4 q1 ) dq2 ( q5 q2 ) dq4 a41 q5 q2, a42 ( q4 q1 ), a43 0, a / q 1 a / q ( q q ) dq not an exact dfferental;.e., t s not an ntegrable constrant. 28

29 Classfcaton: an N partcle system s sad to be: Holonomc - f all constrants are geometrc, or f knematc - are ntegrable (reducble to geometrc). Nonholonomc - f there s a constrant whch s knematc and not ntegrable. Schleronomc - all the constrants, geometrc as well as knetc, are ndependent of tme t explctly. Rheonomc - f at least one constrant depends explctly on tme t. 29

30 Possble and Vrtual Dsplacements Suppose that a system of N partcles, wth poston vectors r1, r2,, r N has d geometrc constrants (,,,...,, ) 0, 1,2,3,...,, r1 r2 r3 r N t d and g knematc constrants N 1 l r D 0, 1,2,..., g j j j Here l l ( r, r, r,..., r, t), etc. j j N 30

31 In dfferental form j N 1 and N 1 j r / t 0, 1,2,..., d (1) j l r D 0, 1,2,..., g (2) j j j For the gven system at tme t, wth poston fxed by the values of r1, r2,..., r N, the veloctes cannot be arbtrary. They must satsfy d + g equatons. 31

32 Possble veloctes: the set of all veloctes whch satsfy the (d + g) lnear equatons of constrants. j 3N > (d + g) nfnty of possble veloctes. One of these s realzed n an actual moton of the system. Let d r r dt, 1,2,..., N These are the possble (nfntesmal) dsplacements. They satsfy N 1 j dr j dt 0, 1,2,..., d (3) t 32

33 N and l d r D dt 0, j 1,2,..., g (4) 1 j j Agan, there are d + g equatons n 3N possble (scalar) dsplacements d r, 1,2,, N. Consder two sets of possble dsplacements at the same nstant at a gven poston of the system: dr v dt and dr v dt, 1,2,..., N Both these dsplacements satsfy the above equatons. 33

34 Takng ther dfferences and N j 1 N 1 ( d r d r ) 0, 1,2,..., d j These are homogeneous relatons not nvolvng (dt). Def: r - vrtual dsplacement d r d r Vrtual dsplacement a possble dsplacement wth frozen tme. (dt set to 0). j l ( d r d r ) 0, 1,2,..., g j j j 34

35 Note: If the constrants are ndependent of tme (schleronomc), a possble dsplacement = vrtual dsplacement. Ex 8: A partcle s movng on a fxed surface defned by f (x, y, z) c = 0. The velocty v s always tangent to the surface d r n r n 0 where f f f n f / f, f j k x y z x z P y dr=vdt Surface f(x,y,z)=c 35

36 Ex 9: A partcle s movng on a surface whch tself moves to the rght wth velocty u. Possble veloctes v v u R ( v R relatve velocty) Possble dsplacements d r vdt ( v R u) dt Two possble dsplacements: d r ( v u) dt, d r ( v u) dt R R x P z v y u v r Surface f(x,y,z)=c(t) 36

37 Thus, a vrtual dsplacement s r d r d r ( v R v R ) dt r R r R n u r dr Note that dr s along absolute velocty drecton, whereas r s along relatve velocty or tangent to the surface (frozen constrant) (set dt = 0). e t u 37

38 Degrees-of-freedom: N - number of partcles (d + g) - geometrc + knematc constrants there are n = 3N - (d + g) ndependent vrtual dsplacements Problem of Dynamcs: Gven a system wth - external forces F F ( r, r, t), 1,2,..., N; Intal postons r o, and ntal veloctes compatble wth constrants; we need to v o 38

39 determne the moton of the system of partcles,.e., the postons ( r( t )), the veloctes r, and the constrant or reacton forces m r F R, 1,2,..., N N j 1 N 1 j (3N equatons) r / t 0, 1,2,..., d j (d equatons) l d r D dt 0, j 1,2,..., g j j R, 1,2,, N. (g equatons) 39

40 In these equatons, the unknowns are: - 6N unknowns Thus, addtonal relatons requred: 6N - (3N + d + g) = 3N - (d + g) (equal to the number of degrees-of-freedom) Need to defne concept of workless constrants. 6.4 Vrtual Work Defnton: A workless constrant s any constrant such that the vrtual work (work done n a vrtual dsplacement) of the constrant forces actng on the system s zero for any reversble vrtual dsplacement. 40 n r, R

41 Ex 10: Consder a double pendulum. The postons are: r x y j r2 x y j 2 2 Constrants are: ( x y ) 0 (1) or 1 1 x x 1 1 y1 y1 0 (dfferental form) ( x x ) ( y y ) 0 (2) O Y 1 l X A m 2 l B m 41

42 or ( x2 x1)( x 2 x 1) ( y2 y1)( y 2 y 1) 0 (dfferental form) Consder FBD s Then, 1 tan ( x / y ) tan ( x x) y y The equatons of moton for A are: x : mx T sn T sn (3) y : my T cos T cos mg (4) T 1 1 mg A T

43 The equatons of moton for B are: x : mx 2 T2 sn 2 (5) y : my mg T cos (6) T 2 mg 2 B 2 N dfferental equatons of moton 2 equatons of constrant varables (unknowns): x 1 ( t), y 1 ( t) x ( t), y ( t) 2 2 T ( t), T ( t)

44 Ex 11: Consder the moton of an deal pendulum The poston s r x y j Newton s Law: r x y j F FBD: mr The reacton force s: R=R x + R y j R y mg A R x O Y l A m X 1 g 44

45 Newton s 2nd law gves x : Rx mx (1) y : R mg my (2) y Constrant on moton s: x y l 0 (3) or xx yy 0 ( r r 0) (dfferentated form) or x dx + y dy = 0 (dfferental form) 45

46 Countng: 3 equatons 4 varables x, y, R, R x y Need one more relaton: - somethng about the nature of the constrant force R R R j x y Hndsght: We know R along the rod normal to the drecton of velocty does no work n moton of the partcle (moton that s consstent wth the constrant). 46

47 Work done n a vrtual dsplacement of the N system W R r 1 Ex 12: Consder a partcle movng on a smooth surface. Then the work done by the constrant force R n a vrtual dsplacement r (consstent wth constrant) s P r R e t e n W R r Rn r 0 47

48 Ex 13: Consder the same stuaton, wth the partcle now movng on a movng surface: Here agan W Rn r 0 Note however that R d r 0 snce dr s not n the tangent drecton. Ex 14: Consder two partcles connected by m a rgd rod: Z 2 X O Y u P m 1 r R l dr e t e n udt 48 u

49 X O Z Y dr 1 R where 1 R2 R2 er R2 R R 2 1 unt vector from m er The length constrant s: ( r1 r2) ( r1 r2) l 0 Dfferentatng, the constrant on possble dsplacements s: ( r1 r2) ( d r1 d r2) 0 Thus, ( r1 r2) ( r1 r2) 0 er ( r1 r2) or e r e r R e R to dr 2 m 2 m 1 R 2 R 1 m 1 R 2 49

50 w vrtual work done on the system (the two partcles) R r R r R e r R e r R 1 2 R 2 R e r e r 2( R 1 R 2) 0 Other examples of workless constrants: hnged constrants; sldng on smooth surfaces; rollng wthout slppng, etc. Remark: Reacton forces correspondng to workless constrants may do work on ndvdual components of the system. 50

51 Ex 15: A partcle moves on a fxed rough surface. f= k R R P e n Clearly, the work done by the f(x,y,z)=0 normal force R n a vrtual dsplacement s R r 0 Note that the frcton force does do work t can be accounted for by treatng as an external force. f k R r e t 51

52 The Prncple of Vrtual Work: Consder a system of N partcles, wth postons r, 1,2,..., N Forces actng on the th partcle of mass m : F R} workless constrant forces external as well as constrant forces not accounted for n workless constrant forces. 52

53 Statc equlbrum for the th partcle F R 0, 1,2,..., N Suppose that the system also satsfes some constrants: N r 0, 1,2,..., d geometrc j 1 j j N 1 l r 0, j 1,2,..., g j knematc (these are requrements wrtten n terms of the vrtual dsplacements) 53

54 Now, vrtual work done by all the forces actng on the system as a result of an arbtrary vrtual dsplacement r at a gven system confguraton s N W ( F R ) r 1 (Note: r are requred to satsfy the constrant relatons,.e., are the possble nfnte dsplacement wth frozen tme). Assume workless constrants: N 1 R r 0 54

55 N 1 F r W (scalar eqn.) If a system of partcles wth workless constrants s n statc equlbrum, the vrtual work of the appled forces s zero for any vrtual dsplacement consstent wth constrants. Also, f the work done at a gven confguraton s zero n any arbtrary vrtual dsplacement from that confguraton, the system must be n statc equlbrum. Prncple of vrtual work. 0 55

56 Ex 16:A nhomogeneous rod AB s restng on two smooth planes. The rod s nonunform wth ts center of mass located at G: AG:GB = k:(1 k). Fnd: The equlbrum poston of the rod. A (l)k G l(1-k) B g The constrants are: the ends must reman n contact wth respectve surfaces. O 56

57 To properly set up the problem, we need to frst defne a coordnate system so that the approprate poston vectors can be defned. Then, we can defne the constrants and the vrtual dsplacements. Let z and z - postons of A and B along A B the nclned surfaces. Also, - the angle of nclnaton of the rod. The constrants are: cos ( Z Z )cos sn ( Z Z )sn A B B A 57

58 The varables are descrbed here on the pcture more clearly: G B h A O z B z A 58

59 A possble set of vrtual dsplacements consstent wth constrants are shown here: h z A A A z A G G O B z B B z B 59

60 FBD: R A A G W B R B Prncple of vrtual work: R A r A R B r B W rg 0 = 0 (workless constrants) W r 0 or W h 0 h 0 W j ( x h j) 0 60

61 Now, h za sn k sn h za sn k cos constrants: za zb l cos /cos, zb za l sn /sn cos sn 1 z A cos sn 2 1 cos sn sn cos 2 cos sn or z lsn( )/sn2 A 61

62 dfferentatng, we get za Thus, h { lcos( )/sn2 }. { lcos( )sn /sn2 } kl cos l cos( ) { sn kl cos } 0 sn2 arbtrary vrtual dsplacement { sn cos( )/sn2 k cos } 0 or tan (2k 1)/ tan 62

63 D Alembert s Prncple: Consder a system wth N partcles, the masses are gven by The external force on th partcle geometrc constrants: f ( r, r,..., r, t) 0, 1,2,..., d, 1 2 N knematc constrants: N l r D ( r, t) 0, 1,2,..., g j 1 j j constrants reacton forces R,( r, r, t), 1,2,..., N m F ( t, r, r ) 63

64 The equatons of moton are: m r F R, 1,2,..., N These are subject to the constrants: (n dfferental form) j j N N 1 1 ( f / r ) d r ( f / t) dt 0, 1,2,..., d j j l d r D ( r, t) dt 0, 1,2,..., g j j 64

65 Then, the relatons satsfed by vrtual dsplacements are and j N j 1 N ( f / r ) r 0, 1,2,..., d 1 j l r 0, 1,2,..., g j j j The condton on constrant forces for the constrants to be workless s: N 1 R r 0 65

66 Newton s law N Workless constrants N 1 m r r ( F R ) r 1 1 ( m r F ) r 0 (a sngle scalar equaton) N D Alembert s Prncple (Note: r are not ndependent. They satsfy the dfferental constrants). 66

67 Ex 17: Sphercal pendulum wth varable length partcle of mass m r ( a b cos t), a b 0. O r(t) e, e r e r n OPZ plane; to OPZ plane X Z e P mg e r e Y 67

68 r P r re re poston of the ball: velocty: P r r or r P re r r e r sn e acceleraton: r ( r r r sn ) e r 2 ( r 2r r sn cos ) e ( r sn 2r sn 2r cos ) e vrtual dsplacement: possble velocty r re r e r sn e P possble dspl. d r drer rd e rd sn e constrant: r a b cos t re r r 68

69 or dr ( b sn t) dt snceconstrant frozen dt 0 dr 0 vrtual dsplacement: r r e r sn e External force actng: F mg cos e mg sn e mgk r D Alembert s Prncple ( mr F) r 0 Note: on the FBD of the partcle, tenson force also acts along the rod - a workless constrant force 69

70 mr g r r r mrsn [ r sn 2r sn 2r cos ] 0, ndependent vrtual dsplacements ( ) ( ) 0 2 ( r 2r r sn cos ) g sn 0 ; r sn 2r sn 2r cos 0 (equatons of moton) Here 2 [ sn ( 2 sn cos )] r a b cos t 0 70

71 6.5 Generalzed Coordnates and Forces Ex 18: consder the double pendulum: the poston vectors are r1 x y j 1 1 r2 ( x x ) ( y y ) j the constrants are x y l x y l O y 1 Y l 1 x 1 y 2 A m 1 (x 1,y 1 ) 2 x2 l B m 2 X 71

72 Thus, there are: 4 (or 6 countng z s) varables or generalzed coordnates 2 (or 4 f z ncluded) constrants n = degrees of freedom = 2. Need only 2 ndependent varables (for geometrc constrants case) to specfy the confguraton at any gven tme e.g.: let y, y 1 2 be the two chosen ndependent coordnates. Then, we can wrte x ( l y ), x ( l y ) ( z 0, z 0)

73 r l y ) y j r ( y ) r { l y ( l y )} ( y y ) j r2( y1, y2) Note: geometrc constrant now automatcally satsfed. Another possble choce of generalzed coordnates are:, angles wth y axs

74 Then x l sn, y l cos x2 l2 sn 2, y2 l2 cos 2 r l (sn cos j) r (, ) r l (sn l sn ) ( l cos l cos ) j r (, ) Agan: geometrc constrants automatcally satsfed. 74

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