Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Transcription

1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the constrant forces. Ths method requres that the system be vared through vrtual dsplacements δq j that are consstent wth the constrants. From ths pont, the equatons can be solved for whatever unknown appled forces exst. But ths method does not apply drectly to dynamcal systems. In ths lecture, we wll dscuss the extenson of the prncple of vrtual work to dynamcal system. Ths extenson s based on the work of D Alembert, and s consdered by many to be the most mportant development n the scence of mechancs after Newton[1]. 1.1 D Alembert s Prncple D Alembert s prncple ntroduces the force of nerta I = m a, thereby convertng problems of dynamcs to problems of statcs F = m a F m a = 0 F + I = 0 (1) where we show the transton from Newton to D Alembert n ths expresson. The force F s sometmes referred to as the real force, whch I wll do so n these lectures to dstngush t from the nertal force. The requrement that the sum of all the forces at each partcle be equal to zero s the necessary condton for statc equlbrum. Snce the prncple of vrtual work apples to systems n statc equlbrum, we wll apply t to ths system of forces ncludng the nertal force. The total work done by the forces n ths system through an arbtray vrtual dsplacement n Cartesan coordnates s δw = [ F a + ] F c m r δ r = 0 (2) where we have splt the real forces nto the appled and constrant forces. If the constrant forces are workless, and the vrtual dsplacements reversble and consstent wth the constrants, the total vrtual work becomes δw = [ ] F a m r δ r = 0 (3) Ths equaton expands upon the prncple of vrtual work from statc to dynamcal system. Note, ths equaton apples to both rheonomc and scleronomc system, provded that the vrtual dsplacements conform to the nstantaneous constrant. 1.2 Example As an example let s consder a wedge of mass M on a frctonless surface, wth a block of mass m on the wedge (see Fg. 1). We wll calculate the equatons of moton for ths system. D Alembert s Prncple and The Lagrangan-1 lec4.tex

2 P. Guterrez PSfrag replacements y y x s x M m Fgure 1: Block sldng down a frctonless nclne, wth nclde also free to slde frctonlessly on a flat surface. Usng the coordnate system specfed n Fg 1, the vrtual work consstent wth the constrants, n Cartesan coordnates s δw = mgδy m(ẍ + ẍ )(δx + δx ) Mẍδx mÿ δy = 0 (4) Snce the varables x and y are not ndependent of each other, because of the constrant of movng along the surface of the wedge, we apply the followng transformaton to take nto account of the constrants δx } = δs cos θ x = s cos θ δy = δs sn θ y = s sn θ ẍ (5) = s cos θ ÿ = s sn θ Applyng ths transformaton and groupng lke terms together, the vrtual work becomes [mg sn θ m( s + ẍ cos θ)] δs [Mẍ + m( s cos θ + ẍ)] δx = 0 (6) Notce at ths pont, we reduced the vrtual work such that there are only two ndependent varatons, whch s the number of degrees of freedom: The wedge s constraned to move n one dmenson, as s the block on the wedge. Snce the varatons are ndependent of each other and arbtrary, the terms n brackets must ndependently be equal to zero. Therefore, the equatons of moton are 1.3 Conservaton of Energy mg sn θ m( s + ẍ cos θ) = 0 (7) Mẍ + m( s cos θ + ẍ) = 0 We return to the conservaton of energy from the pont of vew of D Alembert s prncple. Let s start by consderng the vrtual work assocated wth a collecton of partcles n Cartesan coordnates ( F a m r ) δ r (8) D Alembert s Prncple and The Lagrangan-2 lec4.tex

3 P. Guterrez Next assume that the force can be wrtten as the gradent of a scalar, the vrtual work becomes (δv + m r ) δ r = 0 (9) The vrtual dsplacement can be any arbtrary dsplacement that s consstent wth the constrants. We wll select t to be a real nfntesmal dsplacement (dv + m r ) d r = 0 (10) The second term can be converted to a perfect dfferental of a scalar m r d r = m r d r dt = ( ) d 1 dt 2 m r r dt = dt dt dt = dt (11) where T s the knetc energy as prevously defned. Based on ths expresson, the vrtual work becomes dv + dt = d(t + V ) = 0 T + V = E (12) Therefore, the sum of the knetc and potental energy s a constant. The queston we must ask ourselves before usng ths result s, under what condtons does ths hold? The frst condton s that the force be dervable from a scalar potental. The second condton requres the vrtual dsplacements be the same as the real dsplacement. Ths condton s satsfed f the the problem s tme ndependent. That s the constrants and the potental are scleronomc (tme ndependent). 1.4 The Lagrangan We wll now show the connecton of the Lagrangan to D Alembert s prncple. Let s consder a system subject to a set of constrants ( F c + F a m a ) = 0 (13) The vrtual work s δw = ( F c + F a m a ) δ r = 0 (14) Snce we are assumng the the constrants are workless, the constrant forces are removed from the equaton δw = ( F a m a ) δ r = 0 (15) Notce that all the nformaton s stll n ths equaton, the constrant are now n the vrtual dsplacements. Let s now transform the Eq. 15 to a set of generalzed coordnates q j, wth the transformaton beng 1 r = r(q j ) (16) 1 At ths pont I wll drop the subscrpt and the summaton. Everythng that follows holds for a system of N partcles unless otherwse stated D Alembert s Prncple and The Lagrangan-3 lec4.tex

4 P. Guterrez the velocty s v = d r dt = j r q j + r t v = r (17) q q where the second expresson takes nto account that the coordnate transformatons do not depend explctly on the generalzed veloctes. The vrtual dsplacement s gven by δ r = j r δq j (18) From here we can wrte the vrtual work assocated wth the appled forces as F δ r = j F r δq j = j Q j δq j (19) where Q j s the generalzed force. The force of nerta can also be wrtten n generalzed coordnates m r δ r = m r r δq j (20) The second tme dervatve of the Cartesan coordnates can be wrtten n terms of frst dervatve, ths allows some smplfcaton r r = d ( r r ) dt q r d ( ) r (21) j dt where the tme dervatve of the second term s ( ) d r dt and from Eq. 17, we get = v (22) v q j = r (23) Therefore, Eq. 21 can be wrtten as m r r = d ( m v v ) m v v = d [ ( )] 1 dt q j dt q j 2 mv2 j ( ) 1 2 mv2 Substtutng back nto D Alembert s prncple n terms of generalzed coordnates, we get [ ( ) d T T ] Q j δq j = 0 (25) dt q j where T 1 2 mv2 s the knetc energy. If the force s dervable from a potental ( F = V ), then the generalzed force can be expressed as (24) Q j = F r = V r = V (26) D Alembert s Prncple and The Lagrangan-4 lec4.tex

5 P. Guterrez Therefore D Alembert s prncple becomes [ ( d T dt q j j ) (T V ) ] δq j = 0 (27) f the constrants are holonomc, then the coeffcents of the δq j are ndependently equal to zero ( ) d T (T V ) = 0 (28) dt q j Fnally, snce the potentals of ths form are ndependent of the velocty, D Alembert s prncple can be put nto the form ( ) d (T V (T V ) = d ( ) L L = 0 (29) dt q j dt q j where L = T V. 1.5 Hamlton s Prncple In the prevous secton, we derved an expresson that descrbes the moton at a pont n space. In ths secton we derve an expresson that descrbes the general propertes of the moton through an ntegral relaton. We wll show that the D Alembert s prncple can be as the varaton of an ntegral over tme of a sngle scalar functon. Let s start by takng the tme ntegral of the vrtual work δw dt = [ F a m d v dt ] δ r dt = 0 (30) where we have assumed workless constrants. We take the vrtual work assocated wth the appled force and add the assumpton that t can be derved from a scalar potental F a δ r dt = δv dt = δ V dt (31) Now we work to smplfy the term wth the nertal forces. Ths can be done throught an ntegraton by parts d v t1 m dt δ r d t1 dt = m dt ( v δ r) dt + m v d dt (δ r )dt (32) where we work ths out for a sngle term and then rentroduce the summaton at the end. The frst term on the rght hand sde s a total dfferental and therefore easly ntegrate The second term on the rght hand sde can be wrtten as follows m d dt ( v δ r) dt = [m v δ r ] t 1 t0 (33) m v d dt (δ r )dt = m v (δ v )dt (34) D Alembert s Prncple and The Lagrangan-5 lec4.tex

6 P. Guterrez where we nvert the varaton wth the tme dervatve. The next step s to use the varaton of the square o the velocty to get the product of velocty and varaton of the velocty [ ] 1 t1 [ ] 1 m v (δ v )dt = δ 2 m v v dt = δ 2 m v v dt (35) Next we sum over all partcles and combne all three peces of the ntegral δ V dt + δ [ ] 1 2 m v v dt [m v δ r ] t 1 t0 = 0 (36) The second term s the knetc energy T. On the thrd term, we wll mpose the requrement that the varaton on the endponts be zero. Equaton 36 becomes δ (T V )dt = 0 (37) where L s the same functon we found before, except that now we fnd t n terms a mnmzaton prncple 2. The ntegral s defnes the acton A = Ldt δa = 0 (38) Even thought ths proceedure was carred out n rectangular coordnates, we could have transformed the equatons through a pont transformaton to a new set of coordnates and carred through the proceedure n the new coordnates, and we would have found the same answer. Note that n ths statement the coordnates are assumed to be ndependent, therefore the constrants must be holonomc n nature n order to reduce the number of degrees of freedom through substtutons. The nonholonomc problem wll be dscussed later. We have the functon L (Lagrangan) n two dfferent equatons. One s a dfferental equaton that defnes the dynamcs at a sngle pont, and the second s an ntegral equaton that defnes the global propertes of the moton. The queston that we must now answer s how are the two equatons connected. For ths we wll need to learn somethng about the calculus of varatons. 1.6 Example Lagrangan Consder a bead constraned to move along a wre that makes an angle θ wth respect to the upward vertcal. The wre rotates about the vertcal as shown n the Fg. 2 wth an angular velocty ω. Gravty acts downward. We wsh to determne the Lagrangan usng an approprate set of generalzed coordnates. Before we start settng up the Lagrangan, note that the wre does work on the bead, but the wre forms a workless constrant. The reason the constrant s workless, s that we take the nstantaneous constrant and then apply a vrtual dsplacement. To setup the problem, we start n Cartesan coordnates L = 1 2 m ( ẋ 2 + ẏ 2 + ż 2) mgz (39) 2 Actually at ths pont we are fndng a statonary pont, ether mnmum or maxmum. D Alembert s Prncple and The Lagrangan-6 lec4.tex

7 P. Guterrez ω z θ PSfrag replacements x Fgure 2: Bead on a rotatng wre. The constrants are gven by x = r sn θ sn ωt y = r sn θ cos ωt z = r cos θ ẋ = ṙ sn θ sn ωt + ωr sn θ cos ωt ẏ = ṙ sn θ cos ωt ωr sn θ sn ωt ż = ṙ cos θ (40) where r s along the wre. Workng through the algebra, leads to L = 1 2 m(ṙ2 + ω 2 r 2 sn 2 θ) mgr cos θ (41) References [1] The Varatonal Prncples of Mechancs, C. Lanczos pgs. xx, xxx [2] Classcal Dynamcs, D.T. Greenwood chap. 1 D Alembert s Prncple and The Lagrangan-7 lec4.tex