Three views of mechanics

Transcription

1 Three vews of mechancs John Hubbard, n L. Gross s course February 1, Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental energy. Everythng s requred to be smooth, or at least of class C 2. There are then equatons of moton. whch are a flow n T M,.e., a map Φ : T M R T M. Intutvely, Φ((q, ξ, t should be the poston at tme t of the system startng out at (q, ξ at tme. Ths should be the flow of a frst order dfferental equaton on T M, or of a second-order dfferental equaton on M. I know of three ways of settng up ths equaton; all of them are a lot harder than one mght expect: 1. The Newtonan approach: F = ma. 2. The Lagrangan approach: prncple of least acton. 3. The Hamltonan approach: smplectc manfolds and the symplectc gradent. I wll examne these one at a tme. 1.1 The Newtonan approach Here F = V. Already, we need to be careful: dv s naturally a 1-form on M, but turnng t nto the vector-feld V requres an dentfcaton of tangent spaces and cotangent space. We have one at hand: K s a non-degenerate quadratc form on T q M at every pont q M, provdng us wth such an dentfcaton. Stll, t mght be better to wrte K V, to remember that even to defne V we need to know K. But the real problem s a, the acceleraton. If we have a curve γ : I M, the velocty vector γ (t s a perfectly well defned element of T γ(t M. But γ (t = lm h γ (t + h γ (t h 1

2 s NOT well defned: the two vectors beng subtracted belong to dfferent spaces. The dfferental geometers know how to deal wth ths: they defne the covarant dervatve D γ (tγ (t gven by the Lev-Cvta connecton assocated to K. Ths sn t mmensely hard, but t s non-trval. There s one case where t can be understood wthout any techncaltes: when M R n and K(ξ = m 2 ξ 2 the ordnary nner product. In that case, the formula above for γ (t makes sense, but t doesn t gve a tangent vector to M. We defne (ths s the Lev-Cvta connecton n ths case D γ (tγ (t = the orthogonal projecton of γ (t onto T γ(t M. Ths corresponds to the hgh-school component of the force n the drecton of moton, and I wll consder the dfferental equaton to be the known approach to mechancs. 1.2 The Lagrangan approach md γ (tγ (t = K V (γ(t Defne the Lagrangan functon L : T M R to be L(q, q = K( q V (q. Let P (a,b (M be the space of C 2 curves γ : [, 1] M wth γ( = a, γ(1 = b. Then the prncple of least acton says: A path γ P (a,b (M s a trajectory of classcal mechancs f and only f t s a crtcal ponts of the acton A : P (a,b R gven by A(γ = L(γ(t, γ (tdt. Note the specal case where V = ; n that case a least acton curve s a crtcal pont of the energy, well known to be the geodescs on (M, K. We need to transform the prncple of least acton nto a dfferental equaton: ths s a standard topc n the calculus of varatons. To be rgorous about crtcal ponts of functons on nfnte dmensonal manfolds lke P (a,b (M, we need to be a bt more specfc about just what they are. Let P k (a,b (M be the space of maps γ : [, 1] M wth γ( = a and γ(1 = b of class C k. It s a standard result of global analyss that P(a,b k (M s a Banach manfold, and ts tangent space T γ P(a,b k (M at γ P (a,b k (M s the space of maps δ : Ck ([, 1], T M such that T γ P k (a,b (M = {δ Ck ([, 1], T M δ(t T γ(t M, δ( = δ(1 = }. 2

3 Thus t makes sense ask whether A s dfferentable, and f so what ts dervatve s. Note that ts dervatve should be a lnear functonal on T γ P(a,b k (M. Agan global analyss says that when k 1 the functon A s dfferentable, and ( [DA(γ]δ = (γ(t, γ (tδ(t + q (γ(t, γ (tδ (t dt = for all δ P (a,b (M. Ths computaton defntely makes sense n coordnates. If we have chosen coordnates q 1,..., q m on M so that (q, q (q 1,... q m, q 1 / 1,... q m / m dentfes an open subset R 2m to an open subset of T T M. In these coodnates, DL s a lne matrx of length 2m whose entres are functons of q, q; the frst m are / and the last m are / q. Thus ( [DA(γ]δ = (γ(t, γ (tδ(t + q (γ(t, γ (tδ (t dt = for all δ P (a,b (M. In the standard way, we transform ths usng ntegraton by parts to ( (γ(t, γ (t d dt q (γ(t, γ (t δ(tdt = whch mples that (γ(t, γ (t = d dt q (γ(t, γ (t. Ths s a second-order dfferental for γ, called the Euler-Lagrange equaton. 1.3 The Hamltonan approach The Hamltonan formalsm apples n greater generalty than our mechancal systems: t apples to a functon on a symplectc manfold. Let (N, σ by a symplectc manfold, and H : N R be a functon. The form σ s a non-degenerate closed 2-form on N; n partcular t nduces an somorphsm between the tangent space and the cotangent space at every pont, and we can specfy a vector-feld on N by the formula σ(ξ, σ H = dh(ξ for all tangent vectors ξ to N, called the symplectc gradent of H. 3

4 The dfferental equaton of Hamltonan mechancs s ẋ = σ H(x. In our case, the symplectc manfold s the cotangent bundle T M. cotangent bundles, has a natural symplectc structure, defned as follows. There s a natural 1-form ω on T M: Ths, lke all f (q, p T M,.e., q M and p T q M, and f ξ T (q,p T M, then ω(ξ = p(π ξ where π : T M M s the natural projecton, and π s ts dervatve. Then σ = dω. Suppose q 1,..., q m are local coordnates on M. Then dq 1,..., dq m are 1-forms on M,.e., sectons of the cotangent bundles, so any pont of T M above ths coordnate patch can be wrtten p dq, where q 1,..., q m, p 1,..., q m are coordnate functons on ths regon. Then ( ω(q, p α + β p = ( p j dq j α j = p j α j = ( p j dq j α + j j β p. (1 Thus ω = p dq, and σ = dp dq. Then the equaton ẋ = σ H(x becomes, n these coordnates, q = H p, = 1,..., m ṗ = H, = 1,..., m (2 4

5 Indeed, σ ([ ξ1 ξ 2 ] [ ] ( H/ p ([ξ1, = dp H/ dq ξ 2 H = ξ 2 p + ξ H 1 = dh ] [ H/ p, H/ ([ ξ1 ξ 2 ] ]. (3 In our mechancal system (M, K, V, the functon H s the total energy 1 2 q K q q+v (q. However, ths s a functon on the tangent bundle T M, not the cotangent bundle T M. To vew t as a functon on T M we need to use the somorphsm p = K q q nduced by K. In our coordnates ths gves 1 2 q K q q + V (q = 1 2 (K 1 q p K q (Kq 1 p + V (q = 1 2 p Kq 1 p + V (q Thus H(q, p = 1 2 p Kq 1 p + V (q. 2 Lagrangan mechancs s Newtonan mechancs We can only check ths when M s sometrcally embedded n R n. In that case [DA(γ]δ = ( mδ (t γ (t [DV (γ(t]δ(t dt. Sayng that γ s a crtcal pont of A s sayng that ts dervatve vanshes,.e., that [DA(γ]δ = for all δ T γ P k (a,b (M. Usng an ntegraton by parts, ths means that for all δ T γ P(a,b k (M we have and that means that = [DA(γ]δ = = (δ (t γ (t [DV (γ(t]δ(tdt ( γ (t V (γ(t δ(tdt, (4 γ (t V (γ(t s othogonal to T γ(t M at every pont. Thus the orthogonal projecton of γ (t to T γ(t M s equal to V (γ(t, as was to be shown. 5

6 3 Lagrangan mechancs s Hamltonan mechancs Recall the Euler-Lagrange equaton: (γ(t, γ (t = d dt q (γ(t, γ (t. We wll turn ths second order equaton nto a system of frst order equatons, n almost the standard way. We nee the precse form of L: so that and L(q, p = 1 2 q K q q V (q, = 1 2 q K q V q q = K q q. So we choose as our second varable p = K q q. Note that ths s really the dentfcaton of T M and T M nduced by K. Now f p(t = K γ(t γ (t then the equaton above leads to the frst lne of the followng equatons: d dt p = 1 2 q K q V q = 1 2 (K 1 q p K q = 1 2 p Kq 1 K q = 1 2 (K 1 q K 1 q K 1 p q p V p V p V = H. (5 Ths s the harder of Hamlton s equatons. For the other, recall that H(q, p = 1 2 p K 1 q p + V (q, so H p = K 1 q p. So q = Kq 1 p = H p. Thus the crtcal ponts of A are parametrzed curves satsfyng Hamlton s equatons. 6

7 4 The space pendulum In ths secton we wll set up the equatons of moton for the space pendulum n all three formalsms and show that they gve the same equatons. We wll attach our pendulum, of length l and mass m, at the orgn n R 3. Thus M s the sphere of radus l centered at the orgn, and K(q, q = 1 2 ml2 q 2. We mght want to study the pendulum n a constant gravtatonal feld, wth potental V (q = gmq 3, or n space wth V =. Of course the behavor of the latter s much more elaborate, but t sn t much more dffcult to set of the equatons. The set of postons can be parametrzed by sphercal coordnates; we wll set them up so that the orgn s the south pole x = l sn φ cos θ y = l sn φ sn θ z = l cos φ ẋ = l cos φ cos θ φ l sn φ sn θ θ ẏ = l cos φ sn θ φ + l sn φ cos θ θ ż = l sn φ φ gvng the knetc energy x ẋ K y ẏ = l2 m 2 (ẋ2 + ẏ 2 + ż 2 = l2 m 2 ( φ 2 + sn 2 φ θ 2. z ż and the potental energy V ( φ = mgl cos φ. θ Now we have our manfold wth ts Remannan structure and ts potental functon, everythng wrtten n terms of local parameters The Hamltonan approach As usual, the Hamltonan approach s the least ntutve and the easest to put n practce. The Hamltonan functon s ( φ H = 1 ( θ 2l 2 p p2 2 m sn 2 mgl cos φ φ leadng to φ = H = p 1 p 1 l 2 m θ = H p 2 = p 2 l 2 m sn 2 φ p 1 = H φ = p 2 = H θ = p2 2 cos φ l 2 m sn 3 mgl sn φ φ 7

8 The equaton p 2 = says that p 2 s a constant, whch we call M; the second equaton of the left says M = l 2 m sn 2 φ(θ 2, so t s the angular momentum. Dfferentatng the frst equaton on the left and substtutng n the frst equaton on the rght gves φ = p 1 l 2 m = 4.2 The Euler-Lagrange approach The Lagrangan s L (( φ, θ so the Euler-Lagrange equaton becomes (after cancelng ml 2 ( φ = ml2 θ 2 ( d φ (t dt sn 2 φ(t θ (t M 2 cos φ m 2 l 4 sn 3 φ g sn φ. l ( [ φ 1 θ sn 2 φ d dt q = ] ( φ θ + mgl cos φ ( sn φ cos φ(θ (t 2 g = l sn φ. Agan the second equaton says that sn 2 φ(tθ (t s constant, agan we set ml 2 sn 2 φ(tθ (t = M. Substtutng ths nto the frst equaton gves φ (t = sn φ cos φm 2 m 2 l 4 sn 4 φ g l sn φ = M 2 cos φ m 2 l 4 sn 3 φ g sn φ, l as t should to be consstent wth the Hamltonan approach. 4.3 The Newtonan approach The computatons are pretty panful: I haven t fnshed them. 8