Euler-Lagrange Cohomology, Discrete Versions and Applications

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1 November, 2002 Euler-Lagrange Cohomology, Discrete Versions and Applications Han-Ying Guo In collaboration with J.Z. Pan, S.K. Wang, S.Y. Wu, K. Wu and B. Zhou et al. Contents 1 Introduction 2 2 ELCGs & Sympl c/volume-preserv in CM Lagrangian CM Hamiltonian CM ELCGs on Symplectic Manifolds The first ELCG The Higher ELCGs H 2n 1 EL (M, ω) & Volume-Preserv Equation First ELCG and Liouville s theorem H 1 EL & Liouville s theorem in CM H 1 EL & Liouville s theorem in SP First ELCG & Geometric Quantization 14 5 Discrete ELCG in DCM & Symplectic Algorithm DELC in Discrete CM Discrete LM Discrete HM DELC & SSPP in CM DELC Approach to SA Conclusion & Remarks 22 1

2 1 Introduction The key point I: The Euler-Lagrange cohomology grps. Introduce the EL cohomology grps H 2k 1 EL on (M 2n, ω): H 2k 1 EL (M) := {α dα = 0}/{α α = dβ}, α = i X ω k Ω (2k 1) (M), X T (M). 1. For k = 1, H 1 EL(M) = H 1 dr(m). The null EL 1-forms give rise to the EL eqns. The sympl c structure preserving (SSP) law holds iff the relevant EL 1-forms are closed, i.e. X is symplectic X X S (M). The SSP law holds not only on the slns but also in the fn spaces with relevant EL conditions. 2. For k = n, HEL 2n 1 (M) = HdR 2n 1 (M). The null EL (2n-1)-forms lead to the volume-preserv (v-p) eqns on M. The v-p law holds iff the relevant EL (2n-1)-forms are closed, i.e. X XS 2n 1 (M) X S (M). 3. In general, ELCGs are NOT the same as the DRCGs. The key point II: Sympl c/volume conservation laws. All sympl c conservation laws including Liouville s thm in the phase space of CM shld be extended in the sense of ELCGs. The key point III: Liouville s thm in Statistical Physics. It shld be extended, i.e. d dt ϱ = 0 if the subsystems may move along sympl c flow rather than Hamiltonian flow. The key point IV: Quantization via polarization. All quantum effects are related to EL cohomology via polarization in geometric quantization. The key point V: H 1 EL and its discrete version exist in DCM, CFT and DCFT. H 1 EL approach to sympl c/multi-sympl c algorithms. 2

3 2 ELCGs & Sympl c/volume-preserv in CM 2.1 Lagrangian CM t R 1, M the configuration space on t w crdnts q i, (i = 1,, n), T M the tangent space of M w (q i, q j ), F (T M) the fn space on T M. The Lagrangian: L(q i, q j ). The variat l principle with fixed ending points (a, b) δs = tb gives rise to the EL eqn: t a dt{e q iδq i + d dt { L q iδqi }}, δq i t=ta,t b = 0, E q i := L q d L i dt q = 0. i The exterior derivative d on M of L(q i, q j ): dl = E q idq i + d dt { L q idqi }. Define the EL 1-form and introduce the sympl c potential 1-form E(q i, q i ) := E q idq i, then θ = L q idqi, dl(q i, q i ) = E(q i, q i ) + d dt θ. Due to the nilpotency of d, d 2 L(q, q) = 0 leads to de(q i, q j ) + d dt ω = 0, where ω = dθ = 2 L q i q j dqi dq j + is the symplectic structure 2-form. 2 L q i q j dqi d q j 3

4 Theorem: The sympl c structure preserving (SSP) law wrt t, i.e. d dt ω = 0, holds iff the EL 1-form is closed w.r.t. d, i.e. de(q i, q j ) = 0, which is named the (closed) EL condition. Remarks: 1. The null EL 1-form that gives rise to the EL eqn, of course, satisfies the closed EL condition. The null EL 1-form can be redefined by adding certain exact forms so that the original eqn may be generalized to a type of eqns with different potentials of q i s, say, while the SSP law is the same. 2. In fact, this thm means that sympl c 2-form ω is preserved not only automatically on the slns but also in the fn space with the EL condition in general. 4

5 2.2 Hamiltonian CM Introduce conjugate momentum p j & Hamiltonian H via Legendre trnsfmtn Thus p j = L q j, H(qi, p j ) = p k q k L(q i, q j ). S = tb tb t a dt{p k q k H(q i, p j )}, δs = dt{e q δp i + E pi δq i + d t a dt (p iδq i )}, where E q i & E pi are canonical operators E q i := q i H p i, E pi := H q i ṗ i. Variat l principle leads to the canonical eqns q i = H p i, ṗ j = H q j correspond to E q i = 0 and E pi = 0, respectively. Introduce a pair of EL 1-forms correspond to EL 1-form in LM and sympl c potential 1-form E 1 (q i, p j ) = E q idq i, E 2 (q i, p j ) = E pj dp j, θ = p i dq i, we have where d(p k q k H) = E(q i, p j ) + d dt θ, E(q i, p j ) := E 1 (q i, p j ) + E 2 (q i, p j ). Finally, 0 = d 2 (p k q k H) leads to de(q i, p j ) + d dt ω = 0, ω = dp i dq i. Remarks: The thm in LM can be established in HM. All remarks made in LM can be made in HM. 5

6 2.3 ELCGs on Symplectic Manifolds The first ELCG A symplectic manifold (M 2n, ω) as the phase space. Given an observable f C (M), if the v.f. X f satisfies i Xf ω + df = 0 that corresponds to the canonical eq. for f, X f X H (M). Introduce the EL 1-form wrt vector field X on M E X = i X ω. X becomes a symplectic vector field X X S (M) if E X is closed. see, e.g., D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd Ed. Clarendon Press, Oxford, 1998; p83. A symplectic flow f t, t [0, 1], associates to X X S (M). If E X = df, it becomes Hamiltonian X X H (M). Taking exterior derivative and using L X = di X + i X d, Theorem: de X = L X ω. 1. The 1st EL cohomology group may be nontrivial in general: H 1 EL(M) := {E X de X = 0}/{E X E X = dg}, g C (M); 2. The SSP law wrt t, i.e., d dt ω = L Xω = 0, holds iff the EL 1-form is closed or X X S (M) is symplectic. Remarks: 1. H 1 EL closely links with the space C of all curves, each corresponds to a symplectic flow. 2. It can be proved H 1 EL = H 1 derham on (M, ω). 6

7 2.3.2 The Higher ELCGs On (M 2n, ω), for 1 k n, we can define two sets of v.f.s. Def: X 2k 1 S (M, ω) := { X X (M) L X (ω k ) = 0 }, X 2k 1 H (M, ω) := { X X (M) i X (ω k ) is exact }. Note: 1. They are obviously linear spaces over R. 2. X 1 S (M, ω) = X S (M, ω) and X 1 H(M, ω) = X H (M, ω). 3. It can be found: XS 2n 1 (M, ω) is the space of volume-preserving vector fields. Since, for arbitrary vector field X, there is L X (ω k ) = di X (ω k ). A v.f. X XS 2k 1 (M, ω) iff i X (ω k ) is closed. Therefore, XH 2k 1 (M, ω) XS 2k 1 (M, ω), k. Since and it is also obvious: L X (ω k+1 ) = k + 1 k i X (ω k+1 ) = k + 1 k L X (ω k ) ω i X (ω k ) ω, XS 1 (M, ω)... XS 2k 1 (M, ω) XS 2k+1 (M, ω)... XS 2n 1 (M, ω), XH(M, 1 ω)... XH 2k 1 (M, ω) XH 2k+1 (M, ω)... XH 2n 1 (M, ω). Similarly, [X 2k 1 S (M, ω), X 2k 1 (M, ω)] X 2k 1 (M, ω). S H This indicates: X k H(M, ω) is an ideal of X k S (M, ω). define a quotient Lie algebra. Hence we can 7

8 Def: The 2k 1th E-L cohomology grp H 2k 1 EL which is Abelian for each k. (M, ω) := XS 2k 1 (M, ω)/xh 2k 1 (M, ω) Further, the following theorems can be proved. Theorem: The linear map ν n : X (M) Ω 2n 1 (M) is an isomorphism. Under it, XS 2n 1 (M, ω) and XH 2n 1 (M, ω) are isomorphic to Z 2n 1 (M) and B 2n 1 (M), respectively. Namely, The (2n 1)-th ELCG HEL 2n 1 (M, ω) is linearly isomorphic to HdR 2n 1 (M). Theorem: In general, the 2k 1-th ELCG is not isomorphic to H 1 dr(m) and the 2k 1-th ELCG is not isomorphic to the (2k 1)-th DRCG for 1 < k < n. And the 2k 1-th ELCG is not isomorphic to each other for 1 < k < n. 8

9 2.3.3 HEL 2n 1 (M, ω) & Volume-Preserv Equation The key points: The linear isomorphism btwn X n H(M, ω) (Hamiltonian v-p v.fs) & B 2n 1 (M, ω) (exact (2n 1)-forms on (M 2n, ω)). This is a direct generalization for the ordinary Hamiltonian system (n = 1) the linear isomorphism btwn X 1 H(M, ω) (Hamiltonian (SSP-)v.fs) & B 1 (M, ω) (exact 1-forms on (M, ω)). Namely, a volume-preserv system is described by a volumepreserv v.f.. the latter corresponds to a unique closed (2n 1)-form on (M, ω) and vise versa. Theorem: For a given v.f. X 2n 1 H (M, ω) X = ( P ij q j + Aj j Ai j ) ( p i p j q + Q ij i p j Aj j + Aj i ), q i q j p i where Q ji = Q ij, P ji = P ij. The general set of eqns for a volume-preserv system on (M, ω) loclly read: q i = P ij q j + Aj j p i Ai j p j, ṗ i = Q ij p j Aj j q i + Aj i q j. Remarks: 1. It is a direct generalization of the canonical eqns in Hamiltonian CM. 2. It contains ordinary VPEs in phase space as special cases. 3. It differs to the approach by Y. Nambu. 9

10 3 First ELCG and Liouville s theorem 3.1 HEL 1 & Liouville s theorem in CM Def. Flow: Given a differentiable v.f. X T M, the 1-parameter trnsfmtn grp on (M, ω) ϕ t : M M, p q = ϕ t (p), t [0, 1]; ϕ t : T M T M, X p X q = ϕ t X p s.t. ϕ t (p) is the integral curve of X from p M. The 1-parameter trnsfmtn grp {ϕ t } Diff(M) is called the flow wrt X T M. Def. Phase flow: The 1-parameter trnsfmtn grp on (M, ω) g t : (p i (0), q j (0)) (p i (t), q j (t)), t [0, 1], where p i (t), q j (t) are slns of canonical eqns. g t Diff H (M). Namely, X X H (M), Liouville s thm: The phase flow preserves volume: for any region D M, volume of g t D=volume of D. V.I. Arnold, Mathematical Methods of CM (1989). Def. Symplectic flow: The 1-parameter trnsfmtn grp on (M, ω), {f t } Diff S (M, ω), is called a symplectic flow. If f t : X X t := f X, t X t=0 = X, t [0, 1] where X t X S (M), t [0, 1], i.e. they are symplectic vectors. Extended Liouville s thm: The symplectic flow preserves volume: for any region D M, volume of f t D=volume of D. Proof: v(t) := where τ = 1 n! ω ω. Since }{{} n D(t) τ, D(t) τ = f td(0) τ = 10 D(t) = f t D(0), D(0) f t τ,

11 and for the symplectic map f t f t ω = ω f t τ = τ D(t) τ = D(0) τ. Remarks: 1. In general,. For n 2 it is easy to construct volumepreserving diffeomorphisms of R 2n which are not symplectic, and hence Symp(R 2n ) is a proper subgroup of Diff vol (R 2n ) (D. McDuff et. al. p21). f t v τ = τ L Xv ω = ϖ, f t v Diff vol (R 2n ) X v X V (R 2n ). s.t. ϖ ω = 0 L Xv τ = Both Liouville s thm and extended one are sufficient for the volume conservation but not necessary. 3. Liouville s thm has particularly important applications in SP and allows to apply methods of ergodic theory to the study of CM. 11

12 3.2 H 1 EL & Liouville s theorem in SP Def. Statistical distribution fn ρ(p i, q j ; t) in phase space Γ 6n : The density of the probability distribution in phase space. Normalization condition: Γ ρτ = 1, τ τ the volume element 6n-form. := 1 n! ω } {{ ω }, Liouville s thm: The distribution fn ρ(p i, q j ; t) is constant d dt ρ(p i, q j ; t) = 0, along the phase trajectories of the subsystem. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3nd ed. Oxford, Kerson Huang, Statistical Mechanics, 2nd ed. NY, Extended Liouville s thm: The distribution fn ρ(p i, q j ; t) is constant if the subsystem is moving along the symplectic flow, i.e., the EL 1-form is closed along the flow. To prove the thm, first consider the eqn of continuity. 1. Consider a set of N points moving in a domain D Γ N = D n(p i, q j ; t), n(p i, q j ; t) := ρ(p i, q j ; t) τ. The integral invariant for conservation of the number of points ϕ t (D) n(p i, q j ; t) = D t=0 n(p i, q j ; t = 0), where t ϕ t is a 1-parameter grp, i.e., ϕ t Diff S (Γ), gives the eqn of continuity: 2. Since d dt n = (X t + )n = 0, t X t := xî t 0 = d dt (ρτ) = d dt ρ τ + ρ d dt τ. xî, xî = p i, q i. d dt τ = 0 d dt ω = L X t ω = 0 de Xt (p i, q j ) = 0, 12

13 thus d dt ρ = 0 de X t (p i, q j ) = 0. Remarks: 1. In fact, for Liouville s thm in SP, it also starts with the eq. of continuity first and then restricts to the phase flow. While for extended Liouville s thm in SP, corresponds to the symplectic flow rather than the phase flow. 2. In general, the volume-preserving diffs in R 2n, n 2 may not be symplectic. Therefore, for the distribution fn d dt ρ(p i, q j ; t) = 0, both Liouville s thm and extended one are sufficient only. 3. The extended Liouville s thm in SP, based on the extended Liouville s thm in CM, is more reasonable as one of the fundamental principles of SP. Most general one shld be related w the volumepreserving diffs. 13

14 4 First ELCG & Geometric Quantization Theorem: Quantization theorem in QM Quantization conditions in QM via geometric quantization correspond to the sum of a pair of the canonical EL 1-forms is closed along the taken polarization F, i.e. de(p i, q j ) F = d(e 1 + E 2 )(p i, q j ) F = 0. Proof: In geometric quantization: Quantization Pre-quantization + Polarization. A polarization F Γ satisfies three conditions: Since on Γ 1. [F, F ] F ; 2. dim F= 1 2dim Γ; 3. Along F, ω F = 0. the above condition 3 is equivalent to d(e 1 + E 2 )(p i, q j ) + d dt ω = 0, de(p i, q j ) F = d(e 1 + E 2 )(p i, q j ) F = 0. Remarks: 1. All quantum effects in QM via GQ are in EL cohomology. 2. Conjecture: The thm should be available for all QTs via geometric quantization. 3. What about this condition in other quantization methods? It is an open problem. 14

15 5 Discrete ELCG in DCM & Symplectic Algorithm 5.1 DELC in Discrete CM Discrete LM The DD Lagrangian at t k : L D (k) = L D (q i(k), t q i(k) ). Taking the exterior diff l d of L D (k), it follows dl (k) D = L D (k) q i(k) dqi(k) + L (k) D t q i(k)d( tq i(k) ). Using the modified Leibniz law, introducing the discrete E-L 1-form & discrete symplectic 1-form θ (k) DL E D (k) (q i(k), t q j(k) ) := { L D (k) q i(k) t ( L (k 1) D ( t q i(k 1) ) )}dqi(k), we have θ DL (k) = L D (k 1) ( t q i(k 1) ) dqi(k), dl D (k) = E D (k) + t θ D (k). Due to the nilpotency of d, we get de D (k) + t ω DL (k) = 0, where ω DL (k) is a discrete symplectic 2-form on T (M k ) ω DL (k) = dθ (k) DL = Discrete HM 2 (k 1) L D q i(k) ( t q j(k 1) ) dqi(k) dq j(k) 2 (k 1) L D + ( t q i(k) ) ( t q j(k 1) ) dq t i(k) dq j(k). Define DD canonical conjugate momentum p i (k) = L D (k 1) ( t q i(k 1) ). 15

16 The DD Hamiltonian through the discrete Legendre H D (k) = p i (k+1) t q i(k) L D (k). A set of canonical eqns for the time DD derivative of p (k) i follow from the difference E-L eqns & the above discrete Legendre: t p i (k) = H D (k) q i(k). From the discrete Legendre, another set of canonical eqns for the time DD discrete derivative of q i(k) follow t q i(k) = H D (k) p i (k+1), In terms of z (k), a pair of the canonical eqns become t z (k) = J 1 z H D (k) (z (k) ). The related cohomological issues in difference HM: Releasing all time discrete canonical variables (q i(k), p j (k) ) from the solution space, a pair of discrete E-L 1-forms can be introduced: or in term of z (k) E Dp (k) (q i(k), p j (k) ) = ( t p j (k) + H D (k) E Dq (k) (q i(k), p j (k) ) = ( t q j(k) H D (k) q j(k) )dqj(k), p j (k) )dp j (k), E Dz (k) (z (k) ) = dz (k)t (J t z (k) z H D (k) (z (k) )). By taking the exterior diff l of the discrete E-L 1-forms, we get de Dz (k) (z (k) ) + t ω DH (k) = 0, where ω DH (k) is the difference symplectic 2-form at t k : ω DH (k) = 1 2 dz(k)t Jdz (k). We can also start from the exterior diff l of L D (k) = p i (k+1) t q i(k) H D (k), introduce the discrete E-L 1-forms & the symplectic potential 1-form then take the second exterior differential to get above eqns. 16

17 5.1.3 DELC & SSPP in CM In fact, we have the following correspond relations: Discrete LM Discrete HM L D (q i(k), q i(k) ) p i (k+1) q i(k) H D (k) E D (q i(k), q i(k) ) E Dp (p i (k), q j(k) ) + E Dq (p i (k), q j(k) ) θ DL (k) dl D (k) = E D (k) + d dt θ DL (k) ω DL (k) θ DH (k) d(p (k+1) i (k) ω DH q i(k) H (k) D ) = E (k) Dp + E (k) Dq + d dt θ(k) DH de D (k) + d dt ω DL (k) = 0 d(e Dp (k) + E Dq (k) ) + d dt ω DH (k) = 0 Thus, we have the following theorem. Theorem: 1. There exists a nontrivial difference version of the 1st E-L cohomology in discrete CM: H 1 DCM := {E D de D = 0}/{E D E D = dα}. 2. The difference SSP equation t ω DH (k) = 0, i.e. ω DH (k+1) = ω DH (k) holds iff the discrete E-L forms are closed: de Dz (k) (z (k) ) = 0. 17

18 5.2 DELC Approach to SA How to apply the DELC approach to the symplectic algorithm? In the standard approach: 1. Regarding a numerical scheme as a (time-discrete) mapping. 2. In order to justify whether it is symplectic, the verification is always carried out in the solution space of the scheme, which is assumed to be existed & on which the ordinary diff l calculus could be performed. But, these are questionable! In the DELC approach, it is working on the fn space, which always exists & the diff l calculus is also well defined on it, w the relevant cohomological issues. Analog to the difference CM, there are two slightly different ways to apply the cohomological approach: 1. Based upon the DVP for the schemes & taking second (exterior) diff l of the action functional to get the necessary & sufficient condition for the SSP law of the scheme. 2. Introduce some suitable DD E-L 1-forms associated w the scheme s.t. the null DD E-L 1-forms give rise to the scheme. Then by taking the exterior derivative of the DD E-L 1-forms to see whether follows a time-discrete SSP law. Let us consider some examples to show how the cohomological scenario works. The Euler mid-point scheme: The difference Lagrangian of the scheme: L (k) mdpt = L(q i(k+ 1 2 ), t q i(k) ), q i(k+ 1 2 ) := 1 2 (qi(k+1) + q i(k) ). Note: q i(k+ 1 2 ) & t q i(k) are coordinates & tangents at t k. The discrete canonical momenta: p i (k+ 1 2 ) = L mdpt (k) ( t q i(k) ), p i (k+ 1 2 ) := 1 2 (p(k+1) i + p (k) i ). 18

19 The discrete Legendre transformation: H(q i(k+ 1 2 ), p i (k+ 1 2 ) ) = p i (k+ 1 2 ) t q i(k) L mdpt (k). The difference discrete action functional: S mdpt = k Z {p (k+1/2) i t q i(k) H(q i(k+1/2), p (k+1/2) i }. The variation of this DD action functional, δs mdpt = ɛ S mdptɛ ɛ=0. Eventually, the diff l of each term of S mdpt is given by dl mdpt (k) := dp i (k+1/2) ( t q i(k) p H(qi(k+1/2), p i (k+1/2) )) ( t p i (k) + p H(qi(k+1/2), p i (k+1/2) ))dq i(k+1/2) + t (p i (k) dq i(k) ). Here the generalized modified Leibniz law has been used: t (f (k) g (k) ) := 1 τ (f (k+1) g (k+1) f (k) g (k) ) = 1 τ ( tf (k) g (k+1/2) + f (k+1/2) t g (k) ). Introducing the discrete E-L 1-forms E (k) q = dp (k+1/2) i ( t q i(k) p H(qi(k+1/2), p (k+1/2) i )), E (k) p = ( t p (k) i + p H(qi(k+1/2), p (k+1/2) i ))dq i(k+1/2), & the difference canonical 1-form for the scheme: θ (k) mdpt = p (k) i dq i(k). Then dl mdpt (k) = E q (k) + E p (k) + t θ mdpt (k). By taking exterior diff l of this eqn, due to d 2 L mdpt = 0, it follows d(e q + E p ) (k) + t ω mdpt (k) = 0, where ω mdpt (k) = dp i (k) dq i(k). 19

20 Since the null E-L 1-forms give rise to the scheme & automatically satisfy d(e q + E p ) (k) = 0, while the latter leads to the symplectic conservation law from : t ω mdpt (k) := 1 τ (ω mdpt (k+1) ω mdpt (k) ) = 0. Therefore, the midpoint scheme is symplectic. The mid-point scheme in terms of z (k) : t z (k) = J 1 z H( 1 2 (z(k+1) + z (k) )). Release the scheme form the sln space first, even if it does exist. then introduce a difference E-L 1-form for the scheme E zmdpt (k) = 1 2 d(z(k+1) + z (k) ) T {J t z (k) z H( 1 2 (z(k+1) + z (k) ))} s.t. the null discrete E-L form gives rise to the scheme. Taking the exterior diff l of E zmdpt (k) in the fn space, it follows de zmdpt (k) = 1 2 d(z(k+1) + z (k) ) T J t dz (k). Therefore, the difference SSP law t (dz (k)t Jdz (k) ) = 0 holds iff the discrete E-L form is closed: de zmdpt (k) = 0. The 4-th order symplectic scheme: t z (n) = J 1 z H( 1 2 (z(n+1) + z (n) )) h2 24 J 1 z (( z H) T JH zz J z H)( 1 2 (z(n+1) + z (n) )). Introduce a new Hamiltonian H H = H h2 24 ( zh) T JH zz J z H, then this 4th-order symplectic scheme becomes t z (n) = J 1 z H( 1 2 (z(n+1) + z (n) )). 20

21 The discrete E-L 1-form associated w this case now can be introduced: E z4th (k) = 1 2 d(z(k+1) + z (k) ) T {J t z (k) z H( 1 2 (z(k+1) + z (k) ))}. It is easy to check that these two discrete E-L forms differ by an exact form: E zmdpt (k) E z4th (k) = h2 24 dα, where α = ( z H) T JH zz J z H is a function of 1 2 (z(k+1) + z (k) ). Namely, they are cohomologically equivalent. In addition, this also indicates that the 4-th order midpoint scheme can be established as a variational integrator by the DVP as well. 21

22 6 Conclusion & Remarks The 1st ELCG and SSP law are TWO SIDES OF THE SAME COIN. Applications of the 1st ELCG are as wide as SSP law. Possible applications in other physical topics where Liouville s thm wrt SSP law plays an important role. Are there other directly concrete physical applications? Yes! Mathematically, for finite dim l (M 2n, ω) : HEL/H 1 EL 2n 1 (M, ω) = HdeRham/H 1 derham(m), 2n 1 respectively. Then, due to the Poincarè lemma, In general, HEL(M, 1 ω) = HEL 2n 1 (M, ω). H 2k 1 EL (M, ω) H k derham(m, ω), 1 < k < n. Open questions: Classification? Topological meaning? etc. What about applications for HEL 2k 1 (M, ω)? Mechanically, for k = n BEL 2n 1 (M, ω) leads to the general volumepreserv eqn in phase space. Liouville s theorems in CM and SP shld be further generalized accord to HEL 2n 1 (M, ω). For k 1, n, they are still open. Discrete case of HEL 2k 1 (M, ω): For k = 1, the time discrete version may be applied to DCM & SA. For k 1, it is still open. What about infinite dim l cases for CFT and their discrete versions? 22

GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical