Ma¨
ıdo Rahula
Abstract.We establish a link between the sector-forms of White [10] and the exterior forms of Cartan. We show that the Hamiltonian system on T2M reduces to Lagrange’s equations on the osculating bundle OscM. The structuresTkM and Osck−1M are presented explicitly.
M.S.C. 2010: 58A30, 70H03, 79H05
Key words: tangent structures; analytical mechanics; Hamiltonian systems.
1. Tangent bundles and osculators
The tangent functorT iteratedktimes associates to a smooth manifoldM itsk-fold tangent bundleTkM (thekthlevelofM) and associates to a smooth mapϕ:M
1→
M2 the graded morphism Tkϕ:TkM1 →TkM2, thekth derivative of ϕ. The level
TkM has a multiple vector bundle structure withk projections ontoTk−1M
ρs . =Tk−sπ
s:TkM →Tk−1M, s= 1,2, . . . , k, whereπs is the natural projectionTsM →Ts−1M.
Local coordinates in neighbourhoods
TsU ⊂TsM, s= 1,2, . . . , k, where Ts−1U =π
s(TsU),
are determined automatically by those in the neighbourhoodU ⊂M, the quantities (ui) being regarded either as coordinate functions onU or as the coordinate compo-nents of the pointu∈U:
U: (ui), i= 1,2, . . . , n= dimM, T U: (ui, ui
1), with ui
. =ui◦π
1, ui1
. =dui, T2U: (ui, ui
1, ui2, ui12),
with ui=. ui◦π
1π2, ui1
.
=dui◦π
2, ui2
.
=d(ui◦π
1), ui12
. =d2ui, etc.
We set up the following convention: to introduce coordinates on TkU we take the
coordinates on Tk−1U and repeat them with an additional index k – so that a
tan-gent vector is preceded by its point of origin. This indexing is convenient since
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Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 122-127. c
the symbols with indexs thereby become coordinates in the fibre of the projection ρs, s= 1,2, . . . , k.
Thus, for example, under the projectionsρs:T3U →T2U, s= 1,2,3,the coor-dinates with index 1,2 and 3 are each suppressed in turn:
(ui, ui
1, ui2, ui12, ui3, ui13, ui23, ui123)
ρ1ւ ρ2↓ ցρ3
(ui, ui
2, ui3, , ui23) (ui, ui1, ui3, , ui13) (ui, ui1, ui2, , ui12).
The level TkM is a smooth manifold of dimension 2kn and admits an important subspace of dimension (k+ 1)n called the osculating bundle of M of order k−1 and denoted Osck−1M. The bundle Osck−1M is determined by the equality of the
projections
ρ1=ρ2=. . .=ρk,
meaning that an element of TkM belongs to the bundle Osck−1M precisely when
all itsk projections intoTk−1M coincide. In this case all coordinates with the same
number of lower indices coincide. For example, the first bundle OscMis determined in T2U ⊂T2M by the equationsui dinates by matrix formulae:
The fibres of the bundle OscM are the integral manifolds of the distribution
h∂i1+∂2i, ∂12i i, with ∂i1+∂i2=. ∂
The functions (ui
1−ui2) vanish on OscM.
Historically, osculating bundles were introduced under various names long be-fore the bundles TkM. The systematic study begun 60 years ago by V.Vagner [9] culminated in recent times with Miron-Atanasiu theory [2]. Meanwhile the theme of levels TkM remained unjustly neglected for the obvious reason that the multi-ple fibre bundle structure demands a whole new understanding and new approach: see [5], [7]. Attempts such as [10] and the so-called synthetic formulation ofTkM [3] made progress in that direction.
While an infinitesimal displacement of the pointu∈M is determined by a tangent vectoru1 to M, an infinitesimal displacement of the element (u, u1)∈T M is
interpretation of the elements ofTkM allows us to develop the theory of higher order motion. Clearly the future belongs to these bundles.
White considers on the levelTkM or on ak-multiple vector bundle certain
sector-forms which are functions simultaneously linear in all the fibres of k projections: see [10]. In particular the sector-forms onT2U andT3U can be written as
Φ =ϕijui1u
with coefficients inU. For example, in each term of Ψ we see the index 1 (or 2 or 3) appear exactly once. This means that the function Ψ is linear on the fibres ofρ1
(andρ2 andρ3).
Any scalar function can be lifted from the levelTk−1M to the level TkM by k
different projectionsρs:TkM →Tk−1M. For example, for the sector form Φ above there are three possibilities of lifting toT3M:
Φ◦ρ1=ϕijui2u
Proposition. Every Cartank-form can be regarded as a sector-form in the sense of White, a scalar function onTkM that is constant on the fibres of Osck−1M.
Proof. The sector form Φ is constant on OscM if and only if its derivatives vanish on
OscM. Thus
By definition Φ is an antisymmetric bilinear form and can therefore be expressed in the coordinates (ui, dui) as a 2-form Φ =ϕ
[ij]dui∧duj. Thus the sector-form Φ is
constant on OscM if and only if it is a Cartan 2-form.
In the casek= 3 the fibres Osc2M of dimension 3nare the integral manifolds of
the distribution
h∂i1+∂2i +∂i3, ∂i23+∂i13+∂i12, ∂i123i. For the sector-form Ψ (see above) we have
Ψ =ψijkui1u
The derivatives vanish on the fibres Osc2M when the following conditions hold:
ϕ(ijk)= 0, ψij1 +ψ
2
ij+ψ
3
ij = 0, ψi= 0.
ϕijkdui∧duj∧duk can be regarded as a homogeneous sector-form that is constant on Osc2M.
The argument extends likewise to the casesk >3. ¤
White’s theory of sector-forms is much more extensive than that of Cartan exterior forms. In particular, exterior differentiation is an operation on the set of sector-forms that are constant on the osculating bundles.
There is, however, one inconvenience: sector-forms are represented in natural coordinates in terms which are not invariant. To get rid of this one can use affine connexions and adapted coordinates. InT2U, for example, the ‘bad’ coordinatesui
12
can be replaced by adapted coordinatesUi
12 = Γijku j
1uk2+ui12 using the coefficients
Γi
jkof the affine connection. The sector-form Φ is represented by two invariant terms:
Φ = (ϕij−ϕkΓkij)u i
1u
j
2+ϕiU12i .
In the parentheses we recognize the prototype of the covariant derivative. In fact, for the 1-form Θ =θiui1 the ordinary differential can be written
dΘ =θi,jui1u
j
2+θiui12, θi,j = ∂θi ∂uj,
ordΘ =∇jθiui1u
j
2+θiU12i with the covariant derivative∇jθi=θi,j−θkΓkij.
The connexions play an important role here. The local forms appear in the unified and intrinsic structures
∆h⊕∆v onT M, ∆⊕∆1⊕∆2⊕∆12 onT2M, etc.
The theory extends by iteration to the levelsTkM: see [1], [8].
2. Hamilton, Lagrange, Legendre
The essential importance of the levels T M and T2M for analytical mechanics was
first emphasized by Godbillon [4].
Specifically, Hamiltonian geometry is built on the levelsT M andT2M. Associated to a functionH = H(u, u1) (called the Hamiltonian) is the vector field X on T M
where
X = Σ iHu
i
1∂i−ΣiHu
i∂i1, Hi=.
∂H
∂ui, Hui1 .
= ∂H
∂ui
1
,
for which the flowat= exptX is determined by the system of differential equations (Hamiltonian system)
½
˙ ui =H
ui
1 ˙
ui
1=−Hui
, u˙i =. du i
dt , u˙ i
1
.
=du
i
1
dt .
Under the correspondence
we see this as a section of the bundleπ2:T2M →T M,of dimension 2n. The function
H and the symplectic form Ω =dui∧dui
1[6] are invariant with respect to the vector
fieldX:
XH= 0, LXΩ = 0.
Theorem. The Hamiltonian system reduces to Lagrange’s equations on the os-culating bundleOscM.
Proof. The passage from the Hamiltonian H = H(u, u1) to the Lagrangian L =
L(u, u2) ought to be realized through the equation (Legendre transformation)
H(u, u1)−Σ
i u i
1u
i
2+L(u, u2) = 0.
However, this equation, which should hold identically onT2M, is contradictory:
d(H−Σ i u
i
1ui2+L)≡0 ⇒ Hui+Lui = 0, Hui
1 =u
i
2, Lui
2=u
i
1.
On the other hand, on OscM where ui
1 = ui2 = ˙u, the passage H Ã L is well
determined. On OscM the Hamiltonian system can be written in Lagrangian form:
d dt
³∂L
∂u˙i
´
− ∂L
∂ui = 0.
The Lagrangian system determines a section of the bundle OscM →T M, of the same
dimension 2nas the Hamiltonian system onT2M. ¤
The Hamiltonian geometry on the levels TkM and the Lagrangian geometry on the osculating bundles Osck−1M for k > 2 are structured according to an iterative
scheme.
References
[1] Atanasiu, G., Balan, V., Brˆınzei, N., Rahula, M.,Differential Geometric Struc-tures: Tangent Bundles, Connections in Bundles, Exponential Law in the Jet Space(in Russian), Librokom, Moscow, 2010, 320 pp.
[2] Atanasiu, G., Balan, V., Brˆınzei, N., Rahula, M., Second Order Differential Geometry and Applications: Miron-Atanasiu Theory (in Russian), Librokom, Moscow, 2010, 250 pp.
[3] Bertram, W., Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, Memoirs of AMS, no. 900, 2008, (Zbl pre5250766).
[4] Godbillon, C., G´eom´etrie Diff´erentielle et M´ecanique Analytique, Hermann, Paris, 1969, (Zbl 1074.24602).
[5] Ehresmann, Ch., Cat´egories doubles et cat´egories structur´ees, C.R. Acad. Sci., Paris, 256(1958), 1198-1201.
[6] Kushner, A., Lychagin, V., Rubtsov, V.,Contact Geometry and Nonlinear Dif-ferential Equations, Ser. Encyclopedia of Mathematics and its Applications, No. 101, Cambridge UP, 2007.
[8] Rahula, M., New Problems in Differential Geometry, WSP, 1993, (Zbl 0795.53002).
[9] Vagner, V. V.,Theory of differential objects and foundations of differential geom-etry(in Russian), Appendix in: Veblen, O., Whitehead, J.H.C.,The Foundations of Differential Geometry, IL, Moscow, 1949, 135-223.
[10] White, E. J.,The Method of Iterated Tangents with Applications in Local Rieman-nian Geometry, Monographs and Studies in Mathematics, 13. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.
Author’s address:
Ma¨ıdo Rahula
