A new look at Classical Mechanics

of constrained systems ∗

Enrico MassaDipartimento di Matematica dell’Universita di Genova

Via Dodecaneso, 35 - 16146 Genova (Italia)E-mail: massa@dima.unige.it

Enrico PaganiDipartimento di Matematica dell’Universita di Trento

38050 Povo di Trento (Italia)E-mail: pagani@science.unitn.it

Abstract

A geometric formulation of Classical Analytical Mechanics, especiallysuited to the study of non-holonomic systems is proposed. The ar-gument involves a preliminary study of the geometry of the space ofkinetic states of the system, followed by a revisitation of Chetaev’sdefinition of virtual work, viewed here as a cornerstone for the im-plementation of the principle of determinism. Applications to idealnon-holonomic systems (equivalence between d’Alembert’s and Gauss’principles, equations of motion) are explicitly worked out.

Resume

Nous presentons dans ce papier une formulation de la Mecanique Ana-litique qui permet de traiter des systemes non-holonomes. Cette etudepreliminaire concerne la geometrie de l’espace des etats cinetiques dusysteme. Cela nous a conduit a revoir la definition de Chetaev dutravail virtuel, qui joue ici un role remarquable dans la realisationd’un modele mecanique compatible avec le principe de determinisme.L’ensemble de ce travail nous a permis de l’appliquer aux systemesnon-holonomes ideaux, de comparer les formulations de D’Alembert etde Gauss et d’ecrire de facon explicite les equations de la dynamique.

PACS: 03.20+1, 02.40.+m1991 Mathematical subject classification: 70D10, 70F25Keywords: Lagrangian Dynamics, Non-holonomic Systems.

∗This research was partly supported by the National Group for Mathematical Physicsof the Italian Research Council (CNR), and by the Italian Ministry for Public Education,through the research project “Metodi Geometrici e Probabilistici in Fisica Matematica”.

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1 Introduction

A central issue in the development of Classical Mechanics is the identifi-cation of general statements — often called “principles” — characterizingthe behaviour of the reactive forces for large classes of constraints of actualphysical interest.This is e.g. the role of d’Alembert’s principle of virtual work [1, 2, 3], orof Gauss’ principle of least constraint [4, 5, 6] — both providing equivalentcharacterizations of the class of ideal constraints — or of Coulomb’s laws offriction [7], etc.

In general, of course, the choice of a suitable characterization of thereactive forces is a physical problem, intimately related with the structuralproperties of the devices involved in the implementation of the constraints.In any case, however, a basic condition to be fulfilled by any significantmodel is the requirement of consistency with the principle of determinism,i.e. the ability to give rise to a dynamical scheme in which the evolution ofthe system from given initial data is determined uniquely by the knowledgeof the active forces, through the solution of a well-posed Cauchy problem.

In this paper, we propose a thorough discussion of this point. For gen-erality, we shall deal with arbitrary (finite-dimensional) non-holonomic sys-tems. The analysis will be carried on in a frame-independent language, usingthe standard techniques of jet-bundle theory [8, 9, 10, 11, 12, 13].The main emphasis will be on the intrinsic geometry of the space of kineticstates of the system. On this basis, we shall identify a general definition ofthe concept of virtual work, extending the traditional one to arbitrary non-holonomic systems, along the lines proposed by Chetaev [14, 15, 16, 17, 18].The resulting scheme will throw new light on the interplay between deter-minism and constitutive characterization of the reactive forces, as well as onthe relation between d’Alembert’s principle and Gauss’ principle in the caseof ideal non-holonomic systems.

The plan of presentation is as follows:§2.1 and §2.2 are introductory in nature. In §2.1 we review the basic conceptsinvolved in the construction of a frame-independent model for a mechani-cal system with a finite number of degrees of freedom. In particular, weformalize the concept of configuration space-time, defined as the abstractmanifold t : Vn+1 → < formed by the totality of admissible configurations ofthe system, fibered over the real line < through the absolute time function.

In a similar way, in §2.2 we recall some standard aspects of the geom-etry of the first jet space j1(Vn+1), especially relevant to the subsequentdiscussion [8, 9, 10, 11, 13, 19].

The study of the non-holonomic aspects arising from the presence of ki-netic constraints begins in §2.3. Following [12], the geometrical environmentis now identified with a submanifold i : A → j1(Vn+1), fibered over Vn+1,and representing the totality of admissible kinetic states of the system. In

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addition to the “natural” structures, either implicit in the existence of thefibration A → Vn+1, or obtained by pulling back the analogous structuresover j1(Vn+1), the manifold A is seen to carry further significant geometri-cal objects, depending in a more sophisticated way on the properties of theembedding i : A → j1(Vn+1).

A thorough discussion of this point is presented. Among other topics,the analysis includes the definition of the concept of fundamental tensor ofthe manifold A, as well as the introduction of a distinguished vector bundleof differential 1-forms over A, called the Chetaev bundle.

In §3, the geometrical scheme is applied to the study of the dynamicalaspects of the theory. In §3.1 we show that, in the presence of kinetic con-straints, the Poincare-Cartan 2-form of the system splits into two terms, onlyone of which is effectively significant in the determination of the evolutionof the system.

An analysis of this point provides a hint for a geometrical revisitationof the concepts of virtual displacement and virtual work. The argument isformalized in §3.2. A comparison with the traditional definitions is explicitlyoutlined.

The discussion is completed by a factorization theorem, allowing to ex-press the content of Newton’s 2nd law in a form especially suited to a precisemathematical formulation of the principle of determinism. The interplay be-tween determinism and constitutive characterization of the reactive forcesis considered in §3.3. The geometrical meaning of d’Alembert’s principle, aswell as the relation of the latter with Gauss’ principle of least constraint arediscussed in detail.

The paper is concluded by an analysis of the equations of motion fora general non-holonomic system subject to ideal constraints. Both the in-trinsic formulation, in terms of local fibered coordinates on the submanifoldA, and the extrinsic formulation, based on the “cartesian” representation ofthe embedding i : A → j1(Vn+1) are explicitly outlined.

2 Classical dynamics on jet bundles

2.1 Preliminaries

(i) In this Section we review a few basic aspects of Classical Mechanics,especially relevant to the subsequent applications. Throughout the discus-sion, a major role will be played by the space-time manifold V4, definedas the totality of events, and viewed as the natural environment for theframe-independent formulation of physical laws.

According to the axioms of absolute time and absolute space, V4 has thenature of an affine bundle over the real line <, with projection t : V4 → <(the “absolute–time function”), and standard fibre E3 (euclidean 3-space).

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The totality of vectors tangent to the fibres form a vector bundle V (V4),called the vertical bundle over V4.

Every section ζ : < → V4 — henceforth called a world line — providesthe representation of a possible evolution of a point particle. In this respect,the first jet space j1(V4) has a natural interpretation as the totality of kineticstates of particles.¿From an algebraic viewpoint, we recall that j1(V4) has the nature of anaffine bundle over V4, modelled on the vertical bundle V (V4). The first jetextension of a world-line ζ : < → V4 will be denoted by ζ : < → j1(V4).

(ii) Every trivialization I : V4 → < × E3 mapping each fibre of V4

isometrically into E3 identifies what is usually called a physical frame ofreference. The trivialization I may be “lifted” to the first jet space j1(V4),thus giving rise to the further identification

j1(V4) ' j1(<× E3) ' <× E3 × V3 ' V4 × V3 (2.1)

V3 denoting the space of “free vectors” in E3.The projection x : V4 → E3 associated with I will be called the position

map relative to I. The projection v : j1(V4)→ V3 induced by the represen-tation (2.1) — clearly identical to the restriction to j1(V4) of the tangentmap x∗ : T (V4)→ T (E3), followed by the canonical projection T (E3)→ V3

— will be called the velocity map relative to I.We let the reader verify that, for each world line ζ, the composite mapv · ζ : < → V3 does indeed coincide with the time derivative of the mapx · ζ : < → E3.

(iii) Quite generally, a material system B may be viewed as a measurespace, i.e. as a triple (B,S,m) in which B is an abstract space (the “materialspace”, formed by the totality of points of the system), S is a σ-ring ofmeasurable subsets of B, and m is a finite positive measure over (B,S),assigning to each Ω ∈ S a corresponding inertial mass m(Ω) :=

∫Ωm(dξ).

In the case of discrete systems the simplified notation B = P1, . . . , PN,m(Pi) = mi (whence also m(Ω) =

∑Pi∈Ωmi) will be implicitly understood.

A configuration of the system is defined as a map P : B→ V4 satisfyingthe condition t · P = const., i.e. sending all points ξ ∈ B into one and thesame fiber of V4. In a similar way, a kinetic state of the system is a mapV : B → j1(V4) satisfying t · V = const., t denoting the (pull-back of the)absolute–time function on j1(V4).According to the stated definitions, we may attach a time label to eachconfiguration P (to each kinetic state V), according to the identification

t(P) := t ·P(ξ) ∀ ξ ∈ B (2.2)

(respectively t(V) := t ·V(ξ) ∀ ξ ∈ B).The totality of admissible configurations, or of admissible kinetic states,

does not depend only on the nature of the material space B, but also,

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explicitly, on the constraints imposed on the system.In general, a satisfactory insight into the situation is gained by splitting thedescription into two steps, focussing at first on the positional constraints,and deferring to a subsequent stage the description of the additional kineticones 1.

In connection with the first step, we shall restrict our analysis to theclass of holonomic constraints, completely characterized by the followingproperties:

a. the totality of admissible configurations form an (n + 1)–dimensionaldifferentiable manifold Vn+1, fibered over the real line < through themap (2.2);

b. the correspondence x → Px between points of Vn+1 and admissibleconfigurations Px : B→ V4 has the property that, for each ξ ∈ B, theimage Px(ξ) depends differentiably on x.

As a consequence of a) and b) it is easily seen that every admissibleevolution of the system determines a corresponding section γ : < → Vn+1.Conversely, if no further restrictions are imposed on the system, every suchsection provides the representation of an admissible evolution. In this re-spect, the first jet space j1(Vn+1) has therefore a natural interpretation asthe totality of kinetic states consistent with the given holonomic constraints.

Additional (non-holonomic) constraints, when present, are then easilyinserted into the scheme, in the form of restrictions on the class of admissiblesections, thus shrinking the family of allowed kinetic states to a subset A ⊂j1(Vn+1). This point will be explicitly considered in §2.3.

The manifold Vn+1 will be called the configuration space-time associatedwith the given system; the fibration t : Vn+1 → < will be called the absolutetime function over Vn+1. The vertical bundle over Vn+1 will be denoted byV (Vn+1).

We recall that, by definition, both V (Vn+1) and the first jet space j1(Vn+1)may be identified with corresponding submanifolds of the tangent spaceT (Vn+1), according to the identifications

V (Vn+1) = X |X ∈ T (Vn+1) , < X, dt >= 0j1(Vn+1) = X |X ∈ T (Vn+1) , < X, dt >= 1

¿From these, it is readily seen that j1(Vn+1) has the nature of an affinebundle over Vn+1, modelled on V (Vn+1) [8, 9, 11, 12, 13, 19].

1A more elastic strategy, accounting for the substantial interchangeability betweenpositional constraints and integrable kinetic ones, would be to base step 1 on a pre-selected subfamily of constraints, of strictly positional nature, and to express all otherconstraints in kinetic terms. In this connection, see also [12].

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In view of the requirement b) stated above, each point ξ ∈ B determinesa differentiable map Pξ : Vn+1 → V4 through the prescription

Pξ(x) := Px(ξ) ∀x ∈ Vn+1 (2.3)

We denote by (Pξ)∗ : T (Vn+1) → T (V4) the associated tangent map. Astraightforward check shows that (Pξ)∗ preserves the property of verticality,i.e. it satisfies (Pξ)∗(V (Vn+1)) ⊂ V (V4).

The restriction of (Pξ)∗ to the submanifold j1(Vn+1) ⊂ T (Vn+1), clearlyidentical to the first jet extension of the map (2.3), will be denoted by Pξ.

(iv) Given an arbitrary frame of reference I, let x : V4 → E3 andv : j1(V4) → V3 denote the corresponding position and velocity maps.For each ξ ∈ B the composite maps xξ := x · Pξ : Vn+1 → E3 andxξ := v · Pξ : j1(Vn+1) → V3 are then easily recognized as representa-tions of the position and velocity of ξ relative to I, respectively as functionsof the configuration and of the kinetic state of the system 2.In local fibered coordinates t, q1, . . . , qn over Vn+1, and t, q1, . . . , qn, q1, . . . , qn

over j1(Vn+1), we have the familiar expressions

xξ = xξ(t, q1, . . . , qn) (2.4)

xξ =∂xξ

∂qkqk +

∂xξ

∂t(2.5)

(xi = xi(t, q1, . . . , qn) , xi = ∂xi

∂qk qk + ∂xi

∂t for discrete systems).

By means of eq. (2.4), the ordinary (euclidean) scalar product in V3 maybe lifted to a frame-dependent scalar product in T (Vn+1), according to theprescription

(X,Y )I :=∫B

xξ∗(X) · xξ∗(Y ) m(dξ) ∀X,Y ∈ T (Vn+1) (2.6)

A straightforward check shows that, when both vectorsX and Y are vertical,the right-hand-side of eq. (2.6) is invariant under arbitrary transformationsof the frame of reference I. The restriction of the product (2.6) to thevertical bundle V (Vn+1) is therefore a frame-independent attribute of themanifold Vn+1, henceforth denoted by ( , ), and called the fiber (or vertical)metric [8, 9, 11, 12, 13, 19].

A system is said to be non-singular if and only if the positivity condition(X,X) > 0 holds for all X 6= 0 in V (Vn+1). This condition will be implicitlyassumed throughout the subsequent discussion.

2It goes without saying that, in the presence of non-holonomic constraints, the statedinterpretation applies only to the restriction of the maps xξ to the family of admissiblekinetic states.

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In local coordinates, the representation of the fiber metric is summarizedinto the set of components

ghk :=(∂

∂qh,∂

∂qk

)=∫B

∂xξ

∂qh· ∂xξ

∂qkm(dξ) (2.7)

(ghk :=∑mi

∂xi

∂qh · ∂xi

∂qk for discrete systems).Returning to the general case, it may be seen that the knowledge of

the scalar product (X,Y )I for arbitrary (not necessarily vertical) vectors ismathematically equivalent to the knowledge of the quadratic form (X,X)Iover T (Vn+1), which, in turn, is determined uniquely by the knowledge ofits restriction (z, z)I to the submanifold j1(Vn+1) ⊂ T (Vn+1).The function T defined globally on j1(Vn+1) by T (z) := 1

2(z, z)I will becalled the holonomic kinetic energy relative to the frame of reference I.Comparison with eq. (2.6) yields the representation

T (z) =12

∫B|xξ(z)|2m(dξ) (2.8)

showing that the restriction of T to the class of admissible kinetic statesdoes indeed represent the kinetic energy of the system relative to I.

2.2 Geometry of j1(Vn+1)

For later use, in this Subsection we review the main aspects of the geometryof the first jet space j1(Vn+1). For an exhaustive discussion, see e.g. [8, 9,11, 12, 13], and references therein.

Keeping the same notation as in [13], we denote by V (j1(Vn+1)) the ver-tical bundle over j1(Vn+1) with respect to the fibration π : j1(Vn+1)→ Vn+1,and by j2(Vn+1) the second jet extension of Vn+1, viewed as an affine bun-dle over j1(Vn+1), modelled on V (j1(Vn+1)). The definitions and elementaryproperties of both spaces will be regarded as known [8, 9, 11, 12, 13]. The an-nihilator of the (n+1)–dimensional distribution spanned by j2(Vn+1) will bedenoted by C(j1(Vn+1)), and will be called the contact bundle over j1(Vn+1).

Every sectionX : j1(Vn+1)→ V (j1(Vn+1)) will be called a vertical vectorfield over j1(Vn+1). In a similar way, sections σ : j1(Vn+1) → C(j1(Vn+1))will be called contact 1–forms, while sections Z : j1(Vn+1) → j2(Vn+1),viewed as a vector fields over j1(Vn+1), will be called semi-sprays.

In local fibered coordinates, introducing the notation

ωi := dqi − qidt (2.9)

the situation is summarized into the explicit representations

X vertical vector ⇔ X = Xi ∂

∂qi

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Z semi− spray ⇔ Z =∂

∂t+ qi ∂

∂qi+ Zi ∂

∂qi

σ contact 1−form ⇔ σ = σi ωi

with Xi, Zi, σi ∈ F(j1(Vn+1)).On the basis of the stated definitions, one can prove the following general

results [8, 9, 11, 13, 19].

(i) By regarding each z ∈ j1(Vn+1) as a vector in Tπ(z)(Vn+1), we candefine a linear map Θz : Tz(j1(Vn+1)) → Tπ(z)(Vn+1) according to the pre-scription

Θz(X) = (πz)∗(X)− 〈X, (dt)z〉 z (2.10)

By duality, this gives rise to a map Θz∗ of the cotangent space T ∗

π(z)(Vn+1)into T ∗

z (j1(Vn+1)), on the basis of the equation

Θz∗(ν) = (πz)∗∗(ν)− < z, ν > (dt)z ∀ ν ∈ T ∗

π(z)(Vn+1) (2.11)

In local coordinates, recalling the representation (2.9) for the 1–forms ωi,we have the explicit relations

Θz(X) =⟨X,ωi

|z⟩( ∂

∂qi

)π(z)

(2.12)

Θz∗(ν) =

⟨ν,

(∂

∂qi

)π(z)

⟩ωi

|z (2.13)

Accordingly, we shall call Θz and Θz∗ respectively the vertical push-forward

and the contact pull-back at z.The map (2.13) may be extended in a standard way to fields of 1–forms,

by requiring the condition

(Θ∗(η))|z := Θz∗(η|π(z)

), η ∈ D1(Vn+1) (2.14)

For every f ∈ F(Vn+1), the contact pull back of df will be denoted by dc f ,and will be called the contact differential of f . In local coordinates, eqs.(2.13), (2.14) provide the explicit representations

Θ∗(η) =⟨π∗(η) ,

∂

∂qi

⟩ωi , dc f = Θ∗(df) =

∂f

∂qiωi (2.15)

(ii) According to eq. (2.13), the kernel of the map Θz∗ coincides with

the annihilator of the vertical space Vπ(z)(Vn+1), while the image space

Θz∗(T ∗

π(z)(Vn+1))

is identical to the space Cz(j1(Vn+1)) of contact 1–formsat z.This provides a canonical identification of Cz(j1(Vn+1)) with the dual of

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Vπ(z)(Vn+1), summarized into a bilinear pairing 〈 ‖ 〉 : Cz(j1(Vn+1)) ×Vπ(z)(Vn+1)→ <, defined implicitly by the relation

〈Θ∗z(ν) ‖V 〉 = 〈ν , V 〉 ∀ ν ∈ T ∗

π(z)(Vn+1) , V ∈ Vπ(z)(Vn+1)

In local coordinates, a straightforward comparison with eq. (2.13) yieldsthe explicit representation⟨

νi ωi|z

∥∥∥∥∥V j(∂

∂qj

)|π(z)

⟩= νi V

i (2.16)

(iii) The affine character of the fibration j1(Vn+1)→ Vn+1 ensures that,for each z ∈ j1(Vn+1), the vertical spaces Vz(j1(Vn+1)) and Vπ(z)(Vn+1) arecanonically isomorphic [8, 9, 11, 13, 19]. This gives rise to a vertical liftof vectors from Vn+1 to j1(Vn+1), denoted symbolically by V → V v, andexpressed in coordinates as

V = V i ∂

∂qi7→ V v = V i ∂

∂qi(2.17)

Due to this fact, the scalar product between vertical vectors may be liftedfrom Vn+1 to j1(Vn+1), thus defining a frame-independent structure overthe first jet-space, called once again the fiber (or vertical) metric. In localcoordinates, the evaluation of the scalar products relies on the identifications(

∂

∂qh,∂

∂qk

)|z

=(∂

∂qh,∂

∂qk

)|π(z)

= (ghk)|π(z) (2.18)

with the ghk’s given by eq. (2.7).

(iv) By composing the vertical push forward (2.10) with the vertical lift(2.17), one gets a linear endomorphism J : Tz(j1(Vn+1)) → Tz(j1(Vn+1)),mapping each vector X into the vertical vector

J(X) := (Θz(X))v =⟨X,ωi

|z⟩( ∂

∂qi

)z

(2.19)

By the quotient law, this defines a tensor field of type (1, 1) over j1(Vn+1),known as the fundamental tensor [8, 9, 13]. In local coordinates, eq. (2.19)provides the explicit representation

J =∂

∂qi⊗ ωi (2.20)

(v) The simultaneous presence of the fundamental tensor and of the fibermetric determines a vector bundle isomorphism

V (j1(Vn+1))g−−−−→ C(j1(Vn+1))

π

y yπ

j1(Vn+1) j1(Vn+1)(2.21)

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(process of “lowering the indices”), assigning to each vertical vector X acorresponding contact 1–form g(X) on the basis of the requirement

〈g(X), Y 〉 = (X, J(Y )) ∀Y ∈ T (j1(Vn+1))

In local coordinates, taking eq. (2.18) into account, we have the representa-tion

g

(∂

∂qi

)= gij ω

j

with inverseg−1(ωi) = gij ∂

∂qj

gij denoting the matrix inverse of gij .By means of the isomorphism (2.21), the scalar product (2.18) may be

extended to the bundle of contact 1-forms, by requiring the identification

(σ, τ) :=(g−1(σ), g−1(τ)

)∀σ, τ ∈ C(j1(Vn+1))

In local coordinates, this provides the relations(ωi, ωj

)= girgjs

(∂

∂qr,∂

∂qs

)= gij (2.22)

(vi) The linear endomorphism (2.19) may be extended to an (algebraic)derivation v : D(j1(Vn+1)) → D(j1(Vn+1)) of the entire tensor algebra overj1(Vn+1), commuting with contractions and vanishing on functions, i.e. sat-isfying the conditions

v(f) = 0 , v(X) = J(X) , < v(X), σ > + < X, v(σ) >= 0 (2.23)

∀ f ∈ F(j1(Vn+1)), X ∈ D1(j1(Vn+1)), σ ∈ D1(j1(Vn+1)).By means of v one can construct a special anti-derivation dv of the

Grassmann algebra G(j1(Vn+1)), called the fiber differentiation [13], definedon arbitrary r–forms according to the equation

dvσ := d v(σ)− v(dσ)− r dt ∧ σ (2.24)

This implies, among others, the following relations

dvf = −v df =∂f

∂qkωk ∀f ∈ F(j1(Vn+1))

dv ωk = 0 k = 1, . . . , n

dv · dv σ = 0 ∀σ ∈ G(j1(Vn+1))

For further details, the reader is referred to [13].

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2.3 Non-holonomic constraints

(i) As pointed out in §2.1, when the set of constraints imposed on the systemis larger than the holonomic subset explicitly involved in the definition ofthe configuration space-time Vn+1, the family of admissible kinetic statesdoes no longer fill the entire first-jet space j1(Vn+1), but only a subregionA ⊂ j1(Vn+1). The cases explicitly accounted for in Analytical Mechanicsare those in whichA has the nature of an embedded submanifold of j1(Vn+1),fibered over Vn+1 [12].The situation is summarized into the commutative diagram

A i−−−−→ j1(Vn+1)

π

y yπ

Vn+1 Vn+1

(2.25)

In what follows, we shall stick to the stated assumption. Preserving thenotation t, q1, . . . , qn, q1, . . . , qn for the coordinates on j1(Vn+1), and refer-ring A to local fibered coordinates t, q1, . . . , qn, z1, . . . , zr, the embeddingi : A → j1(Vn+1) will be represented locally in either forms

qi = ψi(t, q1, . . . , qn, z1, . . . , zr) (2.26)

with rank ‖∂(ψ1, . . . , ψn)/∂(z1, . . . , zr)‖ = r (intrinsic representation), or

gσ(t, q1, . . . , qn, q1, . . . , qn) = 0 σ = 1, . . . , n− r (2.27)

with rank ‖∂(g1, . . . , gn−r)/∂(q1, . . . , qn)‖ = n−r (cartesian representation).For each ξ ∈ B, the maps Pξ : Vn+1 → V4 and Pξ : j1(Vn+1) → j1(V4)

retain the meaning already discussed in the holonomic case. In particular,given an arbitrary frame of reference I, the representation of the position andof the velocity of the point ξ as functions respectively of the configurationand of the kinetic state of the system relies on the composite maps

xξ := x ·Pξ : Vn+1 → E3 , vξ := v · Pξ · i = xξ · i : A → V3

expressed in coordinates as

xξ = xξ(t, q1, . . . , qn) (2.28)

vξ =∂xξ

∂qkψk(t, qi, zA) +

∂xξ

∂t(2.29)

(xi = xi(t, q1, . . . , qn) , vi = ∂xi

∂qk ψk + ∂xi

∂t in the discrete case).

(ii) The concepts of vertical bundle and of contact bundle are easilyadapted to the submanifold A. A vector field Z ∈ D1(A), i–related to asemi-spray on j1(Vn+1), will be called a dynamical flow on A.

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In local coordinates, introducing the notation

ωi := i∗(ωi) = dqi − ψi(t, qk, zA) dt (2.30)

we have the obvious identifications

V vertical vector ⇔ V = V A ∂

∂zA

Z dynamical flow ⇔ Z =∂

∂t+ ψi(t, qk, zA)

∂

∂qi+ ZA ∂

∂zA

σ contact 1−form ⇔ σ = σi ωi

In a similar way, by composing the homomorphisms (2.10), (2.11) withthe maps (iz)∗ and (iz)∗

∗, we can extend the notion of vertical push-forwardof vectors, or of contact pull-back of 1–forms at each point z ∈ A, as wellas the subsequent identification between contact 1–forms at z and linearfunctionals on the vertical space Vπ(z)(Vn+1) based on the pairing (2.16).

In addition to the natural structures described above, another impor-tant class of geometrical objects is obtained through the following construc-tion: for each z ∈ A, consider the image space i∗(Tz(A)) under the mapi∗ : T (A) → T (j1(Vn+1)), and denote by Ni(z)(A) ⊂ T ∗

i(z)(j1(Vn+1)) thecorresponding annihilator. Take the image v(Ni(z)(A)) under the deriva-tion induced by the fundamental tensor J , and pull it back to a subspace(iz)∗

∗v(Ni(z)(A)) := χz(A) ⊂ T ∗z (A).

In this way, performing the same construction at each z ∈ A, we end upwith a vector bundle χ(A) =

⋃z∈A χz(A). For reasons that will be clear

later on, we shall call the latter the Chetaev bundle over A. Every sectionν : A → χ(A) will be called a Chetaev 1–form over A. The role of theChetaev bundle in the study of the integrability conditions for a given setof kinetic constraints is briefly analysed in Appendix A.

A description of χ(A) in local coordinates is easily obtained, startingwith an arbitrary cartesian representation (2.27) for the submanifold A,and observing that, by definition, the differentials (dgσ)i(z), σ = 1, . . . , n− rspan Ni(z)(A). Recalling eq. (2.24), we conclude that the virtual differentials(dvg

σ)i(z) = −v(dgσ)i(z) span v(Ni(z)(A)), which is the same as saying thatthe differential 1–forms

i∗(dvgσ)|z =

(∂gσ

∂qk

)|i(z)

ωk|z (2.31)

span χz(A). Moreover, due to the assumption on the rank of the Jacobian‖∂(g1, . . . , gn−r)/∂(q1, . . . , qn)‖, the 1–forms (2.31) are linearly independent,so that they form a basis in χz(A). To sum up, every Chetaev 1–form ν onA may be expressed locally as

ν = λσ i∗(dvg

σ) (2.32)

12

with λσ ∈ F(A).Switching to the intrinsic description (2.27) for the submanifold A, it is

easily seen that the characterization (2.32) is mathematically equivalent toa representation of the form ν = νk ω

k, with the components νk(t, qi, zA)subject to the conditions

νk∂ψk

∂zA= 0 A = 1, . . . , r (2.33)

(iii) At each z ∈ A, the vertical space Vz(A) is canonically isomorphicto its image i∗Vz(A) ⊂ Vi(z)(j1(Vn+1)). The fiber metric on j1(Vn+1) maytherefore be used to induce a scalar product on Vz(A), as well as an “or-thogonal projection” PA : Vi(z)(j1(Vn+1)) → Vz(A). Both operations areentirely straightforward. The geometric structure determined by the scalarproduct on V (A) will be called the fiber metric over A.

In local coordinates, making use of the intrinsic representation (2.26)for the embedding i, and omitting all unnecessary subscripts, we have theexplicit relations (

∂

∂zA,∂

∂zB

):= GAB = gij

∂ψi

∂zA

∂ψj

∂zB(2.34)

PA(∂

∂qi

)= GAB

(∂

∂qi, i∗

∂

∂zB

)∂

∂zA= GABgij

∂ψj

∂zB

∂

∂zA(2.35)

with GAB GBC = δAC .

By means of PA , recalling the definition (2.19) of the fundamental tensorof j1(Vn+1), we construct a linear map J : Tz(A)→ Tz(A) according to theequation

J(X) := PA(J(i∗X)) ∀X ∈ Tz(A) (2.36)

By the quotient law, the latter identifies a tensor field J of type (1, 1) overA, henceforth called the fundamental tensor of A. In local coordinates,recalling eqs. (2.20), (2.35), we have the explicit expression

J(X) =⟨i∗(X), ωi

⟩PA

(∂

∂qi

)=⟨X, ωi

⟩GABgij

∂ψj

∂zB

∂

∂zA

resulting in the representation

J = σA ⊗ ∂

∂zA(2.37)

with

σA = GAB ∂ψj

∂zBgij ω

i (2.38)

13

In connection with the previous definition, it is worth noticing that,unlike the fundamental tensor J of j1(Vn+1) — whose construction had auniversal character, involving only the definition of the concept of first jetspace — the tensor J depends also explicitly on the fiber metric, i.e., througheq. (2.7), on the material properties of the system in study.In this respect, rather than as a geometrical attribute of the manifold A,the field J is therefore to be regarded as a mechanical object over A. 3.

By means of J , parallelling the procedure followed in j1(Vn+1), we mayset up a correspondence between vertical vectors and contact 1–forms overA, by assigning to each X ∈ Vz(A) the unique element g(X) ∈ Cz(A)satisfying the condition

〈g(X), Y 〉 = (X, J(Y )) ∀Y ∈ Tz(A) (2.39)

A straightforward check shows that the previous definition is consistent withthe identification

g(X) = (iz)∗∗ g (iz∗X)

g denoting the isomorphism between vertical vectors and contact 1–formsover j1(Vn+1) described in §2.2.

The image of the vertical bundle V (A) under the map g will be denotedby V ∗(A). Every element η |z ∈ V ∗(A), or, more generally, every sectionη : A → V ∗(A) will be called a virtual differential form over A.

In local coordinates, eqs. (2.37), (2.38), (2.39) imply the representation

g

(∂

∂zA

)= GAB σ

B ⇔ σA = GAB g

(∂

∂zB

)(2.40)

(iv) The linear endomorphism J : T (A)→ T (A) described by eq. (2.36)may be extended to an algebraic derivation of the tensor algebra D(A),commuting with contractions and vanishing on functions. The resultingoperation, henceforth denoted by v, is characterized by properties formallyanalogous to eqs. (2.23), namely

v(f) = 0 , v(X) = J(X) , < v(X), η > + < X, v(η) >= 0

for all f ∈ F(A), X ∈ D1(A), η ∈ D1(A).Exactly as we did in §2.2, by means of v we construct an anti-derivation

dv of the Grassmann algebra G(A), called once again the fiber differentiation,acting on arbitrary r–forms according to the equation

dvη := d v(η)− v (dη)− rdt ∧ η (2.41)3An alternative proposal for the construction of a “fundamental tensor” over the con-

straint submanifold A is outlined in Ref. [20]. The argument relies on the preliminaryassignment of a fibration of the configuration space time Vn+1 over an (r +1)-dimensionalbase manifold Vr+1. The resulting geometrical framework — intrinsically different fromthe one discussed here — is especially suited to the study of coupled systems of second-and first-order differential equations.

14

In local coordinates, the evaluation of dv relies on the pair of relations

dvf = −v df =∂f

∂zAσA (2.42)

dv df = d v df − dt ∧ df = −d(∂f

∂zAσA − f dt

)(2.43)

for all f ∈ F(A)

(v) At each z ∈ A, the pull-back (iz)∗∗ sets up a 1-1 correspondencebetween contact 1–forms in T ∗

i(z)(A) and contact 1–forms in T ∗z (A) . This

fact, together with eq. (2.22), determines a scalar product in the vectorbundle C(A), expressed locally by the relations(

ωi, ωj)

= gij (2.44)

with gij = gij(t, q1, . . . , qn, z1, . . . , zr) obtained by pulling back the quan-tities (2.22) to the submanifold A. In particular, by eqs. (2.38), (2.44) wehave the relations (

ωi, σA)

= GAB ∂ψi

∂zB(2.45)

whence, recalling eq. (2.34)

(σA, σB

)= GAM GBN ∂ψi

∂zM

∂ψj

∂zNgik gjl

(ωk, ωl

)= GAB (2.46)

as it was to be expected, on the basis of the identifications (2.40).The previous results are completed by the following

Proposition 2.1 The bundle V ∗(A) = g(V (A)) of virtual 1–forms over Acoincides with the orthogonal complement of the Chetaev bundle χ(A) in thevector bundle C(A) of contact 1–forms over A.

Proof. The orthogonality between V ∗(A) and χ(A) under the scalar prod-uct (2.44) is easily checked by direct computation. Indeed, if ν denotes anarbitrary Chetaev 1–form, eqs. (2.33), (2.40), (2.45) imply(

ν , g

(∂

∂zA

))= νiGAB

(ωi, σB

)= νi

∂ψi

∂zA= 0

The required conclusion then follows from a dimensionality argument, basedon the fact that, at each z ∈ A, the space V ∗

z (A) is r–dimensional, while theChetaev space χz(A) is (n− r)–dimensional. 2

(vi) In view of Proposition 2.1, every contact 1–form η over A admits aunique representation as the sum of a virtual 1–form and a Chetaev 1–form.

15

In local coordinates, setting η = ηi ωi, the required splitting is accomplished

through the decomposition

η = ηi dvψi + ηi

(ωi − dvψ

i)

(2.47)

Indeed, in view of eqs. (2.40), (2.42), (2.45), (2.46), it is easily seen that thefiber differentials dvψ

i are automatically in V ∗(A), while the differences

χi := ωi − dvψi (2.48)

satisfy the orthogonality conditions

(χi , σA

)=

(ωi − ∂ψi

∂zBσB, σA

)= 0 A = 1, . . . , r (2.49)

i.e. they are all Chetaev 1–forms.

Remark 2.1 The previous discussion points out that the 1–forms χi , i =1, . . . , n generate (locally) the Chetaev bundle. Needless to say, these gen-erators are not independent. By eqs. (2.34), (2.38) they are in fact subjectto the linear relations

gij∂ψi

∂zAχj = gij

∂ψi

∂zAωj −GAB σ

B = 0 A = 1, . . . , r (2.50)

mathematically equivalent to eqs. (2.49).In a similar way, the fiber differentials dvψ

i , i = 1, . . . , n generatelocally the bundle V ∗(A) of virtual 1–forms over A. Once again, a straight-forward dimensionality argument indicates that these generators are not in-dependent, but are subject to n−r linear relations. In terms of an arbitrarycartesian representation (2.27) for the submanifold A, these are summarizedinto the system

i∗(∂gσ

∂qk

)dv ψ

k =∂gσ

∂qk

∂ψk

∂zAσA ≡ 0 σ = 1, . . . , n− r (2.51)

3 Dynamics

3.1 Mechanical quantities

In this Section we shall discuss the dynamical aspects of the theory of con-strained systems, within the framework introduced in §2. As compared withthe analysis proposed in [12], the present contribution is aimed at a betterunderstanding of the role of d’Alembert’s principle as a general tool for theconstitutive characterization of the ideal constraints, independently of anyrequirement of holonomy, or of linearity.

16

Needless to say, the analysis is much indebted to the pioneering work ofN. Chetaev on the extension of the concept of virtual displacement to theclass of non-holonomic systems [14, 15, 16, 17, 18].

In what follows, we shall restate Chetaev’s ideas in a rigourous geo-metrical setting, especially suited to the formal developments of AnalyticalMechanics.

To keep the language as close as possible to the traditional one, we shalldiscuss everything in the euclidean three-space associated with an (arbitrar-ily chosen) frame of reference I, sticking to eqs. (2.28), (2.29) in order toexpress the (relative) positions and velocities of the points of the system.More precisely, for notational uniformity, we shall regard the functions (2.28)as representing the maps xξ · π : A → E3, i.e. we shall regard them as beingdefined on A, rather than on Vn+1.

The (pull-back of the) contact differentials of the functions xξ will bedenoted by dcxξ. Comparison with eq. (2.15), (2.28), (2.29), (2.30) yieldsthe explicit representation

dcxξ =∂xξ

∂qkωk = dxξ − vξdt (3.1)

By eqs. (2.29), (2.42), (2.48), the latter may be splitted into the sum

dcxξ =∂xξ

∂qk(dvψ

k + χk) = dvvξ + dχ xξ (3.2)

withdχ xξ :=

∂xξ

∂qkχk (3.3)

The expression (3.3) will be called the Chetaev differential of xξ.The contact differentials (3.1), together with the position and velocity

maps (2.28), (2.29), are explicitly involved in the representation of the me-chanical quantities associated with the system B in the frame of referenceI. Among these, especially relevant in a Lagrangian context are the kineticenergy

T =12

∫B|vξ|2m(dξ) (3.4)

and the kinetic momenta pk, defined collectively by the equation

pk ωk :=

∫B

vξ · dcxξ m(dξ) =(∫

Bvξ ·

∂xξ

∂qkm(dξ)

)ωk (3.5)

A straightforward comparison with the representation (2.8) of the holonomickinetic energy T yields the identifications

T = i∗(T ) , pk = i∗(∂T∂qk

)(3.6)

17

In a similar way, by eqs. (2.7), (2.29), (3.4), (3.5) we have the identities

∂T

∂zA=∫B

vξ ·∂vξ

∂zAm(dξ) =

∫B

vξ ·∂xξ

∂qk

∂ψk

∂zAm(dξ) = pk

∂ψk

∂zA(3.7)

∂pk

∂zA=∫B

∂vξ

∂zA· ∂xξ

∂qkm(dξ) =

∫B

(∂xξ

∂qr

∂ψr

∂zA

)· ∂xξ

∂qkm(dξ) = gkr

∂ψr

∂zA(3.8)

The exterior 2–form

Ω :=∫Bdvξ ∧ dcxξ m(dξ) (3.9)

will be called the kinetic Poincare-Cartan 2–form of the system relative tothe frame of reference I 4. Eqs. (3.1), (3.4), (3.5) imply the relation

Ω = d(pk ω

k)

+(∫

Bvξ · dvξ m(dξ)

)∧ dt = d

(pk ω

k + T dt)

(3.10)

Together with eqs. (2.30), (3.7), the latter gives rise to the representation

Ω = dpk ∧ ωk + pk dωk + dT ∧ dt =

[dpk +

(pr∂ψr

∂qk− ∂T

∂qk

)dt

]∧ ωk (3.11)

Alternatively, by eqs. (2.30), (3.6), (3.10), we have the identification

Ω = i∗ d

(∂T∂qk

ωk + T dt)

= i∗(d

(∂T∂qk

)− ∂T∂qk

dt

)∧ ωk (3.12)

A better insight into the nature of the kinetic Poincare-Cartan 2–formis provided by the splitting (3.2). The latter allows to express eq. (3.9) inthe equivalent form

Ω = Ω + Ωχ (3.13)

with

Ω :=∫Bdvξ ∧ dvvξ m(dξ) =

(∫Bdvξ ·

∂xξ

∂qkm(dξ)

)∧ dvψ

k (3.14)

Ωχ :=∫Bdvξ ∧ dχ xξ m(dξ) =

(∫Bdvξ ·

∂xξ

∂qkm(dξ)

)∧ χk (3.15)

The meaning of the representation (3.13) is clarified by the following

Theorem 3.1 For every vertical vector field V over A, the interior productV Ω satisfies the identity

V Ω = V Ω = g(V ) (3.16)

g : V (A) → V ∗(A) denoting the process of lowering the indices induced bythe fundamental tensor of the manifold A through eq. (2.39).

4The term “kinetic” is meant to point out the fact that, as compared with the standarddefinition [21, 22], eq. (3.9) leaves out all contributions coming from the forces acting onthe system.

18

Proof. By the definition (3.5) of the kinetic momenta, setting V = V A ∂∂zA ,

and recalling the identity (3.8), we have easily∫BV (vξ) ·

∂xξ

∂qkm(dξ) = V (pk) = V A gkr

∂ψr

∂zA

Together with eqs. (2.34), (2.40), (2.50), taking the representations (3.14),(3.15) into account, this implies the relations

V Ω = V (pk) dv ψk = V A gkr

∂ψr

∂zA

∂ψk

∂zBσB = V AGAB σ

B = g(V )

V Ωχ = V (pk)χk = V A gkr∂ψr

∂zAχk = 0

mathematically equivalent to eq. (3.16). 2

If, rather than a vertical vector field V , we consider a dynamical flow Zover A, eq. (3.13) provides a splitting of the interior product Z Ω into thesum of a virtual 1–form and a Chetaev one, namely

Z Ω = Z Ω + Z Ωχ (3.17)

In analogy with Theorem 3.1 we have then the following

Corollary 3.1 Let Z denote an arbitrary dynamical flow. Then:

a. the Chetaev 1–form Z Ωχ is independent of Z, i.e. it is determineduniquely in terms of the mechanical properties of the system;

b. the knowledge of the virtual 1–form Z Ω is mathematically equivalentto the knowledge of Z.

Proof. Starting with an arbitrarily chosen dynamical flow Z0, set Q0 :=Z0 Ω, and recall that the most general dynamical flow Z is obtained byadding to Z0 an arbitrary vertical vector field V . In view of Theorem 3.1we have then the relation

(Z − Z0) Ωχ = V Ωχ = 0

proving assertion a), and

(Z − Z0) Ω = V Ω = g(V )

whence also

Z = Z0 + g−1[(Z − Z0) Ω

]= Z0 + g−1

(Z Ω−Q0

)(3.18)

showing that the knowledge of Z Ω is indeed equivalent to the knowledgeof Z. 2

19

In local coordinates, recalling eqs. (2.48), (3.11), (3.17) (or also by directcomputation, starting with eqs. (3.14), (3.15)) one can easily verify thevalidity of the expressions

Z Ω =[Z(pk)−

∂T

∂qk+ pr

∂ψr

∂qk

]dvψ

k (3.19)

Z Ωχ =[Z(pk)−

∂T

∂qk+ pr

∂ψr

∂qk

]χk (3.20)

As a concluding remark we observe that, as a consequence of Corollary3.1, strictly associated with the kinetic Poincare-Cartan 2–form — and thus,ultimately, with the choice of the frame of reference I — is a distinguisheddynamical flow ZI , defined by the condition

ZI Ω = 0 (3.21)

In view of eq. (3.18), every other dynamical flow Z is then determineduniquely by the knowledge of the virtual 1–form Q = Z Ω, according tothe relation

Z = ZI + g−1(Q) (3.22)

The expression (3.22) resembles very closely the situation discussed in[12], where the term g−1(Q) was taken axiomatically as a representation ofthe forces acting on the system in the frame of reference I.

In local coordinates, taking eqs. (2.34), (3.8), (3.19) into account, thesolution of eq. (3.21) is easily recognized to be

ZI =∂

∂t+ψk ∂

∂qk−GAB

(∂pk

∂t+ ψr ∂pk

∂qr+ pr

∂ψr

∂qk− ∂T

∂qk

)∂ψk

∂zB

∂

∂zA(3.23)

3.2 Virtual work

The (frame-dependent) correspondence between dynamical flows and virtual1–forms described in Corollary 3.1, namely

Z ←→ Z Ω =∫BZ(vξ) · dvvξ m(dξ)

plays a crucial role in the geometrization of Dynamics.To ensure consistency with the traditional language, we introduce the

following

Definition 3.1 For each z ∈ A, the map B → V ∗z (A) assigning to each

ξ ∈ B the vector valued virtual 1–form

(dv vξ)|z =(∂xξ

∂qk

)|z

(dv ψk)|z (3.24)

will be called the virtual displacement of the system at z.

20

Before discussing the dynamical significance of Definition 3.1, a compar-ison with the conventional concept of virtual displacement is in order. Tothis end, we make use of the fact that every contact 1–form at z may beviewed as a linear functional on the vertical space Vπ(z)(Vn+1), through thepairing (2.16).In particular, if we let the functionals associated with the 1–forms (dv vξ)|zand (dv ψ

k)|z be denoted respectively by δxξ and δqk, k = 1, . . . , n, eq. (3.24)takes the traditional form 5

δxξ =(∂xξ

∂qk

)|π(z)

δqk (3.25)

In general, however, as pointed out in Remark 2.1, the fiber differen-tials dvψ

k — and thus also the functionals δqk — are not linearly indepen-dent, but are subject to the inner identities (2.51), gσ(t, qi, qi) = 0 , σ =1, . . . , n − r denoting any cartesian representation for the submanifold A.When expressed in terms of the functionals δqk these reproduce the so-calledChetaev conditions(

∂gσ

∂qk

)|zδqk = 0 σ = 1, . . . , n− r (3.26)

It goes without saying that in the holonomic case (corresponding tor = n), the conditions (3.26) are empty, and the functionals δqk are linearlyindependent.

To sum up, Definition 3.1, translated into the language of linear func-tionals over Vπ(z)(Vn+1), yields back the traditional notion of virtual dis-placement, with the Chetaev conditions (3.26) automatically embodied intothe formalism.

To understand the role of Definition 3.1 in a dynamical context, we nowturn our attention to the description of the interactions. To this end, we relyon the usual classification of the mechanical forces into active and reactiveones, as well as on the identification — already outlined in §2.1 — of thematerial system B with a measure space (B,S,m), S denoting the σ-ring ofmeasurable subsets of the material space B. By definition, a representationof the active forces is then a map F : S ×A → V3, assigning to each Ω ∈ S acorresponding vector valued function FΩ(z) := F(Ω, z), expressing the total(active) force on Ω in terms of the kinetic state of the system.

With the standard notation of measure theory, we shall write

FΩ =∫Ω

F(dξ) (3.27)

5The viewpoint of regarding both sides of eq. (3.25) as functionals on vertical vectorsreflects the classical notion of virtual displacement as an “infinitesimal change of theconfiguration of the system, resulting from arbitrary infinitesimal changes in the positionsof all points, consistent with the restrictions imposed by the constraints at the giveninstant t” [3].

21

the dependence of both sides on the variable z being implicitly understood.The quantities

Qk :=∫B

∂xξ

∂qk· F(dξ) (3.28)

(Qk =∑N

i=1 Fi ·∂xξ

∂qk in the discrete case) will be called the Lagrangiancomponents of F.

Recalling the representation (3.1) for the contact differentials dcxξ , wehave the obvious identification∫

Bdcxξ · F(dξ) =

(∫B

∂xξ

∂qk· F(dξ)

)ωk = Qk ω

k (3.29)

The description of the reactive forces follows a similar pattern, the only (sub-stantial!) difference being that, in general, the associated map Φ : S × A→ V3 is a-priori unknown. In place of it, one has a statement — hence-forth called a constitutive characterization of the constraints — indicatingwhich reactive forces are effectively allowed and which are not, the selectiondepending on the physical properties of the devices involved in the imple-mentation of the constraints. We shall return on this point in §3.3. Thetotality of admissible maps Φ : S ×A → V3 will be indicated by H(S ×A).

Exactly as we did with the active forces, we introduce the notation ΦΩ =∫Ω Φ(dξ), and define the Lagrangian components ϕk of Φ collectively through

the equation ∫Bdcxξ · Φ(dξ) =

(∫B

∂xξ

∂qk· Φ(dξ)

)ωk = ϕk ω

k (3.30)

Remark 3.1 In the case of distributed forces (defined by the conditionFΩ = ΦΩ = 0 whenever m(Ω) = 0), the measures F,Φ may be expressed interms of the corresponding Radon-Nikodym derivatives f = dF/dm , ϕ =dΦ/dm (specific forces, or forces per unit mass) in the standard form F =fm, Φ = ϕm [23, 24]. This possibility breaks down in the presence ofconcentrated forces, i.e. of force measures not absolutely continuous withrespect to m.

Remark 3.2 Various instances of vector-valued, z-dependent measures onB, absolutely continuous with respect to m, have already been met in thedescription of the mechanical quantities. The most significant exampleis provided by the “momentum measure” P, related to the velocity mapv(ξ, z) := vξ(z) by the identification P = vm. For each measurable domainΩ ⊂ B, the quantity

PΩ(z) :=∫Ω

v(ξ, z)m(dξ) (3.31)

expresses the total linear momentum of Ω as a function of the kinetic stateof the system.

22

Let us now denote by M(S × A) (M for short) the totality of vectorvalued, z-dependent measures U : S×A → V3 such that, for each Ω ∈ S, thefunction UΩ(z) := U(Ω, z) depends differentiably on z. From an algebraicviewpoint,M has the nature of a module over the ring F(A) of differentiablefunctions over A.

Definition 3.2 For each U ∈M, the virtual 1–form

Λ(U) :=∫Bdvvξ ·U(dξ) =

(∫B

∂xξ

∂qk·U(dξ)

)dvψ

k (3.32)

(Λ(U) =∑N

i=1 Ui ·dvvi in the discrete case), will be called the virtual contentof U. In the special case when U is a force measure, the expression (3.32)will be called the virtual work of U.

Comparison with eq. (3.2) shows that the virtual work of the active forcescoincides with the virtual part of the 1–form (3.29). In local coordinates,recalling eq. (2.48), we have the explicit representations

Λ(F) =(∫

B

∂xξ

∂qk· F(dξ)

)dv ψ

k = Qk∂ψk

∂zAσA := QA σ

A (3.33)

Qk ωk = Qk (dvψ

k + χk) = Λ(F) +Qk χk (3.34)

In particular at each z ∈ A, keeping the same notation as in the discus-sion following Definition 3.1, and denoting by δW the linear functional overVπ(z)(Vn+1) associated with Λ(F), we get the familiar expression

δW =(∫

B

∂xξ

∂qk· F(dξ)

)δqk = Qk δq

k

with the Chetaev conditions (3.26) automatically embodied into the formal-ism. Fairly similar conclusions apply to the virtual work Λ(Φ) of the reactiveforces.

We conclude this Subsection by proving a factorization theorem, whichwill play a major role in the discussion of the constitutive characterizationof the reactive forces. To this end, within the moduleM of vector valued, z-dependent measures defined above, we single out the submoduleM⊥ formedby the totality of measures with vanishing virtual content, namely

M⊥ = U | U ∈M , Λ(U) = 0 (3.35)

Moreover, we introduce a correspondence between vertical vector fieldsand vector valued measures, based on the prescription

V = V A ∂

∂zA7→ V (v)m = V A ∂v(ξ, z)

∂zA(3.36)

23

In view of eqs. (3.14), (3.32), taking Theorem 3.1 into account, thisimplies the identity

Λ (V (v)m) =∫B

(V dvξ) · dv vξ m(dξ) = V Ω = g (V ) (3.37)

showing that the knowledge of the measure V (v)m is mathematically equiv-alent to the knowledge of the field V .

Collecting all previous definitions, we can now state

Theorem 3.2 Every measure U ∈M admits a unique factorization of theform

U := U⊥ + VU(v)m (3.38)

in terms of a measure U⊥ ∈M⊥, and of a vertical vector field VU over A.

Proof. Simply observe that, on account of eqs. (3.35), (3.37), eq. (3.38) ismathematically equivalent to the condition VU = g−1(Λ(U)). 2

For dynamical purposes, the content of Theorem 3.2 is conveniently sum-marized into the pair of assertions

a. the correspondence U→ U⊥ = U− VU(v)m determines an F-linearprojection of the module M ontoM⊥;

b. every measure U ∈M is uniquely determined by the knowledge of itsprojection U⊥ ∈M⊥, and of the virtual 1–form Λ(U).

The proof is entirely straightforward, and is left to the reader.

3.3 Equations of motion

To complete our analysis, we shall now discuss the role of the previous def-initions in the study of the problem of motion for the system B.As widely explained in the literature (see, e.g. [12] and references therein),the problem can be stated geometrically as follows: given the active forces,expressed by a vector-valued, z-dependent measure F ∈ M, as well as theconstitutive characterization of the constraints, summarized into the assign-ment of a suitable class H(S × A) of admissible reactive forces, determinea dynamical flow Z over A in such a way that, for each z ∈ A, the integralcurve of Z through z coincides with the first jet extension of the sectionγ : < → Vn+1 describing the evolution of the system from the initial kineticstate z.Within the stated framework, the principle of determinism has then an obvi-ous mathematical counterpart in the requirement that, for each assignmentof F, the constitutive characterization of the constraints be sufficient todetermine a unique such Z.

24

Let us examine this point in detail: with the notation of §3.2, Newton’s2nd law requires that, for each measurable subset Ω ∈ S, the total linearmomentum PΩ, viewed as a vector-valued function over A, be related to thetotal (active + reactive) force FΩ + ΦΩ by the evolution equation

Z(PΩ) = FΩ + ΦΩ (3.39)

Z denoting the (so far unknown) dynamical flow. Recalling the representa-tions (3.27), (3.31), as well as the analogous expression for ΦΩ, eq. (3.39)may be written in the equivalent form∫

ΩZ(v)m(dξ) =

∫Ω

(F(dξ) + Φ(dξ)) ∀Ω ∈ S

i.e. as a relationF + Φ = Z(v)m (3.40)

between vector valued measures in the class M.Keeping the same notation as in Theorem 3.2, we can then state

Corollary 3.2 Eqs. (3.40) are mathematically equivalent to the system

Z Ω = Λ(F + Φ) (3.41)

ZI(v)m = F⊥ + Φ⊥ (3.42)

ZI denoting the dynamical flow associated with the frame of reference Ithrough the prescription (3.21).

Proof. For every dynamical flow Z, eqs. (3.14), (3.32) yield the identity

Λ(Z(v)m) =∫BZ(vξ) · dvvξ m(dξ) = Z Ω

Setting V = g−1(Z Ω), and recalling eqs. (3.21), (3.22), this implies therelations

Λ(ZI(v)m) = 0 ; Z(v)m = ZI(v)m+ V (v)m

which, together with eqs. (3.35), (3.38), provide the further identification

ZI(v)m = (Z(v)m)⊥

The system (3.41), (3.42) is therefore equivalent to the pair of conditions

Λ(Z(v)m) = Λ(F + Φ)

ZI(v)m = F⊥ + Φ⊥

25

The conclusion is then a direct consequence of assertion b) following Theo-rem 3.2. 2

Corollary 3.2 allows a precise mathematical formulation of the concept ofdeterminism. This is readily understood by recalling that, according to eq.(3.21), the vector field ZI is a datum of the problem, completely determinedby the knowledge of the frame of reference I, and expressed locally througheq. (3.23).Taking eqs. (3.41), (3.42) as well as Corollary 3.1 into account, and recallingthe comments following Theorem 3.2, one is then led to the conclusion thatthe only freedom left by Newton’s 2nd law in the determination of both thedynamical flow Z and the reactive force measure Φ is the value of the virtualwork Λ(Φ).

The missing information is of course to be found in the constitutivecharacterization of the constraints. A criterion for the latter to be consistentwith the principle of determinism is therefore the requirement that, for eachchoice of the active force measure F, the condition Φ ∈ H(S ×A), possiblycompleted by eq. (3.42), be sufficient to determine the virtual 1-form Λ(Φ)uniquely in terms of F, and of the kinetic state of the system.

By far the simplest and most significant application of the stated crite-rion is provided by the celebrated d’Alembert’s principle, summarized intothe following

Definition 3.3 A set of constraints is said to be ideal if and only if thecorresponding constitutive characterization is expressed by the condition

Φ ∈ H(S ×A) ⇔ Λ(Φ) =∫Bdvvξ · Φ(dξ) ≡ 0 (3.43)

i.e. if and only if the class of admissible reactive forces coincides with thesubmodule M⊥ ⊂M.

As pointed out above, the constitutive characterization (3.43) is automati-cally deterministic.

An important insight into the dynamical content of d’Alembert’s princi-ple is provided by an equally celebrated statement, known as the principleof least constraint, of K. F. Gauss. In the traditional formulation, valid fordiscrete systems, the latter reads:

For a mechanical system subject to ideal constraints, the actualmotion under the action of given active forces, is the one forwhich, at any instant t, the quantity

C =12

N∑i=1

|Φi|2

mi(3.44)

26

attains a minimum within the class of the kinematically admis-sible evolutions [3, 5, 6, 12, 17, 25].

In strictly logical terms, rather than as a characterization of the class ofadmissible reactive forces, Gauss’ principle is to be seen as a prescriptionindicating how to handle the dynamical equations Fi + Φi = mi ai, in orderto determine the evolution of the system — and thus also, indirectly, thereactive forces Φi — in terms of the active forces Fi.To this end, one has simply to replace each vector Φi , i = 1, . . . , N in eq.(3.44) by the difference mi ai−Fi, looking then, at each instant t, for the val-ues ai that minimize the resulting expression within the class of admissibleaccelerations.

In order to compare the content of d’Alembert’s and Gauss’ principles,we shall first extend the latter to the case of reactive forces described byvector valued measures.An apparent limitation here arises from the fact that the expression (3.44)has no natural counterpart in the case of force measures not absolutely con-tinuous with respect to the mass measure m. The difficulty, however, iseasily overcome by observing that, as far as the dependence on the acceler-ations is concerned, the function (3.44) has the same extremal points as thedifference

C := C − 12

N∑i=1

|Fi|2

mi=

12

N∑i=1

(Φi + Fi)mi

· (Φi − Fi)

Unlike the original expression (3.44), the newer one can be extended toarbitrary force measures, due to the fact that, as a consequence of Newton’s2nd law, the sum Φ + F is always absolutely continuous with respect to m.Introducing the Radon-Nikodym derivative d(Φ + F)/dm, we can thereforerestate Gauss principle as a minimality request for the functional

C =12

∫B

d(Φ + F)dm

· [Φ(dξ)− F(dξ)] (3.45)

at any instant t, within the class of admissible accelerations.We can now prove

Theorem 3.3 When formulated in terms of the functional (3.45), Gauss’characterization of the class of ideal constraints is mathematically equivalentto d’Alembert’s constitutive characterization (3.43).

Proof. Taking Newton’s 2nd law (3.40) into account, the functional (3.45)may be written as

C =12

∫BZ(vξ) · [Z(vξ)m(dξ)− 2F(dξ)]

27

Viewed as a prescription for the evaluation of the dynamical flow Z in termsof F, Gauss minimality request may therefore be stated in the form∫

BZ(vξ) · [Z(vξ)m(dξ)− 2F(dξ)] ≤

∫BZ ′(vξ) ·

[Z ′(vξ)m(dξ)− 2F(dξ)

]for all admissible flows Z ′. Setting Z ′ − Z := V , this amounts to the re-quirement∫B

V (vξ) · [Z(vξ)m(dξ)− F(dξ)] + |V (vξ)|2m(dξ)

≥ 0 ∀V = V A ∂

∂zA

whence, by the arbitrariness of V A

0 =∫B

∂vξ

∂zA· [Z(vξ)m(dξ)− F(dξ)] , A = 1, . . . , r

The previous relations may be summarized into the single equation

0 =(∫

B

∂vξ

∂zA· [Z(vξ)m(dξ)− F(dξ)]

)σA =

∫Bdvvξ·[Z(vξ)m(dξ)− F(dξ)]

clearly identical to the prescription that would arise by inserting Newton’s2nd law (3.40) directly into d’Alembert’s condition (3.43). 2

The equations of motion for a mechanical system subject to ideal con-straints are obtained by inserting d’Alembert’s characterization (3.43) di-rectly into eq. (3.41), namely

Z Ω = Λ(F) (3.46)

In local coordinates, making use of the representations (3.19), (3.33), thisgives rise to the system(

Z(pk) + pr∂ψr

∂qk− ∂T

∂qk

)∂ψk

∂zA= QA A = 1, . . . , r (3.47)

Alternatively, on the basis of eq. (3.22), the solution of eq. (3.46) maybe written in compact form as

Z = ZI + g−1 Λ(F) (3.48)

From this, recalling eqs. (3.23), (3.33), we conclude that the evolution of thesystem is determined by the n+ r ordinary differential equations

dqi

dt= ψi(t, qk, zA)

dzA

dt= GAB

(−∂pk

∂t− ψr ∂pk

∂qr− pr

∂ψr

∂qk+∂T

∂qk+Qk

)∂ψk

∂zB

(3.49)

28

for the unknowns qi = qi(t) , zA = zA(t).The previous arguments provide a complete solution of the problem of

motion, valid whenever the embedding i : A → j1(Vn+1) is expressed in theintrinsic form (2.26).

Of course, the same problem can also be tackled starting with a cartesianrepresentation for the submanifold A, of the form

gσ(t, q1, . . . , qn, q1, . . . , qn) = 0 σ = 1, . . . , n− r

with rank ‖∂(g1, . . . , gn−r)/∂(q1, . . . , qn)‖ = n− r.Under the stated assumption, rather than looking for a direct evaluation

of the dynamical flow Z over A, it is more convenient to consider the push-forward i∗(Z), viewed as the restriction to the submanifold i(A) of a semi-spray Z = ∂

∂t + qi ∂∂qi + Zi ∂

∂qi , defined in a neighbourhood of i(A), andi–related to Z.By definition, this implies the identification Z |i(z) = iz∗(Z z) ∀ z ∈ A,showing that the evaluation of Z at each point i(z) is indeed equivalent tothe solution of the problem of motion.

A first set of conditions on Z comes from the relations

i∗(Z(gσ)

)= Z(i∗(gσ)) = 0 σ = 1, . . . , n− r (3.50)

expressing the requirement that the field Z be everywhere tangent to thehypersurface i(A).

In addition to this, recalling the representation (3.12) of the Poincare-Cartan 2–form on A, we have the identification

Z Ω = Z i∗(d

(∂T∂qk

)− ∂T∂qk

dt

)∧ ωk = i∗

[Z

(∂T∂qk

)− ∂T∂qk

]ωk (3.51)

T denoting the holonomic kinetic energy (2.8).On the other hand, in view of eqs. (3.9), (3.29), (3.30), Newton’s 2nd law

(3.40) implies the relation

Z Ω =∫BZ(vξ) · dcxξ m(dξ) =

=∫Bdcxξ · [F(dξ) + Φ(dξ)] = (Qk + ϕk) ωk (3.52)

Qk and ϕk denoting the Lagrangian components of the active and reactiveforces.

All this holds independently of any constitutive assumption concerningthe nature of the constraints. In particular, on account of eqs. (3.2), (3.32),the 1–form (3.30) may be splitted into

ϕk ωk =

∫Bdcxξ · Φ(dξ) = Λ(Φ) +

∫Bdχ xξ · Φ(dξ) (3.53)

29

The content of d’Alembert’s principle (3.43) is then that, within theclass of ideal constraints, the expression at the left-hand side of eq. (3.53) isalways a Chetaev 1–form over A.Noting further that the differential 1–forms i∗(dv g

σ) span the Chetaev bun-dle, the constitutive characterization (3.43) may therefore be rephrased as

Φ ∈ H(S ×A) ⇔ ϕk ωk = λσ i

∗(dv gσ) = λσ i

∗(∂gσ

∂qk

)ωk (3.54)

the coefficients λσ denoting n− r a-priori unspecified functions over A.The rest is now entirely straightforward: by eqs. (3.51), (3.52), taking

the characterization (3.54) into account, we get the n independent relations

i∗[Z

(∂T∂qk

)− ∂T∂qk

]= Qk + λσ i

∗(∂gσ

∂qk

)(3.55)

These, together with eqs. (3.50), form a system of 2n−r equations for the ncomponents Zi of the semi-spray Z along i(A), and for the n−r coefficientsλσ, thus providing once again a complete solution for the problem of motion.

Example 3.1 As an illustration of the methods discussed above, we shallsketch the familiar problem of a rigid disc, rolling without sliding on a hor-izontal plane, under the action of given active forces. For simplicity, thepositional constraints will be assumed to include the requirement that theplane of the disk remains vertical throughout the evolution.Denoting by m and R the mass and radius of the disk, and introducingLagrangian coordinates x, y, ϕ, θ, expressing respectively the cartesian coor-dinates of the point of contact of the disk with the plane, the angle betweenthe normal to the disk and the x axis, and the angle of rotation of the diskaround its axis, the holonomic kinetic energy is given by the equation

T =12m(x2 + y2) +

12

(mR2

4ϕ2 +

mR2

2θ2

)(3.56)

while the rolling condition is summarized into the pair of relations

x+Rθ sinϕ = 0 y −Rθ cosϕ = 0

Choosing the generalized velocities z1 := ϕ , z2 := θ as independent vari-ables, one can easily write down the explicit representation of the constraintmanifold A

x = ψ1(t, q, z) = −Rz2 sinϕ y = ψ2(t, q, z) = Rz2 cosϕ

ϕ = ψ3(t, q, z) = z1 θ = ψ4(t, q, z) = z2

30

as well as the (pull-back of the) contact 1-forms

ω1 = dx+Rz2 sinϕdt ω2 = dy −Rz2 cosϕdt

ω3 = dϕ− z1dt ω4 = dθ − z2dt

The Chetaev bundle is generated by the pair of independent 1-forms

ν1 = ω1 +R sinϕ ω4 = dx+R sinϕdθ

ν2 = ω2 −R cosϕ ω4 = dy −R cosϕdθ

Taking the expression (3.56) for the holonomic kinetic energy into account,and denoting by Qx, Qy, Qϕ, Qθ the Lagrangian components of the activeforces, a straightforward comparison with eqs. (3.55) yields the equations ofmotion

md2x

dt2= Qx + λ1 m

d2y

dt2= Qy + λ2

mR2

4d2ϕ

dt2= Qϕ

mR2

2d2θ

dt2= Qθ + λ1R sinϕ− λ2R cosϕ

These determine the evolution of the system, as well as the Lagrangiancomponents of the reactive forces, summarized into the Chetaev 1-form

ϕkωk = λ1ν

1 + λ2ν2 = λ1ω

1 + λ2ω2 + (λ1 sinϕ− λ2 cosϕ)Rω4

In order to write down the equations of motion in the intrinsic form, wenotice that, on account of eqs. (3.6), (3.56), the (pull-back of the) kineticenergy and of the kinetic momenta pk over A are given by the expressions

T = i∗(T ) =12mR2(z2)2 +

12

[mR2

4(z1)2 +

mR2

2(z2)2

]and

p1 = −mRz2 sinϕ , p2 = mRz2 cosϕ , p3 =mR2

4z1 , p4 =

mR2

2z2

In view of eq. (3.49) the required equations are then summarized into thesystem

dx

dt= −Rz2 sinϕ

dy

dt= Rz2 cosϕ

dϕ

dt= z1 dθ

dt= z2

dz1

dt=

4mR2

Qϕdz2

dt=

23mR2

[−R sinϕQx +R cosϕQy +Qθ] .

31

A Appendix

Keeping the same notation as in §2.3, the fibration

A i−−−−→ j1(Vn+1)

π

y yπ

Vn+1 Vn+1

(A.1)

will be said to be integrable (or more precisely, to identify a set of inte-grable kinetic constraints) if and only if, in a neighbourhood of each pointi(z) , z ∈ A, the submanifold i(A) ⊂ j1(Vn+1) admits at least one cartesianrepresentation (2.27) of the special form

gσ =∂fσ

∂qkqk +

∂fσ

∂t= 0 σ = 1, . . . , n− r (A.2)

involving (the pull-back of) n − r differentiable functions fσ(t, q1, . . . , qn)defined on Vn+1. Recalling the definition of the Chetaev bundle over A, weprove

Theorem A.1 The fibration (A.1) is integrable if and only if the ideal I

generated by the module of Chetaev 1-forms over A is a differential ideal.

Proof. Necessity: the existence of a local representation of the form (A.2)for the submanifold A implies that the Chetaev bundle is generated locallyby the family of 1-forms

νσ = i∗ (dvgσ) =

∂fσ

∂qkωk σ = 1, . . . , n− r (A.3)

Comparison with eqs. (2.30), (A.2) provides the identifications

νσ = dfσ − i∗(∂fσ

∂qkqk +

∂fσ

∂t

)dt = dfσ

showing that, under the stated assumption, I is a differential ideal.Sufficiency: according to Frobenius theorem, the assumption that I is

a differential ideal is mathematically equivalent to the assertion that theChetaev bundle is generated locally by n− r exact 1-forms νσ = dfσ.Since — by definition — the latter are automatically contact 1-forms, thecondition Z dfσ = 0 for all dynamical flows Z implies the validity of therelations

∂fσ

∂zA= 0 ⇒ fσ = fσ(t, q1, . . . , qn) (A.4)

32

(∂

∂t+ ψk ∂

∂qk

)dfσ = 0 ⇒ i∗

(∂fσ

∂qkqk +

∂fσ

∂t

)= 0 (A.5)

On the basis of eqs. (A.4), (A.5), a straightforward dimensionality argumentshows that, under the stated assumption, the submanifold A is representedlocally in the form

∂fσ

∂qkqk +

∂fσ

∂t= 0 σ = 1, . . . , n− r

i.e. it satisfies the integrability requirement (A.2). 2

33

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