A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems

Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the ba...

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Дата:	2007
Автор:	Benenti, S.
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Опубліковано:	Інститут математики НАН України 2007
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems / S. Benenti // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ.

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spelling	irk-123456789-1478262019-02-17T01:23:51Z A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems Benenti, S. Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vector calculus on Euclidean 3-space and on tangent bundles are needed. Elementary examples are illustrated. 2007 Article A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems / S. Benenti // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J60; 70F25 http://dspace.nbuv.gov.ua/handle/123456789/147826 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vector calculus on Euclidean 3-space and on tangent bundles are needed. Elementary examples are illustrated.
format	Article
author	Benenti, S.
spellingShingle	Benenti, S. A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Benenti, S.
author_sort	Benenti, S.
title	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
title_short	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
title_full	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
title_fullStr	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
title_full_unstemmed	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems
title_sort	'user-friendly' approach to the dynamical equations of non-holonomic systems
publisher	Інститут математики НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/147826
citation_txt	A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems / S. Benenti // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-11T02:55:05Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 036, 33 pages A ‘User-Friendly’ Approach to the Dynamical Equations of Non-Holonomic Systems? Sergio BENENTI Department of Mathematics, University of Turin, Italy E-mail: sergio.benenti@unito.it URL: http://www2.dm.unito.it/∼benenti/ Received November 29, 2006, in final form February 13, 2007; Published online March 01, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/036/ Abstract. Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vector calculus on Euclidean 3-space and on tangent bundles are needed. Elementary examples are illustrated. Key words: non-holonomic systems; dynamical systems 2000 Mathematics Subject Classification: 37J60; 70F25 1 Preamble The classical theory of non-holonomic dynamical systems, even in recent times, is treated in a growing number of papers. Most of them require the use of modern differentiable and algebraic structures which are not familiar to non-mathematicians working on concrete applications. On the other hand, several papers are dedicated to the analysis of special non-holonomic mechanical systems, quite interesting but treated with ad hoc methods. These are the reasons why I think useful to propose a ready to use approach to the dynamics of non-holonomic systems, requiring the knowledge of the basic notions of the vector calculus on the Euclidean three-space and on tangent bundles only, and avoiding the use of cotangent bundles (Hamiltonian framework) and jet-bundles. In the present paper I illustrate two different, general, effective and concise methods for writing the dynamical equations of a given non-holonomic system. These methods correspond to the two ways of representing kinematical constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints and lead to different (but obviously equivalent) dynamical systems. In order to make this paper self-contained, a straightforward approach to the Gauss principle and to the Gibbs–Appell equations is illustrated. Our starting point will be the well under- standable Newton dynamical equations for a system of massive points. The tutorial character of this paper does not exclude the presence of some novelties. Many articles and books have been consulted in writing this paper. To cite all them would make the list of references quite long. On the other hand, such a long list would not fit with the purposes of this paper. Anyway, I must mention the excellent recent books [2, 4], from which I have got many useful hints. They are very well readable and advisable to non-experts which would like to go into this matter in more depth, and to know its present developments and applications. ?This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html mailto:sergio.benenti@unito.it http://www2.dm.unito.it/~benenti/ http://www.emis.de/journals/SIGMA/2007/036/ http://www.emis.de/journals/SIGMA/kuznetsov.html 2 S. Benenti 2 Introduction Let us consider a mechanical system with a well defined configuration manifold Qn. The di- mension n of Q is the number of degrees of freedom of the system. Let us denote by q = (qi) any (Lagrangian) coordinate system on Q and by (q, q̇) = (qi, q̇i) the associated coordinates on the tangent bundle TQ. This tangent bundle is the space of the kinematical states. A kinematical constraint is given by a subset C ⊂ TQ. A special case is that of regular constraint : Definition 2.1. A kinematical constraint C ⊂ TQ is said to be regular if • C is a submanifold of dimension n+m, m < n; • for all q ∈ Q, Fq = C ∩ TqQ is a submanifold of dimension m; • the restriction to C of the tangent fibration τQ : TQ→ Q is a surjective submersion1. The constraint is linear if each Fq is a subspace (see Fig. 1). Figure 1. A regular constraint is a fibration onto Q. A regular constraint can be represented in two ways: • Parametric representation: it is described by m equations, q̇i = ψi(q, z), (2.1) where z = (zα), α = 1, . . . ,m < n are called parameters. Note that (q, z) can be interpreted as local coordinates on C. • Implicit representation: it is described by r = n−m independent equations2, Ca(q, q̇) = 0, a = 1, . . . , r. (2.2) 1If this notion is not understood by the reader, he can look at the equivalent conditions (2.3) and (2.4) below. The definition of regularity is taken from [10] and [6]. In [6] an extension of this definition is given: it requires the existence of a submanifold Q1 ⊂ Q such that C is a submanifold of TQ1 and the restriction to C of the tangent fibration is a submersion. For our purposes we do not need to consider this more general case. 2This means that the differentials dCa are linearly independent at each point of C. Dynamical Equations of Non-Holonomic Systems 3 In these two representations, the regularity of the constraint is represented by conditions rank [ψi α]n×m = m, rank [Ca i ]n×r = r, (2.3) respectively, where ψi α .= ∂ψi ∂zα , Ca i .= ∂Ca ∂q̇i . (2.4) Note that the regularity conditions may be not fulfilled at certain states, that we call sin- gular states. They always occurs, for instance, for non-linear homogeneous constraints (see Remark 7.5). In the following, with the exception of special remarks, the constraints are assumed to be non- linear. However, in the case of linear constraints, we shall assume – without loss of generality – that the functions ψi(q, z) are linear in the parameters, so that ψi = ψi α(q) zα, or that the functions Ca(q, q̇) are linear in the q̇, so that Ca = Ca i (q) q̇i. The leading idea of the f irst method is to consider the parametric equations (2.1) of the constraint as a first set of dynamical equations, to be completed by a second set of equations of the kind żα = Zα(q, z): q̇i = ψi(q, z), żα = Zα(q, z). (2.5) This is a first-order system of ODE’s. Once the initial conditions (q0, z0) are fixed, they give a unique actual motion of the mechanical system. By actual motion we mean a physical motion obeying to the constraints. The explicit expressions of the functions Zα(q, z) depend of course on the given functions ψi(q, z). But they depend also on the dynamical principles we assume. The Newton dynamical equation for a material point will be the only physical principle on which we shall base our approach. Furthermore, we shall assume that the constraints are ideal or perfect, according to a suitable mathematical definition: this means to accept a certain constitutive condition on the constraint as a postulate. In this way, the Gauss ‘principle’ will follow as a ‘theorem’ from the Newton equations of a system of particles. The second method presented here is based on the implicit representation of the con- straints. It is a revisitation, with improvements and simplifications, of the well-known Lagrange- multipliers method. In this context we get the explicit expression of the reactive forces. This is useful, in the concrete applications, for measuring the stress that the constraints have to support. These two methods lead to different first-order dynamical equations i.e., to different vector fields: • The first method (parametric representation) produces a vector field Z on the constraint manifold C. The integral curves of Z give all actual motions – see Fig. 2. • The second method (implicit representation) produces a vector field D on the whole TQ but tangent to C. Only its restriction to C has in fact a physical meaning i.e., only its integral curves which start from a point of C (and which will lie on C) represent actual motions – see Fig. 3. As a byproduct, this method gives the explicit expressions of the reactive forces, which can be estimated along any actual motion. 4 S. Benenti Figure 2. Parametric representation of C – The vector field Z. Figure 3. Implicit representation of C – The vector field D. 3 Ideal constraints At a microscopical level, a mechanical system is made of a collection of material points (Pν ,mν), ν = 1, . . . , N . The position vector rν of each point Pν in the Euclidean three-space is a func- tion rν(q) of the chosen Lagrangian coordinates. A motion of the system is then represented by a time-parametrized curve qi = qi(t) on Q, and at each configuration q ∈ Q all possible velocities of the points are given by vν = ∂rν ∂qi q̇i, (vi) ∈ R. (3.1) A kinematical state of the system is the collection of all pairs position-velocity (rν ,vν) of the points. The collection of all possible states is then the tangent bundle TQ of Q. At any fixed state the accelerations aν are given by aν = ∂2rν ∂qi∂qj q̇iq̇j + ∂rν ∂qi dq̇i dt . (3.2) Let us consider the parametric representation (2.1) of the constraint. Then the velocity and the acceleration of each point, compatible with the kinematical constraint, are vν(q, z) = ∂rν ∂qi ψi, aν(q, z, ż) = ∂2rν ∂qi∂qj ψiψj + ∂rν ∂qi ( ∂ψi ∂qj ψj + ∂ψi ∂zα żα ) . Let us write these equations in the more compact form, according to the above-given notation: vν(q, z) = ∂irν ψ i, aν(q, z, ż) = ∂ijrν ψ iψj + ∂irν ( ψi j ψ j + ψi α ż α ) . (3.3) As we shall see, the additional parameters ż play a crucial role. Note that, at any fixed configuration q ∈ Q, the parameters z determine all kinematical states (q, v) compatible with Dynamical Equations of Non-Holonomic Systems 5 the constraints. Intuitively, the parameters ż determine a small displacement from a state (q, q̇) to another close state (q, q̇′) (with the same configuration q). For a precise definition, let us decompose the acceleration (3.3) into the sum aν = a0ν + aαν ż α, (3.4) where a0ν .= ∂ijrν ψ iψj + ∂irν ψ i j ψ j , aαν .= ∂irν ψ i α. (3.5) We observe that the vectors a0ν depend only on the state of the system. We say that Definition 3.1. The second vector in (3.4) wν .= aαν ż α, (3.6) represents the virtual displacements of the point Pν at the given kinematical state. Remark 3.1. Note that the parameters żα span all possible virtual displacements at a given state. The mechanical meaning of virtual displacement, which is strictly related to that of ideal constraint (Definition 3.2) would require a detailed discussion. We can skip it simply by accepting equation (3.6) as a mathematical definition, since it will be justified first, by the fact that with such a definition the Gauss principle becomes a consequence of the Newton dynamical equations (Theorem 4.1) and second, by the fact that the reactive forces of ideal constraints are not dissipative (Remark 3.3). It is customary to associate the intuitive idea of ‘virtual displacement’ with that of ‘virtual velocity’, as a limit of a ‘small’ displacement between two configurations’ of the system. Instead, within the present context, a ‘virtual displacement’ is a ‘small’ displacement between kinema- tical states (configurations plus velocities), so it is associated with the intuitive idea of ‘virtual acceleration’. This viewpoint is in fact coherent with the philosophy of the Gauss principle, which deals with accelerations. We assume for the dynamics of each point (Pν ,mν) the Newton equation mν aν = Aν + Rν , (3.7) where Aν is the active force (due to external fields and internal interactions) and Rν is the reactive force: it has the role of making the constraint fulfilled. Remark 3.2. The idea of ‘reactive force’ arises from the Newtonian philosophy, according which any action deviating a point from the uniform rectilinear motion (in an inertial reference frame) is a ‘force’, mathematically represented by a vector. Thus, the presence of a kinematical constraint must be represented by a vector, called ‘reactive force’, to be summed to the ‘active force’, which in turns represents the action of fields present in the space and independent from the constraints (gravitational, electromagnetical, centrifugal, Coriolis, etc.). Definition 3.2. Non-holonomic constraints are said to be ideal or perfect if∑ ν Rν ·wν = 0 (3.8) for all virtual displacements wν . 6 S. Benenti We consider equation (3.8) as a constitutive condition for the constraint: it says which kind of reactive forces the constraint is able to supply in order to be satisfied along any motion. It is straightforward to check that for linear constraints, or simply for holonomic constraints (which do not involve velocities), equation (3.8) reduces to the classical virtual work principle. The validity of such a constitutive condition is a matter of theoretical and experimental analysis of the behavior of the mechanical system one is dealing with. Theorem 3.1. Let Ri .= ∑ ν Rν · ∂rν ∂qi , (3.9) be the Lagrangian reactive forces. Then the definition (3.8) of ideal constraint is equivalent to the following equations, Ri ψ i α = 0, (3.10) Ri = λaC a i , (3.11) in the parametric and in the implicit representation, respectively. Equation (3.11) means that the components Ri are linear combinations of the functions Ca i (q, q̇). Proof. Put equation (3.6) and (3.5) in equation (3.8), 0 = ∑ ν Rν · aαν ż α = ∑ ν Rν · ∂irν ψ i α ż α = Ri ψ i α ż α. This proves equation (3.10). By differentiating the identity Ca(q, ψ(q, z)) = 0, we get the following relations between the two representations, Ca i ψ i α = 0, ∂iC a + Ca j ψ j i = 0, (3.12) where Ca i .= ∂Ca ∂q̇i , ψj i .= ∂iψ j , ∂i .= ∂ ∂qi . (3.13) Due to the first identity (3.12), equation (3.10) is then equivalent to (3.11) � Remark 3.3. Equations (3.13)1 and (3.11) show that the ideal constraints do not dissipate ener- gy: the power of the reactive forces is zero (we are dealing with time-independent constraints). See [3]. Remark 3.4. By a well-known process, we pass from the microscopical level to the ‘macrosco- pical ’ one i.e., to the Lagrange equations d dt ( ∂K ∂q̇i ) − ∂K ∂qi = Ai +Ri, (3.14) where K = 1 2 gij q̇ iq̇j , gij .= ∑ ν mν ∂irν · ∂jrν . (3.15) At the right hand side of the Lagrange equations we find the sum of the active Lagrangian forces Ai .= ∑ ν Aν · ∂rν ∂qi , (3.16) and the reactive Lagrangian forces (3.9). Dynamical Equations of Non-Holonomic Systems 7 4 The Gauss principle At the microscopical level we introduce the quantity G .= 1 2 ∑ ν mν ( aν − Aν mν )2 . (4.1) The active forces Aν are known functions of the state (q, q̇). Thus, due to the parametric equations of the constraints, G becomes a function of (q, z). Moreover, even for active forces depending on the velocities, Aν does not depend on ż, ∂Aν ∂żα = 0. Thus, due to equations (3.4) and (3.5), along any motion satisfying the constraints we have ∂G ∂żα = ∑ ν mν ( aν − Aν mν ) · ∂aν ∂żα = ∑ ν mν ( aν − Aν mν ) · aαν . (4.2) Furthermore, ∂2G ∂żα∂żβ = ∑ ν mν ∂aν ∂żβ · aαν = ∑ ν mν aβν · aαν = ∑ ν mν ψ i α ψ j β ∂irν · ∂jrν = gij ψ i α ψ j β. Then, if we introduce the functions Gαβ .= gij ψ i α ψ j β , (4.3) we get ∂2G ∂żα∂żβ = Gαβ . (4.4) Since the matrix [ψi α] has maximal rank, the symmetric matrix [Gαβ ] is regular and positive- definite as well as [gij ]. Theorem 4.1. Assume the Newton equations mν aν = Aν + Rν for each point Pν . Then, at any state of any actual motion the quantity G takes a minimal value (Gauss principle) if and only if the constraints are ideal. Proof. Write the Newton equations in the form mν ( aν − Aν mν ) = Rν . Then, due to equations (3.4), (3.5) and (3.9), ∂G ∂żα = ∑ ν Rν · ∂aν ∂żα = ∑ ν Rν · aαν = Ri ψ i α. (4.5) This shows that for ideal constraints, see equation (3.10), the Newton equations imply ∂G ∂żα = 0, (4.6) at any state along any actual motion. Due to equation (4.4), being [Gαβ ] positive, at the statio- nary states for which equation (4.6) holds, the function G has a strong minimum. (ii) Conversely, assume that the Gauss principle holds true. Then equation (4.6) is satisfied, so that from (4.5) we get Ri ψ i α = 0. This means that the constraint is ideal (Theorem 3.1). � 8 S. Benenti Remark 4.1. The vector Aν/mν is the acceleration of the point Pν in a free motion, free from the constraints. Let us denote it by af ν . As a consequence, the function G can be also defined as G .= 1 2 ∑ ν mν ( aν − af ν )2 , (4.7) and Theorem 4.1 can be reformulated as follows: Theorem 4.2. Let rν(t) and rf ν (t) be two motions of the system Pν such that for t = t0 the corresponding states coincide i.e., rν(t0) = rf ν (t0), vν(t0) = vf ν (t0). Assume that rf ν (t) is a free motion. Then, at this state, and for any motion compatible with ideal constraints, the actual accelerations aν(t0), are such that G takes a minimal value. Remark 4.2. For any arbitrary motion, vν = ∂rν ∂qi q̇i, aν = ∂rν ∂qi∂qj q̇iq̇j + ∂rν ∂qi q̈i. Then, at any fixed state compatible with the constraints we have aν − af ν = ∂rν ∂qi (q̈i − q̈i f ). Due to the definition (3.15) of gij , from (4.7) we get the so-called Lipschitz expression of the function G: G = 1 2 gij (q̈i − q̈i f )(q̈j − q̈j f ). (4.8) Note that in this expression the Christoffel symbols are not involved. 5 The Gibbs–Appell equations Let us go back to the definition (4.1) of the function G. If we introduce the functions S .= 1 2 ∑ ν mν a2 ν , S1 .= 1 2 ∑ ν 1 mν A2 ν , S2 .= ∑ ν Aν · aν , (5.1) then we have the decomposition G = S + S1 − S2. The function S is called the energy of the accelerations. We observe that ∂S1 ∂żα = 0 and that, due to the second equation (3.3) and the definition (3.16) of active Lagrangian force, ∂S2 ∂żα = ∑ ν Aν · ∂irν ψ i α = Ai ψ i α. Thus, ∂G ∂żα = ∂S ∂żα −Ai ψ i α. Due to the Gauss principle (Theorem 4.1), this proves Dynamical Equations of Non-Holonomic Systems 9 Theorem 5.1. The Gauss principle is equivalent to equations ∂S ∂żα = Aα, (5.2) where the function S(q, z, ż) .= 1 2 ∑ ν mν a2 ν (5.3) is determined by the expression (3.3) of the accelerations, and Aα .= Aiψ i α. (5.4) Remark 5.1. Equations (5.2) are the celebrated Gibbs–Appell equations. The equivalence between these equations and the Gauss principle is highlighted within the framework of the parametric representation (2.1) of the constraints. Remark 5.2. The quantities Aα can be computed by writing the the virtual power of the active forces: W .= ∑ ν Aν ·wν = ∑ ν Aν · ∂irν ψ i α ż α = Ai ψ i α ż α = Aα ż α. (5.5) 6 The explicit form of the Gibbs–Appell equations Both sides of the Gibbs–Appell equations (5.2) are functions of (q, z, ż). Let us solve them w.r.to the variables żα. To this end, it is crucial to observe that by using the second equations (3.3) we get for the function S (5.3) the expression S = 1 2 gij ψ i αψ j β ż αżβ + ∑ ν mν ∂ijrν · ∂krν ψ iψjψk α ż α + S0, where S0 is a function dependent on (q, z) only. Then, this function is not involved by the Gibbs–Appell equations and S can be replaced by S∗ = 1 2 gij ψ i αψ j β ż αżβ + ∑ ν mν ∂ijrν · ∂krν ψ iψjψk α ż α. (6.1) Now we show that this new function S∗ assumes a very interesting expression. Let us introduce the functions ξijk(q) .= ∑ ν mν ∂ijrν · ∂krν . Since ξijk = ∑ ν mν ∂i(∂jrν · ∂krν)− ∑ ν mν (∂jrν · ∂ikrν) = ∂igjk − ξikj , by a cyclic permutation of the indices we get ξijk + ξikj = ∂igjk, ξjki + ξjik = ∂jgki, ξkij + ξkji = ∂kgij . By summing the first two equations and subtracting the third one, since ξijk is symmetric in the first two indices, we get ξijk = Γijk, where Γijk .= 1 2 (∂igjk + ∂jgki − ∂kgij) 10 S. Benenti are the Christoffel symbols of the metric tensor gij (the coefficients of the Levi-Civita connec- tion). As a consequence, if we recall the definition (4.3) of Gαβ , the function S∗ (6.1) can be written as S∗ = 1 2 Gαβ ż αżβ + Γijk ψ iψjψk α ż α, and the Gibbs–Appell equations (5.2) assume the form Gαβ ż β + Γijk ψ iψjψk α = Aα. (6.2) Then we can prove Theorem 6.1. The Gibbs–Appell equations (5.2) are equivalent to equations żα = Gαβ(Aβ − Γijk ψ iψjψk β), (6.3) where [Gαβ ] the inverse matrix of [Gαβ ]. Proof. Indeed, as we remarked in Section 4, the matrix [Gαβ ] is regular. If we apply the inverse matrix [Gαβ ] to equations (6.2), then we get equations (6.3). � Equations (6.3) are the explicit form (or normal form) of the Gibbs–Appell equations (5.2). 7 The dynamical equations of the first kind By setting żα = dzα dt , equations (6.3) together with the constraint equations (2.1) build up a first-order differential system, in normal form, in the unknown functions qi(t) and zα(t): dqi dt = ψi(q, z), dzα dt = Gαβ (Aβ − Γijk ψ iψjψk β). (7.1) Hence, we can summarize the results so far obtained in the following Theorem 7.1. Let Q be the configuration n-manifold of a mechanical system, with local La- grangian coordinates q = (qi). Let TQ be the tangent bundle of Q with canonical coordinates (q, q̇) = (qi, q̇i). Let C ⊂ TQ be a submanifold representing kinematical time-independent con- straints. Assume that the constraint submanifold C is locally described by parametric equations q̇i = ψi(q, z) with m < n parameters z = (zα), such that the n×m matrix [ψi α] .= [ ∂ψi ∂zα ] has maximal rank m. If the constraints are ideal, then the actual motions are represented by functions qi = qi(t) satisfying the differential system (7.1)3, where Γijk are the Christoffel symbols of the metric tensor gij associated with the kinetic energy K = 1 2 gij q̇ iq̇j and Aα = Ai ψ i α, where Ai are the Lagrangian active forces. 3Any solution of the differential system (7.1) is of course a set of functions qi(t) and zα(t). But, after the integration, we can get rid of the functions zα(t). Dynamical Equations of Non-Holonomic Systems 11 Remark 7.1. The dynamical system (7.1) is the first-order system associated with the vector field Z on the constraint submanifold C, whose components (Zi, Zα) w.r.to the coordinates (q, z) are given by Zi = ψi(q, z), Zα = Gαβ (Aβ − Γijk ψ iψjψk β). (7.2) The Zα are the ‘vertical components’ of Z w.r.to the projection onto Q. The actual motions are the projections onto Q of the integral curves of Z. Remark 7.2. This theorem provides a first ‘recipe’ for writing the dynamical equations for non-holonomic systems with linear or non-linear ideal constraints: 1. Choose Lagrangian coordinates (qi), write the kinetic energy of the system K = 1 2 gij q̇ iq̇j , and extract the n× n matrix [gij ]. 2. Choose parametric equations q̇i = ψi(q, z) of the constraints, compute the m × n ma- trix [ψi α], and check its rank. If the constraints are initially expressed by implicit equations, then use (for instance) the method illustrated in Remark 7.3 below for finding parametric equations. 3. Compute the m×m matrix [Gαβ ] = [gijψ i αψ j β], and the inverse matrix [Gαβ ] = [Gαβ ]−1.4 4. Write the Lagrange equations of the free motions (i.e., with only active forces Ai) in the form gij q̈ j = Li(q, q̇) (7.3) (note that the formal expression of Li is Li = Ai(q, q̇)− Γhki q̇ h q̇k) and compute Z̄i(q, z) = Li(q, ψ) = Ai(q, ψ)− Γhki(q)ψhψk. (7.4) 5. Compute Zα = Z̄iψ i α and Zα = Gαβ Zβ. 6. Write the dynamical system dqi dt = ψi(q, z), dzα dt = Zα(q, z). (7.5) Remark 7.3. When the constraint submanifold C ⊂ TQ is described by a system of implicit independent equations of the kind Ca(q, q̇) = 0, a = 1, . . . , r, r = n−m, (7.6) then we have to transform these equations into parametric equations. The choice of the pa- rameters zα is completely free and it is only a matter of convenience, depending on the explicit concrete form of the dynamical equations (7.5) we get. Anyway, since the matrix [ψi α] does not have the maximal rank, equations (7.6) can be solved w.r.to m of the n Lagrangian velocities q̇i, say – up to a reordering – w.r.to q̇α, α = 1, . . . ,m. This process leads to considering as param- eters these Lagrangian velocities: zα = q̇α. It works very well for linear or affine constraints, where equations (7.6) have the form Ca i (q) q̇i − bi(q) = 0. Another possible choice of the parameters, for the linear constraints only, is that related to the use of quasi-velocities or quasi-coordinates – see for instance [9]. 4Note that according to this recipe we do not have to compute the inverse n× n-matrix of [gij ], but only the inverse of [Gαβ ], whose dimension is m < n. 12 S. Benenti Remark 7.4. For the analysis of the qualitative (or quantitative) behavior of a non-holonomic mechanical system (like stability, equilibrium states, small oscillations, numerical integration, etc.) we can apply to Z all the known theorems about dynamical systems. For instance, a f irst integral is a function F (q, z) such that ψi(q, z) ∂F ∂qi + Zα(q, z) ∂F ∂zα = 0. (7.7) Remark 7.5. The singular points of the dynamical system (7.1) are the solutions (q, z) of the simultaneous equations ψi(q, z) = 0, Zα(q, z) .= Z̄i ψ i α = 0. (7.8) In the case of a homogeneous quadratic constraint, ψi(q, z) = 1 2 ψ i αβ(q) zαzβ , equations (7.8) become ψi αβ(q) zαzβ = 0, Z̄i ψ i αβ z β = 0, (7.9) being ψi α = ψi αβ(q) zβ. This shows that, whatever q and Z̄i, singular points are given by zα = 0. But for zα = 0 the matrix [ψi α] does not have the maximal rank, since all its elements vanish. Hence, at these singular points the constraint C is not regular. Remark 7.6. The geometrical picture of the above results gives an intrinsic meaning of the objects we have introduced. Any vector v ∈ TC can be represented by a sum v = vi ∂i + vα ∂α, where ∂i = ∂/∂qi and ∂α = ∂/∂zα are interpreted as pointwise independent vector fields on TC: at each point x of C they span the tangent space TxC. The vectors ∂α are ‘vertical ’ i.e., they are tangent to the fibers Fq of C. Hence, we call vα the vertical components of v, while we call vi the basic components. For instance, the basic components of Z are Zi = ψi(q, z) and the vertical components are Zα. The functions ψi α have the role of transforming Latin components, labeled by Latin indices h, i, j, k, . . ., into Greek components, labeled by Greek indices α, β, . . .. For instance, when it is applied to a one-form (covariant vector) Z̄i dq i, we get a vertical one form Zα dz α, and when it is applied to the covariant metric tensor gij , we get a metric tensor Gαβ on the fibers of C, so that, by raising the indices of Zα by the contravariant metric Gαβ , we get a vertical vector field Zα ∂α. Note 7.1. I did not find equations (7.1) in the recent and old articles I have consulted. In fact, it is rather surprising that the simple idea of considering the parametric representation of the constraints as a part of the dynamical equations does not appear in the major textbooks and treatises on non-holonomic mechanics. Only recently this idea appeared in a paper of Massa and Pagani [8]. Their general approach, which is based on the jet-bundle theory and deals with time- dependent constraints, leads to the introduction of the vector field Z. The elementary approach presented here is of course quite different and leads, for instance, to different expressions of the vertical part of Z. Our equations (7.1) should be compared with equations (3.5b) and (3.15) of [8]. The difference is that the second equation (7.1) is written in terms of the Euclidean vectors Fν , vν , while equation (3.15) of [8] Z is written in terms of the Lagrange equations, but still in an implicit form. Dynamical Equations of Non-Holonomic Systems 13 Remark 7.7. For linear constraints, the approach presented here is more general than that of Čaplygin – see [9], Ch. III, § 3, where the coordinates qi are divided into two groups, say (qa, qα), with a = 1, . . . ,m and α = m + 1, . . . , n. The constraint equations are assumed to be of the form q̇α = ∑ a bαa q̇ a, where the coefficients bαa and the Lagrangian L are assumed to be independent from the coor- dinates (qa).5 8 The dynamical equations of the second kind About the method for writing the dynamical equations of a non-holonomic mechanical system so far illustrated two remarks are in order: • It lies on a parametric representation of the constraint C (however, the vector field Z does not depend on the chosen parametrization). • It does not give any information about the reactive forces. Here, we propose an alternative method for writing the dynamical equations which is based on any implicit representation of C by a system of independent equations Ca(q, q̇) = 0, and which provides a way for evaluating the reactive forces. The Lagrange equations (3.14) – see Remark 3.4 – are equivalent to the dynamical system Xλ =  dqi dt = q̇i, dq̇i dt = − Γi hk q̇ h q̇k +Ai +Ri λ, (8.1) on the tangent bundle TQ of the configuration manifold Q. Here, Ai = gijAj and Ri λ = gij λaC a j are the contravariant components of the Lagrangian active and reactive forces, respectively. The label λ points out that the reactive Lagrangian forces depend on the a priori unknown Lagrangian multipliers λ = (λa), according to Theorem 3.1 and equation (3.11). For a better understanding of what we are going to do, it is useful to consider the following Definition 8.1. We say that a vector V on TQ is vertical if it is tangent, at each point where it is defined, to the corresponding fiber of TQ. This is equivalent to say that it has the form V = V i ∂ ∂q̇i . (8.2) As a consequence, the vertical part of the vector Xλ is given by( Ai +Ri λ − Γi hk q̇ h q̇k ) ∂ ∂q̇i , 5In the history of the non-holonomic systems we can find the famous equations of Maggi, Volterra, Voronec and Čaplygin, dealing with linear constraints. The comparison of these equations with our approach is left to the reader, who can find a detailed discussion in in [9], Ch. 3. A neat illustration of the non-holonomic dynamical equations, with the essential classical and recent bibliography, can be found in the book [2]. 14 S. Benenti where the active and reactive forces are represented by the vertical vectors A = Ai ∂ ∂q̇i , Rλ = Ri λ ∂ ∂q̇i , Hence, the vector Xλ is decomposed into the sum Xλ .= XG + A + Rλ, (8.3) where XG represents the geodesic f low, XG =  dqi dt = q̇i, dq̇i dt = − Γi hk q̇ h q̇k. (8.4) We can consider Xλ = XG+A+Rλ as a family of vector fields, depending on the Lagrangian multipliers. However, it is a remarkable fact that we can obtain an explicit form of them, as functions of the kinematical states (q, q̇) only. Theorem 8.1. Let [Gab] be the symmetric matrix defined by Gab .= gij Ca i C b j . (8.5) Let [Gab] = [Gab]−1 be its inverse, and Cai .= gij Ca j . If the constraints are ideal, then the Lagrangian multipliers and the Lagrangian reactive forces are well determined functions of (q, q̇): λa(q, q̇) = Gab ( Cb i (Γi hk q̇ h q̇k −Ai)− q̇i ∂iC b ) , (8.6) Ri(q, q̇) = GabC ai ( Cb j (Γj hk q̇ h q̇k −Aj)− q̇j ∂jC b ) . (8.7) Proof. First of all, we observe that the matrix [Gab] is regular, since the vector fields Ca are independent. Hence, the inverse matrix [Gab] is well defined. In order to satisfy the constraints, the vector field Xλ must be tangent to the constraint submanifold C. This condition is expressed by equations 〈Xλ, dC a〉 = 0, (8.8) to be satisfied at least on C. In components, these equations read q̇i ∂iC a + (Ai − Γi hk q̇ h q̇k +Ri)Ca i = 0, i.e., RiCa i = Ca i (Γi hk q̇ h q̇k −Ai)− q̇i ∂iC a. Note that the right hand side Λa(q, q̇) .= Ca i (Γi hk q̇ h q̇k −Ai)− q̇i ∂iC a (8.9) is a known function of (q, q̇). Then equation RiCa i = Λa assumes the form λbC a i C b j g ij = Λa, i.e., λbG ab = Λa. By applying the inverse matrix [Gab] we get equation (8.6) and equation (8.7). � The explicit form (8.7) of the reactive forces allows us to state Dynamical Equations of Non-Holonomic Systems 15 Theorem 8.2. The actual motions of a mechanical system with regular and ideal non-holonomic constraints represented by a submanifold C ⊂ TQ are the integral curves based on C of the vector field D .= XG + A + R, (8.10) where the components Ri of the vertical vector R are defined by (8.7). If we introduce the symbols πij .= GabC aiCbj , Cai .= gij Ca j , Ci a .= GabC bi, (8.11) then the explicit expressions of the first-order differential system associated with D and of the reactive forces are D =  dqi dt = q̇i, dq̇i dt = (gij − πij) (Aj − Γhkj q̇ h q̇k)− q̇j ∂jC aCi a, (8.12) and Ri = πij (Γhkj q̇ h q̇k −Aj)− q̇j ∂jC aCi a, (8.13) respectively. Remark 8.1. This last theorem provides a second ‘recipe’ for writing the dynamical equations for non-holonomic systems with linear or non-linear ideal constraints: 1. Choose Lagrangian coordinates (qi), write the kinetic energy of the system K = 1 2 gij q̇ iq̇j , extract the n× n matrix [gij ], and compute the inverse matrix [gij ]. 2. Take the constraint equations Ca(q, q̇) = 0 and compute, in the order, the following ma- trices: [Ca i ] .= [ ∂Ca ∂q̇i ] (seek the singular states), [Cai] .= [gij Ca j ], [Gab] = [Gba] .= [gij Ca i C b j ] = [CaiCb i ], [Gab] .= [Gab]−1, [Ci a] .= [GabC bi], [πij ] = [πji] .= [CaiCj a], [gij − πij ], [∂iC a] .= [ ∂Ca ∂qi ] , [∂jC aCi a]. 3. Write the Lagrange equations for the free motions in the form gij q̈ j = Ai − Γhki q̇ h q̇k, and keep in evidence the functions Li(q, q̇) .= Ai − Γhki q̇ h q̇k. 16 S. Benenti 4. Compute the vector Di .= (gij − πij)Lj − q̇j ∂jC aCi a. 5. Write the differential system (8.12), D =  dqi dt = q̇i, dq̇i dt = Di. 6. Its solutions qi(t), q̇i(t), with initial conditions belonging to C, describe the actual motions of the system. 7. Evaluate the reactive forces along any actual motion by means of equation (8.13), Ri = − πij Lj − q̇j ∂jC aCi a. (8.14) Remark 8.2. The case of single constraint equation C(q, q̇) = 0. In this case the above- given recipe can be applied by setting a = b = 1. Items 1 and 2 of the general recipe still hold. However, since some of the above matrices reduces to scalar functions or to vectors, the index 1 can be omitted or replaced by ∗: [Ci] .= [ ∂C ∂q̇i ] , [Ci] .= [gij Cj ], G .= [gij CiCj ] = CiCi, Ci ∗ .= G−1Ci, [πij ] .= [CiCj ∗ ] = G−1 [CiCj ], [gij − πij ], ∂iC .= ∂C ∂qi , [∂jC C i ∗]. Then, follows items 3–7 of the general recipe. 9 Illustrative examples As shown above, for writing the dynamical equations of a non-holonomic system we can apply two methods: the first method is established by Theorem 7.1 and the corresponding recipe is il- lustrated in Remark 7.2; the second method is established by Theorem 8.2 and the corresponding recipe is illustrated in Remark 8.1. Let us see how these two methods work by concrete examples. We begin with two paradig- matic and simple examples of linear non-holonomic constraints, the ‘skate’ and the ‘vertical rolling disc’. Then, we shall consider two more demanding examples: ‘two co-axial rolling discs’ and ‘two points with parallel velocities’. This last one is a genuine non-linear non-holonomic system. In illustrating examples of application of a theory it is not customary, in general, to provide detailed calculations – which usually are left to the reader. Here, however, it is worthwhile to disregard such a custom in order to compare the effectiveness of the two methods (mainly the length of the calculations) applied to a same mechanical system. Dynamical Equations of Non-Holonomic Systems 17 9.1 The skate This mechanical system is made of a homogeneous rod (material segment) sliding without friction on a plane6. The configurations of the skate are determined by the Cartesian coordinates (x, y) of the center of mass (i.e., of the segment) G and by the angle θ of the rod w.r.to the x- axis. The configuration manifold Q is R × S1 and natural ordered Lagrangian coordinates are (q1, q2, q3) = (x, y, θ). The velocity vG = [ẋ, ẏ] of the mass-center is constrained to be parallel to the rod. This constraint is then represented by a single equation: ẋ sin θ − ẏ cos θ = 0. (9.1) The kinetic energy is given by K = 1 2 m (ẋ2 + ẏ2) + 1 2 I θ̇, where m and I are the mass and the moment of inertia w.r.to G (i.e., w.r.to the line orthogonal to the plane through G), respectively. (i) First method. Since dim(TQ) = 6 and dim(C) = 3, we need two parameters (z1, z2) for describing C. We can consider the parametric equations ẋ = z1 cos θ, ẏ = z1 sin θ, θ̇ = z2. Then, we compute the necessary matrices and vectors: [gij ] =  m 0 0 0 m 0 0 0 I  , [ψi] = [z1 cos θ , z1 sin θ , z2], [ψi α] = [ cos θ sin θ 0 0 0 1 ] . This matrix has maximal rank everywhere: the constraint is regular. [Gαβ ] = [ m 0 0 I ] , [Gαβ ] = [ 1 m 0 0 1 I ] . Since,[ ∂K ∂q̇i ] = [mẋ , m ẏ , I θ̇], [ ∂K ∂qi ] = [0 , 0 , 0], the Lagrange equations for the free motions gij q̈ j = Ai − Γhki q̇ h q̇k read mẍ = A1, m ÿ = A2, I θ̈ = A3. Hence, Li = Ai(q, q̇), Z̄i = Ai(q, ψ) and [Zα] = [ψi α Z̄i] = [ A1 cos θ +A2 sin θ A3 ] . The dynamical equations are dx dt = z1 cos θ, dy dt = z1 sin θ, dθ dt = z2, dz1 dt = 1 m (A1 cos θ +A2 sin θ), dz2 dt = A3 I . 6Quite similar classical examples are that of two material points linked by a massless rigid segment [5], p. 23 and 63, and the Čaplygin sleigh [9], Ch. III, § 3, Examples 2 & 5, and Ch. V, § 4. Another example of this kind is examined in [4], § 4.1 & § 4.2. 18 S. Benenti (ii) Second method, for a single constraint equation – Remark 8.2: C .= ẋ sin θ − ẏ cos θ = 0. [gij ] =  1 m 0 0 0 1 m 0 0 0 1 I  , [Ci] .= [ ∂C ∂q̇i ] = [sin θ , − cos θ , 0] , [Ci] .= [gij Cj ] = [ sin θ m , − cos θ m , 0 ] , G .= CiCi = 1 m , [Ci ∗] .= G−1[Ci] = [sin θ , − cos θ , 0] = [Ci], [πij ] .= G−1[CiCj ] = 1 m  sin2 θ − sin θ cos θ 0 − sin θ cos θ cos2 θ 0 0 0 0  , [gij − πij ] = 1 m  cos2 θ sin θ cos θ 0 sin θ cos θ sin2 θ 0 0 0 m I  , [∂iC] .= [ ∂C ∂qi ] = [0 , 0 , ẋ cos θ + ẏ sin θ] , [q̇j ∂jC C i ∗] = θ̇ (ẋ cos θ + ẏ sin θ) [Ci ∗]. Since [ ∂K ∂q̇i ] = [mẋ , m ẏ , I θ̇], [ ∂K ∂qi ] = [0 , 0 , 0], the Lagrange equations read mẍ = A1, m ÿ = A2, I θ̈ = A3. Hence, Li = Ai. We have all the ingredients for computing the vector Di .= (gij − πij)Lj − q̇j ∂jC C i ∗: D1 = cos θ m (A1 cos θ +A2 sin θ)− θ̇ (ẋ cos θ + ẏ sin θ) sin θ, D2 = sin θ m (A1 cos θ +A2 sin θ) + θ̇ (ẋ cos θ + ẏ sin θ) cos θ, D3 = 1 I A3, and the differential system (8.12) reads dx dt = ẋ, dy dt = ẏ, dθ dt = θ̇, dẋ dt = cos θ m (A1 cos θ +A2 sin θ)− θ̇ (ẋ cos θ + ẏ sin θ) sin θ, dẏ dt = sin θ m (A1 cos θ +A2 sin θ) + θ̇ (ẋ cos θ + ẏ sin θ) cos θ, dθ̇ dt = I−1A3. Dynamical Equations of Non-Holonomic Systems 19 9.2 The vertical rolling disc A material disc of radius R running on a plane is kept perpendicular to it by massless and frictionless devices. The configuration manifold is Q4 = R2 × S1 × S1, with coordinates (q1, q2, q3, q4) = (x, y, θ, ψ), where (x, y) are Cartesian coordinates of the center P of the disc (i.e., of the point C in contact with the plane), θ a rotation angle of the disk around its axis, and ψ an angle giving the orientation of the axis (see Figure 4, with θ = θ1). Constraint: the disc rolls on the plane without sliding. Let (i, j,k) be the unitary vectors associated with the (x, y, z)-axes. The unitary vector k is associated with the oriented angle ψ. Let u be the unitary vector associated with the oriented angle θ. Then, u = cosψ j − sinψ i. The angular velocity ω is given by ω = θ̇ u + ψ̇ k. The velocity vC of the point C is given by vC = vP + ω × PC, where PC = −Rk. Hence, vC = ẋ i + ẏ j− (θ̇ u + ψ̇ k)×Rk = ẋ i + ẏ j−R θ̇ u× k. Since u× k = cosψ j× k− sinψ i× k, we get vC = ẋ i + ẏ j−R θ̇ (cosψ j× k− sinψ i× k) = ẋ i + ẏ j−R θ̇ (cosψ i + sinψ j) = (ẋ−R θ̇ cosψ) i + (ẏ −R θ̇ sinψ) j. The kinematical constraint vC = 0 is then represented by the following two linear equations C1 .= ẋ−R cosψ θ̇ = 0, C2 .= ẏ −R sinψ θ̇ = 0. (9.2) (i) First method. Assume that the center of mass of the disc coincides with its geometrical center. Then the kinetic energy is given by K = 1 2 m(ẋ2 + ẏ2) + 1 2 (Aθ̇2 +Bψ̇2), (9.3) where m is the mass, A and B are the moments of inertia w.r.to the axis of rotation and a diameter, respectively. Thus, [gij ] =  m 0 0 0 0 m 0 0 0 0 A 0 0 0 0 B  . From the constraint equations (9.2) we get the parametric equations ẋ = R cosψ z1, ẏ = R sinψ z1, θ̇ = z1, ψ̇ = z2. (9.4) Thus, [ψi] = [R cosψ z1 , R sinψ z1 , z1 , z2], [ψi α] = [ R cosψ R sinψ 1 0 0 0 0 1 ] (α = 1, 2, index of line). This matrix has maximal rank, thus the constraint is regular. It follows that Gαβ = gijψ i αψ j β = mψ1 αψ 1 β +mψ2 αψ 2 β +Aψ3 αψ 3 β +B ψ4 αψ 4 β , G11 = mψ1 1ψ 1 1 +mψ2 1ψ 2 1 +Aψ3 1ψ 3 1 +B ψ4 1ψ 4 1 = mR2 cos2 ψ +mR2 sin2 ψ +A = mR2+A, 20 S. Benenti G12 = mψ1 1ψ 1 2 +mψ2 1ψ 2 2 +Aψ3 1ψ 3 2 +B ψ4 1ψ 4 2 = 0, G22 = mψ1 2ψ 1 2 +mψ2 2ψ 2 2 +Aψ3 2ψ 3 2 +B ψ4 2ψ 4 2 = B, [Gαβ ] = [ mR2 +A 0 0 B ] , [Gαβ ] =  1 mR2 +A 0 0 1 B  . Since,[ ∂K ∂q̇i ] = [mẋ , m ẏ , A θ̇ , B ψ̇], [ ∂K ∂qi ] = [0 , 0 , 0 , 0], the Lagrange equations for the free motions gij q̈ j = Ai − Γhki q̇ h q̇k read mẍ = A1, m ÿ = A2, A θ̈ = A3, B ψ̈ = A4. This shows that Li = Ai. Hence, Z̄i = Ai(q, ψ), and [Zα] = [Z̄i ψ i α] = [ A1R cosψ +A2R sinψ +A3 A4 ] , [Zα] = [Gαβ Zβ ] =  A1R cosψ +A2R sinψ +A3 mR2 +A A4 B  . Thus, the dynamical equations are dx dt = R cosψ z1, dy dt = R sinψ z1, dθ dt = z1, dψ dt = z2, dz1 dt = (mR2 +A)−1 (A1R cosψ +A2R sinψ +A3), dz2 dt = B−1A4. (9.5) (ii) Second method. Recall the constraint equations (9.2). Then, [Ca i ] .= [ ∂Ca ∂q̇i ] = [ 1 0 −R cosψ 0 0 1 −R sinψ 0 ] , [Cai] .= [ gij Ca j ] = [ 1 m 0 − R A cosψ 0 0 1 m − R A sinψ 0 ] , [Gab] .= [ CaiCb i ] =  1 m + R2 A cos2 ψ R2 A sinψ cosψ R2 A sinψ cosψ 1 m + R2 A sin2 ψ  , G .= det[Gab] = 1 m2 + R2 mA , G−1 = m2A mR2 +A , [Gab] = G−1  1 m + R2 A sin2 ψ − R2 A sinψ cosψ − R2 A sinψ cosψ 1 m + R2 A cos2 ψ  , Dynamical Equations of Non-Holonomic Systems 21 [Ci a] = [GabC bi] = G−1 [ 1 m ( 1 m + R2 A sin2 ψ) − R2 mA sinψ cosψ ∗C3 1 0 − R2 mA sinψ cosψ 1 m ( 1 m + R2 A cos2 ψ) ∗C3 2 0 ] , where ∗C3 1 = ( 1 m + R2 A sin2 ψ ) ( −R A cosψ ) + ( −R 2 A sinψ cosψ ) ( −R A sinψ ) = − R mA cosψ − R3 A2 sin2 ψ cosψ + R3 A2 sin2 ψ cosψ = − R mA cosψ, ∗C3 2 = ( −R 2 A sinψ cosψ ) ( −R A cosψ ) + ( 1 m + R2 A cos2 ψ ) ( −R A sinψ ) = ( −R 2 A sinψ cosψ ) ( −R A cosψ ) + ( 1 m + R2 A cos2 ψ ) ( −R A sinψ ) = − R mA sinψ. Hence, [Ci a] = G−1  1 m ( 1 m + R2 A sin2 ψ) − R2 mA sinψ cosψ − R mA cosψ 0 − R2 mA sinψ cosψ 1 m ( 1 m + R2 A cos2 ψ) − R mA sinψ 0  . Let us compute πij .= CaiCj a = πji. Let us set ∗πij .= Gπij . Then, [πij ] = G−1  1 m2 ( 1 m + R2 A sin2 ψ) − R2 m2A sinψ cosψ ∗π14 0 − R2 m2A sinψ cosψ 1 m2 ( 1 m + R2 A cos2 ψ) ∗π24 0 ∗π31 ∗π32 ∗π33 0 ∗π41 ∗π42 0 0  , where ∗π13 = 1 m ( 1 m + R2 A sin2 ψ ) ( −R A cosψ ) + ( − R2 mA sinψ cosψ ) ( −R A sinψ ) = 1 m ( 1 m + R2 A sin2 ψ ) ( −R A cosψ ) + R3 mA2 sin2 ψ cosψ = − R m2A cosψ, ∗π23 = ( − R2 mA sinψ cosψ ) ( −R A cosψ ) + 1 m ( 1 m + R2 A cos2 ψ ) ( −R A sinψ ) = − R m2A sinψ. It follows that [πij ] = G−1  1 m2 ( 1 m + R2 A sin2 ψ) − R2 m2A sinψ cosψ − R m2A cosψ 0 − R2 m2A sinψ cosψ 1 m2 ( 1 m + R2 A cos2 ψ) − R m2A sinψ 0 − R m2A cosψ − R m2A sinψ R2 mA2 0 0 0 0 0  , 22 S. Benenti [gij − πij ] = G−1  G m − 1 m2 ( 1 m + R2 A sin2 ψ) R2 m2A sinψ cosψ R m2A cosψ 0 R2 m2A sinψ cosψ G m − 1 m2 ( 1 m + R2 A cos2 ψ) R m2A sinψ 0 R m2A cosψ R m2A sinψ G A − R2 mA2 0 0 0 0 G B  = m2A mR2 +A  G m − 1 m2 ( 1 m + R2 A sin2 ψ) R2 m2A sinψ cosψ R m2A cosψ 0 R2 m2A sinψ cosψ G m − 1 m2 ( 1 m + R2 A cos2 ψ) R m2A sinψ 0 R m2A cosψ R m2A sinψ G A − R2 mA2 0 0 0 0 G B  = 1 mR2 +A  R2 cos2 ψ R2 sinψ cosψ R cosψ 0 R2 sinψ cosψ R2 sin2 ψ R sinψ 0 R cosψ R sinψ 1 0 0 0 0 mR2+A B  . Recall once more equations (9.2). Then, [∂iC a] .= [ ∂Ca ∂qi ] 0 0 0 R sinψ θ̇ 0 0 0 −R cosψ θ̇  , [q̇i ∂iC a] =  R sinψ θ̇ ψ̇ −R cosψ θ̇ ψ̇  . Let us set Xi .= q̇j ∂jC aCi a. Then, GX1 = (R sinψ θ̇ ψ̇) ( 1 m ( 1 m + R2 A sin2 ψ )) + (−R cosψ θ̇ ψ̇) ( − R2 mA sinψ cosψ ) = R m2 sinψ θ̇ ψ̇ + R3 mA sinψ θ̇ ψ̇ = R m2 sinψ θ̇ ψ̇ ( 1 + mR2 A ) = R (mR2 +A) m2A sinψ θ̇ ψ̇, GX2 = (R sinψ θ̇ ψ̇) ( − R2 mA sinψ cosψ ) + (−R cosψ θ̇ ψ̇) ( 1 m ( 1 m + R2 A cos2 ψ )) = − R m2 cosψ θ̇ ψ̇ − R3 mA cosψ θ̇ ψ̇ = −R (mR2 +A) m2A cosψ θ̇ ψ̇, GX3 = (R sinψ θ̇ ψ̇) ( − R mA cosψ ) + (−R cosψ θ̇ ψ̇) ( − R mA sinψ ) = 0, GX4 = (R sinψ θ̇ ψ̇)(0) + (−R cosψ θ̇ ψ̇)(0) = 0. Since G−1 = m2A mR2+A , we get [Xi] = [q̇j ∂jC aCi a] = G−1 [ R (mR2 +A) m2A sinψ θ̇ ψ̇ , −R (mR2 +A) m2A cosψ θ̇ ψ̇ , 0 , 0 ] = R θ̇ ψ̇ [sinψ , − cosψ , 0 , 0] . Dynamical Equations of Non-Holonomic Systems 23 Now we are able to compute the components Di .= (gij − πij)Lj − q̇j ∂jC aCi a of the vector D: D1 = 1 mR2 +A ( A1R 2 cos2 ψ +A2R 2 sinψ cosψ +A3R cosψ ) −R θ̇ ψ̇ sinψ = R cosψ mR2 +A (A1R cosψ +A2R sinψ +A3)−R θ̇ ψ̇ sinψ, D2 = 1 mR2 +A ( A1R 2 sinψ cosψ +A2R 2 sin2 ψ +A3R sinψ ) +R θ̇ ψ̇ cosψ = R sinψ mR2 +A (A1R cosψ +A2R sinψ +A3) +R θ̇ ψ̇ cosψ, D3 = 1 mR2 +A (A1R cosψ +A2R sinψ +A3) = 1 mR2 +A (A1R cosψ +A2R sinψ +A3) , D4 = A4 B . The resulting dynamical system (8.12) is dx dt = ẋ, dy dt = ẏ, dθ dt = θ̇, dψ dt = ψ̇, dẋ dt = R cosψ mR2 +A (A1R cosψ +A2R sinψ +A3)−R θ̇ ψ̇ sinψ, dẏ dt = R sinψ mR2 +A (A1R cosψ +A2R sinψ +A3) +R θ̇ ψ̇ cosψ, dθ̇ dt = 1 mR2 +A (A1R cosψ +A2R sinψ +A3) , dψ̇ dt = A4 B . (9.6) By introducing the new variables X .= mR2 +A R x, Y .= mR2 +A R y, Θ .= (mR2 +A) θ, it assumes the more compact form dX dt = Ẋ, dY dt = Ẏ , dΘ dt = Θ̇, dψ dt = ψ̇, dẊ dt = cosψ (A1R cosψ +A2R sinψ +A3)− Θ̇ ψ̇ sinψ, dẎ dt = sinψ (A1R cosψ +A2R sinψ +A3) + Θ̇ ψ̇ cosψ, dΘ̇ dt = (A1R cosψ +A2R sinψ +A3) , dψ̇ dt = A4 B . (9.7) Note that in these new variables, by considering also Z1 .= (mR2 +A) z1, the differential system (9.5) obtained by the first method reads dX dt = cosψ Z1, dY dt = sinψ Z1, dΘ dt = Z1, dψ dt = z2, dZ1 dt = A1R cosψ +A2R sinψ +A3, dz2 dt = B−1A4. (9.8) 24 S. Benenti The two systems (9.7) and (9.8) are in perfect agreement. The above detailed calculations show that for the rolling disc the first method is much shorter than the second one. 9.3 Two co-axial rolling discs Two identical material discs of radius R running on a plain are joined by a massless common axis, along with they can slide without friction. The configuration manifold is Q6 = R2 × S1 × S1 × S1 × R, with Lagrangian coordinates (q1, q2, q3, q4, q5, q6) = (x, y, θ1, θ2, ψ, a), where (x, y) are Cartesian coordinates of the center P1 of one of the two discs, θ1 and θ2 are the angles of rotations around the common axis, ψ is the angle giving the orientation of the axis, and a is the distance between the centers (see Fig. 4). Figure 4. Co-axial rolling discs. Constraint: the discs roll on the plane without sliding. For each disc this constraint is represented by linear equations of the kind (9.2), ẋ1 −R cosψ θ̇1 = 0, ẏ1 −R sinψ θ̇1 = 0, ẋ2 −R cosψ θ̇2 = 0, ẏ2 −R sinψ θ̇2 = 0. (9.9) However, the coordinates of the two centers are related by equations x2 = x1 + a sinψ, y2 = y1 − a cosψ. By differentiating these equations we get the link between the velocities, ẋ2 = ẋ1 + a cosψ ψ̇ + sinψ ȧ, ẏ2 = ẏ1 + a sinψ ψ̇ − cosψ ȧ. (9.10) By inserting these relations into equations (9.9), with x1 = x and y1 = y, we get the final constraint equations ẋ−R cosψ θ̇1 = 0, ẏ −R sinψ θ̇1 = 0, ẋ+ a cosψ ψ̇ + sinψ ȧ−R cosψ θ̇2 = 0, ẏ + a sinψ ψ̇ − cosψ ȧ−R sinψ θ̇2 = 0. (9.11) Since for a single rolling disc the first method is faster, we limit ourselves to apply the first method to the case of two discs. Dynamical Equations of Non-Holonomic Systems 25 Equations (9.11) show that the constraint submanifold C ⊂ TQ6 has dimension 12− 4 = 8. Hence, we need two parameters zα for the parametric representation. Let us choose z1 = θ̇1 and z2 = θ̇2. This means to solve the linear system (9.11) w.r.to (ẋ, ẏ, ψ̇, ȧ). The result is ẋ = R cosψ z1, ẏ = R sinψ z1, ψ̇ = R a (z2 − z1), ȧ = 0, θ̇1 = z1, θ̇2 = z2. Equation ȧ = 0, a = constant, exhibits the intuitive fact that, under the pure-rolling con- dition, the distance a between the two discs remains constant. Hence, we can reduce the configuration manifold Q6 to Q5 = R2 × S1 × S1 × S1, with coordinates (q1, q2, q3, q4, q5) = (x, y, θ1, θ2, ψ). The last equations reduce to ẋ = R cosψ z1, ẏ = R sinψ z1, ψ̇ = R a (z2 − z1), θ̇1 = z1, θ̇2 = z2, (9.12) with a = constant. Then, [ψi] = [ R cosψ z1 , R sinψ z1 , R a (z2 − z1) , z1 , z2 ] and [ψi α] .= [ ∂ψ ∂zα ] = [ R cosψ R sinψ − R a 1 0 0 0 R a 0 1 ] . (9.13) This matrix has maximal rank, thus the constraint submanifold C is regular. The kinetic energy of the system is the sum of the kinetic energies of the two discs. According to equations (9.3) and (9.10), K = 1 2 m(ẋ2 1 + ẏ2 1) + 1 2 (Aθ̇2 1 +Bψ̇2) + 1 2 m(ẋ2 2 + ẏ2 2) + 1 2 (Aθ̇2 2 +Bψ̇2) = 1 2 m(ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ2 2) + 1 2 (Aθ̇2 1 +Bψ̇2) + 1 2 (Aθ̇2 2 +Bψ̇2) = 1 2 m ( ẋ2 1 + ẏ2 1 + (ẋ1 + a cosψ ψ̇)2 + (ẏ1 + a sinψ ψ̇)2 ) + A 2 (θ̇2 1 + θ̇2 2) +Bψ̇2 = 1 2 m ( ẋ2 1 + ẏ2 1 + ẋ2 1 + a2 cos2 ψ ψ̇2 + 2a ẋ1 cosψ ψ̇ + ẏ2 1 + a2 sin2 ψ ψ̇2 + 2a ẏ1 sinψ ψ̇ ) + A 2 (θ̇2 1 + θ̇2 2) +Bψ̇2 = 1 2 m ( 2ẋ2 1 + 2ẏ2 1 + a2 ψ̇2 + 2a ψ̇ (ẋ1 cosψ + ẏ1 sinψ) ) + A 2 (θ̇2 1 + θ̇2 2) +Bψ̇2 = m (ẋ2 + ẏ2) + A 2 (θ̇2 1 + θ̇2 2) + (1 2 ma2 +B) ψ̇2 +ma ψ̇ (ẋ cosψ + ẏ sinψ). Thus, [gij ] =  2m 0 0 0 ma 2 cosψ 0 2m 0 0 ma 2 sinψ 0 0 A 0 0 0 0 0 A 0 ma 2 cosψ ma 2 sinψ 0 0 ma2 + 2B  . 26 S. Benenti Moreover, Gαβ = gij ψ i α ψ j β = g11 ψ 1 α ψ 1 β + g22 ψ 2 α ψ 2 β + g33 ψ 3 α ψ 3 β + g44 ψ 4 α ψ 4 β + g55 ψ 5 α ψ 5 β + 2g12 ψ 1 α ψ 2 β + 2g13 ψ 1 α ψ 3 β + 2g14 ψ1 α ψ 4 β + 2g15 ψ1 α ψ 5 β + 2g23 ψ2 α ψ 3 β + 2g24 ψ2 α ψ 4 β + 2g25 ψ 2 α ψ 5 β + 2g34 ψ 3 α ψ 4 β + 2g35 ψ3 α ψ 5 β + 2g45 ψ4 α ψ 5 β = 2mψ1 α ψ 1 β + 2mψ2 α ψ 2 β +Aψ3 α ψ 3 β +Aψ4 α ψ 4 β + (ma2 + 2B)ψ5 α ψ 5 β +ma cosψ ψ1 α ψ 5 β +ma sinψ ψ2 α ψ 5 β , G11 = 2mψ1 1 ψ 1 1 + 2mψ2 1 ψ 2 1 +Aψ3 1 ψ 3 1 +Aψ4 1 ψ 4 1 + (ma2 + 2B)ψ5 1 ψ 5 1 +ma cosψ ψ1 1 ψ 5 1 +ma sinψ ψ2 1 ψ 5 1 = 2mR2 cos2 ψ + 2mR2 sin2 ψ +A R2 a2 +A = 2mR2 +A ( 1 + R2 a2 ) , G22 = 2mψ1 2 ψ 1 2 + 2mψ2 2 ψ 2 2 +Aψ3 2 ψ 3 2 +Aψ4 2 ψ 4 2 + (ma2 + 2B)ψ5 2 ψ 5 2 +ma cosψ ψ1 2 ψ 5 2 +ma sinψ ψ2 2 ψ 5 2 = A R2 a2 +ma2 + 2B, G12 = 2mψ1 1 ψ 1 2 + 2mψ2 1 ψ 2 2 +Aψ3 1 ψ 3 2 +Aψ4 1 ψ 4 2 + (ma2 + 2B)ψ5 1 ψ 5 2 +ma cosψ ψ1 1 ψ 5 2 +ma sinψ ψ2 1 ψ 5 2 = −A R2 a2 +ma cosψR cosψ +ma sinψR sinψ = maR−A R2 a2 , and we obtain [Gαβ ] =  2mR2 +A ( 1 + R2 a2 ) maR−AR2 a2 maR−AR2 a2 A R2 a2 +ma2 + 2B  . It follows that G .= det[Gαβ ] = [ 2mR2 +A ( 1 + R2 a2 )] [ A R2 a2 +ma2 + 2B ] − [ maR−A R2 a2 ]2 = 2mA R4 a2 + 2m2R2a2 + 4mR2B +A2 R 2 a2 +ma2A+ 2AB +A2R 4 a4 +ma2A R2 a2 + 2AB R2 a2 −m2a2R2 −A2R 4 a4 + 2mA R3 a = R2 a2 ( 2mAR2 +A2 +ma2A+ 2AB + 2mAaR−m2a4 + 2m2a4 + 4ma2B ) +A (ma2 + 2B) = R2 a2 ( 2mAR2 +A2 +ma2A+ 2AB + 2mAaR+m2a4 + 4ma2B ) +A (ma2 + 2B) = R2 a2 ( m (2AR2 + a2A+ 2AaR+ 4a2B +ma4) +A2 + 2AB ) +A (ma2 + 2B). We observe that the determinant G is a constant. The inverse matrix is [Gαβ ] = G−1  A R2 a2 +ma2 + 2B AR2 a2 −maR AR2 a2 −maR 2mR2 +A ( 1 + R2 a2 )  . From the expression of the kinetic energy, K = m (ẋ2 + ẏ2) + A 2 (θ̇2 1 + θ̇2 2) + (1 2 ma2 +B) ψ̇2 +ma ψ̇ (ẋ cosψ + ẏ sinψ), Dynamical Equations of Non-Holonomic Systems 27 we obtain [ ∂K ∂q̇i ] =  2mẋ+ma cosψ ψ̇ 2mẏ +ma sinψ ψ̇ Aθ̇1 Aθ̇2 (ma2 + 2B)ψ̇ +ma(ẋ cosψ + ẏ sinψ)  and [ ∂K ∂qi ] =  0 0 0 0 maψ̇ (ẏ cosψ − ẋ sinψ)  . The Lagrange equations for the free motions are 2mẍ+ma cosψ ψ̈ −ma sinψ ψ̇2 = A1, 2mÿ +ma sinψ ψ̈ +ma cosψ ψ̇2 = A2, Aθ̈1 = A3, Aθ̈2 = A4, (ma2 + 2B)ψ̈ +ma(ẍ cosψ + ÿ sinψ)−ma(ẋ sinψ − ẏ cosψ)ψ̇ = maψ̇(ẏ cosψ − ẋ sinψ) +A5. They show that L1 = A1 +ma sinψ ψ̇2, L2 = A2 −ma cosψ ψ̇2, L3 = A3, L4 = A4, L5 = A5 +maψ̇ (ẏ cosψ − ẋ sinψ) +ma(ẋ sinψ − ẏ cosψ) ψ̇ = A5. Thus, due to the parametric equations (9.12), Z̄1 = A1 + mR2 a (z2 − z1)2 sinψ, Z̄2 = A2 − mR2 a (z2 − z1)2 cosψ, Z̄3 = A3, Z̄4 = A4, Z̄5 = A5. Let us compute Zα = Z̄i ψ i α – recall (9.13): Z1 = (A1 + mR2 a (z2 − z1)2 sinψ)R cosψ + (A2 − mR2 a (z2 − z1)2 cosψ)R sinψ − R a A3 +A4 = A1R cosψ +A2R sinψ − R a A3 +A4 = R (A1 cosψ +A2 sinψ − 1 a A3) +A4, Z2 = R a A3 +A5. 28 S. Benenti Thus, the dynamical equations (7.5) associated with the vector field Z are dx dt = R cosψ z1, dx dt = R sinψ z1, dψ dt = R a (z2 − z1), dθ1 dt = z1, dθ2 dt = z2, dz1 dt = R (A1 cosψ +A2 sinψ − 1 a A3) +A4, dz2 dt = R a A3 +A5. 9.4 Two points with parallel velocities Two material points P1 = (x1, y1) and P2 = (x2, y2) running on the Cartesian plane R2 = (x, y) are constrained to have parallel vector-velocities v1 and v2. This is an example of non-linear non-holonomic constraint, since it is expressed by the quadratic homogeneous equation C = ẋ1 ẏ2 − ẋ2 ẏ1 = 0. (9.14) The configuration manifold is Q4 = R4 with ordered Lagrangian coordinates (q1, q2, q3, q4) = (x1, y1, x2, y2). The kinetic energy is K = 1 2 m1 (ẋ2 1 + ẏ2 1) + 1 2 m2 (ẋ2 2 + ẏ2 2). Hence, [gij ] =  m1 0 0 0 0 m1 0 0 0 0 m2 0 0 0 0 m2  . (i) First method. Since dim(Q) = 4 and dim(C) = 7, for a parametric representation of the constraint ẋ1 ẏ2−ẋ2 ẏ1 = 0 we need three parameters (z1, z2, z3). Let us consider the parameters (zα) = (z1, z2, z3) = (ρ, σ, θ) and the parametric equations ẋ1 = ρ cos θ, ẏ1 = ρ sin θ, ẋ2 = σ cos θ, ẏ2 = σ sin θ. (9.15) The meaning of the parameters is the following: ρ2 = v2 1 = ẋ2 1 + ẏ2 1, σ 2 = v2 2 = ẋ2 2 + ẏ2 2, and θ is the angle of the two vector velocities w.r.to the x-axis. Then we find: [ψi α] =  cos θ sin θ 0 0 0 0 cos θ sin θ − ρ sin θ ρ cos θ − σ sin θ σ cos θ  (α index of line), [Gαβ ] .= [gij ψ i αψ j β] =  m1 0 0 0 m2 0 0 0 m1ρ 2 +m2σ 2  , [Gαβ ] =  1 m1 0 0 0 1 m2 0 0 0 1 m1ρ2 +m2σ2  . Dynamical Equations of Non-Holonomic Systems 29 Since,[ ∂K ∂q̇i ] = [m1ẋ1 , m1ẏ1 , m2ẋ2 , m2ẏ2], [ ∂K ∂qi ] = [0 , 0 , 0 , 0], the Lagrange equations for the free motions gij q̈ j = Ai − Γhki q̇ h q̇k read m1 ẍ1 = A1, m1 ÿ1 = A2, m2 ẍ2 = A3, m2 ÿ2 = A4. (9.16) They show that Z̄i = Ai. Hence, [Zα] .= [ψi αZ̄i] =  A1 cos θ +A2 sin θ A3 cos θ +A4 sin θ ρ (A2 cos θ −A1 sin θ) + σ (A4 cos θ −A3 sin θ)  , [Zα] .= [Gαβ Zβ ] =  A1 cos θ +A2 sin θ m1 A3 cos θ +A4 sin θ m2 ρ (A2 cos θ −A1 sin θ) + σ (A4 cos θ −A3 sin θ) m1ρ2 +m2σ2  , and the differential system associated with Z is dx1 dt = ρ cos θ, dy1 dt = ρ sin θ, dx2 dt = σ cos θ, dy2 dt = σ sin θ, dρ dt = A1 cos θ +A2 sin θ m1 , dσ dt = A3 cos θ +A4 sin θ m2 , dθ dt = ρ (A2 cos θ −A1 sin θ) + σ (A4 cos θ −A3 sin θ) m1ρ2 +m2σ2 , (9.17) where the Lagrangian active forces are in general known functions of (x1, y1, x2, y2) and (ρ, σ, θ). In the special case of an inclined plane we have A1 = m1 g, A3 = m2 g, A2 = A4 = 0, and the system (9.17) becomes dx1 dt = ρ cos θ, dy1 dt = ρ sin θ, dx2 dt = σ cos θ, dy2 dt = σ sin θ, dρ dt = g cos θ, dσ dt = g cos θ, dθ dt = − g sin θ m1 ρ+m2 σ m1ρ2 +m2σ2 . (9.18) Note that the last three equations are separated from the first four. This occurs in general when the Lagrangian active forces do not depend on the position of the point, but only on their velocities. For equal masses m1 = m2, we have a further simplification: dx1 dt = ρ cos θ, dy1 dt = ρ sin θ, dx2 dt = σ cos θ, dy2 dt = σ sin θ, dρ dt = g cos θ, dσ dt = g cos θ, dθ dt = − g sin θ ρ+ σ ρ2 + σ2 . (9.19) 30 S. Benenti (ii) Second method for the single constraint equation (9.14). In this case, [Ci] = [ẏ2 , − ẋ2 , − ẏ1 , ẋ1] does not have the maximal rank for v1 = v2 = 0. This is a singular state for whatever configuration (see Remark 7.3); the set of the singular states is the zero-section of TQ. Moreover, since [gij ] =  1 m1 0 0 0 0 1 m1 0 0 0 0 1 m2 0 0 0 0 1 m2  , we have: [Ci] .= [gij Cj ] = [ ẏ2 m1 , − ẋ2 m1 , − ẏ1 m2 , ẋ1 m2 ] , G = ẋ2 2 + ẏ2 2 m1 + ẋ2 1 + ẏ2 1 m2 = 2K m1m2 , G−1 = m1m2 2K , [Ci ∗] = G−1 [ ẏ2 m1 , − ẋ2 m1 , − ẏ1 m2 , ẋ1 m2 ] , [πij ] = G−1  ẏ2 2 m2 1 − ẋ2ẏ2 m2 1 − ẏ1ẏ2 m1m2 ẋ1ẏ2 m1m2 − ẋ2ẏ2 m2 1 ẋ2 2 m2 1 ẋ2ẏ1 m1m2 − ẋ1ẋ2 m1m2 − ẏ1ẏ2 m1m2 ẏ1ẋ2 m1m2 ẏ2 1 m2 2 − ẋ1ẏ1 m2 2 ẋ1ẏ2 m1m2 − ẋ1ẋ2 m1m2 − ẋ1ẏ1 m2 2 ẋ2 1 m2 2  , [gij − πij ] =  1 m1 0 0 0 0 1 m1 0 0 0 0 1 m2 0 0 0 0 1 m2  − m1m2 2K  ẏ2 2 m2 1 − ẋ2ẏ2 m2 1 − ẏ1ẏ2 m1m2 ẋ1ẏ2 m1m2 − ẋ2ẏ2 m2 1 ẋ2 2 m2 1 ẋ2ẏ1 m1m2 − ẋ1ẋ2 m1m2 − ẏ1ẏ2 m1m2 ẏ1ẋ2 m1m2 ẏ2 1 m2 2 − ẋ1ẏ1 m2 2 ẋ1ẏ2 m1m2 − ẋ1ẋ2 m1m2 − ẋ1ẏ1 m2 2 ẋ2 1 m2 2  , g11 − π11 = 1 m1 − m2 2K ẏ2 2 m1 = 1 m1 ( 1− m2 ẏ 2 2 2K ) = 2K −m2 ẏ 2 2 2m1K = m1 (ẋ2 1 + ẏ2 1) +m2 ẋ 2 2 2m1K , g12 − π12 = m1m2 2K ẋ2ẏ2 m2 1 = m2 ẋ2ẏ2 2m1K , Dynamical Equations of Non-Holonomic Systems 31 g13 − π13 = m1m2 2K ẏ1ẏ2 m1m2 = ẏ1ẏ2 2K , g14 − π14 = − m1m2 2K ẋ1ẏ2 m1m2 = − ẋ1ẏ2 2K , g22 − π22 = 1 m1 − m2 2K ẋ2 2 m1 = 1 m1 ( 1− m2 ẋ 2 2 2K ) = 2K −m2 ẋ 2 2 2m1K = m1 (ẋ2 1 + ẏ2 1) +m2 ẏ 2 2 2m1K , g23 − π23 = − m1m2 2K ẋ2ẏ1 m1m2 = − ẋ2ẏ1 2K , g24 − π24 = m1m2 2K ẋ1ẋ2 m1m2 = ẋ1ẋ2 2K , g33 − π33 = 1 m2 − m1 2K ẏ2 1 m2 = 1 m2 ( 1− m1 ẏ 2 1 2K ) = 2K −m1 ẏ 2 1 2m2K = m2 (ẋ2 2 + ẏ2 2) +m1 ẋ 2 1 2m2K , g34 − π34 = m1m2 2K ẋ1ẏ1 m2 2 = m1 ẋ1ẏ1 2m2K , g44 − π44 = 1 m2 − m1 2K ẋ2 1 m2 = 1 m2 ( 1− m1 ẋ 2 1 2K ) = 2K −m1 ẋ 2 1 2m2K = m2 (ẋ2 2 + ẏ2 2) +m1 ẏ 2 1 2m2K , [∂iC] = [0 , 0 , 0 , 0], [∂jC C i ∗] = [0]. As in this case Li = Ai, we have Di .= (gij − πij)Lj − q̇j ∂jC C i ∗ = (gij − πij)Lj = (gij − πij)Aj , D1 = 1 2K ( A1 m1 (ẋ2 1 + ẏ2 1) +m2 ẋ 2 2 m1 +A2 m2 ẋ2ẏ2 m1 +A3 ẏ1ẏ2 −A4 ẋ1ẏ2 ) , D2 = 1 2K ( A1 m2 ẋ2ẏ2 m1 +A2 m1 (ẋ2 1 + ẏ2 1) +m2 ẏ 2 2 m1 −A3 ẋ2ẏ1 +A4 ẋ1ẋ2 ) , D3 = 1 2K ( A1ẏ1ẏ2 −A2ẋ2ẏ1 +A3 m2 (ẋ2 2 + ẏ2 2) +m1 ẋ 2 1 m2 +A4 m1 ẋ1ẏ1 m2 ) , D4 = 1 2K ( −A1ẋ1ẏ2 +A2ẋ1ẋ2 +A3 m1 ẋ1ẏ1 m2 +A4 m2 (ẋ2 2 + ẏ2 2) +m1 ẏ 2 1 m2 ) . Then the dynamical system (8.12) reads dx1 dt = ẋ1, dy1 dt = ẏ1, dx2 dt = ẋ2, dy2 dt = ẏ2, dẋ1 dt = 1 2K ( A1 m1 (ẋ2 1 + ẏ2 1) +m2 ẋ 2 2 m1 +A2 m2 ẋ2ẏ2 m1 +A3 ẏ1ẏ2 −A4 ẋ1ẏ2 ) , dẏ1 dt = 1 2K ( A1 m2 ẋ2ẏ2 m1 +A2 m1 (ẋ2 1 + ẏ2 1) +m2 ẏ 2 2 m1 −A3 ẋ2ẏ1 +A4 ẋ1ẋ2 ) , dẋ2 dt = 1 2K ( A1ẏ1ẏ2 −A2ẋ2ẏ1 +A3 m2 (ẋ2 2 + ẏ2 2) +m1 ẋ 2 1 m2 +A4 m1 ẋ1ẏ1 m2 ) , dẏ2 dt = 1 2K ( −A1ẋ1ẏ2 +A2ẋ1ẋ2 +A3 m1 ẋ1ẏ1 m2 +A4 m2 (ẋ2 2 + ẏ2 2) +m1 ẏ 2 1 m2 ) . For two points running on an inclined plane, D1 = g m1 (ẋ2 1 + ẏ2 1) +m2 (ẋ2 2 + ẏ1ẏ2) m1 (ẋ2 1 + ẏ2 1) +m2 (ẋ2 2 + ẏ2 2) , D2 = g m2ẋ2 (ẏ2 − ẏ1) m1 (ẋ2 1 + ẏ2 1) +m2 (ẋ2 2 + ẏ2 2) , D3 = g m2 (ẋ2 2 + ẏ2 2) +m1 (ẋ2 1 + ẏ1ẏ2) m1 (ẋ2 1 + ẏ2 1) +m2 (ẋ2 2 + ẏ2 2) , D4 = g m1 ẋ1 (ẏ1 − ẏ2) m1 (ẋ2 1 + ẏ2 1) +m2 (ẋ2 2 + ẏ2 2) . 32 S. Benenti For equal masses, m1 = m2, D1 = g ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ1ẏ2 ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ2 2 , D2 = g ẋ2 (ẏ2 − ẏ1) ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ2 2 , D3 = g ẋ2 2 + ẏ2 2 + ẋ2 1 + ẏ1ẏ2 ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ2 2 , D4 = g ẋ1 (ẏ1 − ẏ2) ẋ2 1 + ẏ2 1 + ẋ2 2 + ẏ2 2 . Note 9.1. It is easy (and obvious) to propose examples of non-linear constraints: it is sufficient to choose any set of non-linear independent equations Ca(q, q̇) = 0. However, any example of a non-linear constraint remains meaningless unless we know how to realize it physically by means of realizable devices. The famous Appell–Hamel example gives a matter of discussion (see [9], Ch. 4, § 2). Indeed, in order to be really a non-linear device, a certain distance of the Appell–Hamel device must be infinitesimally small. This fits with the thought of Hertz: non- linear constraints can be realized by passing to the limit x→ 0 of certain physical quantities x (masses, lengths, etc.) in devices realizing linear constraints. The same kind of problem arises in trying to ‘realize’ two mass-points moving with parallel velocities. A tentative project has been presented in [1]. In fact, for an effective project, we have to invent devices for 1. Realizing a mass-point. 2. Realizing a parallel transport on the plane. 3. Transforming the direction of the velocity of a point into a solid segment. 4. Applying forces of special kind to the points (the weight is of course always present). This research is a work in progress. Updated information will be found on my personal web-page. Acknowledgments A preliminary version of this paper has been elaborated and exposed at the University of Linköping, Department of Mathematics, on May 27, 2005. I wish to thank Stefan Rauch and all the Linköping school for their warm hospitality. I wish also to thank: Waldyr Oliva, Willy Sarlet, David Martin de Diego for making me aware of their contributions to the theory; Enrico Pagani and Enrico Massa, for the enlightening discussions during a workshop on dynamical systems held in Torino in April 2005 – their papers [8, 7] have been of great help; Beppe Gaeta, for pointing me out some errors in my first manuscript; Franco Cardin for his kind invitation to give a seminar in Padova on the contents of this paper (November 2006). References [1] Benenti S., Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino 54 (1996), 203–212. [2] Bullo F., Lewis A.D., Geometric control of mechanical systems, Texts in Applied Mathematics, Vol. 49, Springer, Berlin, 2004. [3] Carathéodory C., Sur les équations de la mécanique, Actes Congrès Interbalcanian Math. (1934, Athènes), 1935, 211–214. [4] Cortés Monforte J., Geometrical, control and numerical aspects of nonholonomic systems, Lecture Notes in Mathematics, Vol. 1793, Springer, Berlin, 2002. [5] Gantmacher F., Lectures in analytical mechanics, Mir, Moscow, 1970. [6] Marle C.-M., Reduction of constrained mechanical systems and stability of relative equilibria, Comm. Math. Phys. 174 (1995), 295–318. Dynamical Equations of Non-Holonomic Systems 33 [7] Massa E., Pagani E., A new look at classical mechanics of constrained systems, Ann. Inst. H. Poincaré Phys. Théor. 66 (1997), 1–36. [8] Massa E., Pagani E., Classical dynamics of non-holonomic systems: a geometric approach, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), 511–544. [9] Neimark J.I., Fufaev N.A., Dynamics of nonholonomic systems, Translations of Mathematical Monographs, Vol. 33, American Mathematical Society, Providence, Rhode Island, 1972. [10] Oliva W.M., Kobayashi M.H., A note on the conservation of energy and volume in the setting of nonholo- nomic mechanical systems, Qual. Theory Dyn. Syst. 4 (2004), 383–411. 1 Preamble 2 Introduction 3 Ideal constraints 4 The Gauss principle 5 The Gibbs-Appell equations 6 The explicit form of the Gibbs-Appell equations 7 The dynamical equations of the first kind 8 The dynamical equations of the second kind 9 Illustrative examples 9.1 The skate 9.2 The vertical rolling disc 9.3 Two co-axial rolling discs 9.4 Two points with parallel velocities References

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