PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 91(105) (2012), 19–24 DOI: 10.2298/PIM1205019B

EHRESMANN CONNECTION IN THE GEOMETRY OF NONHOLONOMIC SYSTEMS

Aleksandar Bakša

Abstract. This article deals with a dynamic system whose motion is constrained by nonholonomic, reonomic, affine constraints. The article analyses the geometrical properties of the “reactions" of nonholonomic constraints in Voronets’s equations of motion. The analysis shows their link with the torsion of the Ehresmann connection, which is defined by the nonholonomic constraints.

1. Differential Equations of Motion

We consider nonholonomic mechanical system on a configuration manifold Q^{n}, where the local coordinates are q = (q^{1}, …, q^{n}). The motion of the system is constrained by the affine constraints

(1)  q̇^{ν} = a_{α}^{ν} q̇^{α} + a^{ν}, α = 1, …, m; ν = m + 1, …, n

where a_{α}^{ν} : (q, t) ↦ a_{α}^{ν} (q, t) and a^{ν} : (q, t) ↦ a^{ν}(q, t) are smooth functions. At each point (q, t) ∈ Q^{n} × R, the constraints (1) define a (n − m)-dimensional timedependent tangent subspace D_{q,t} ⊂ T_{q}Q^{n} × R (assuming that the subspace dimensions are the same at each point), i.e., (n − m)-dimensional distribution.

A curve γ : (a, b) ∋ t ↦ γ(t) ∈ Q^{n}, is said to satisfy the constraints if γ̇(t) ∈ D_{γ(t)} for all t. The set 𝔐 comprising all the points at which the tangent vector belongs to the distribution is called the integral set of the distribution.

The nonholonomic constraints (1) define the 1-forms ω on an extended configuration manifold Q^{n} × R in the following manner: ω^{ν} = dq^{ν} − a_{α}^{ν} dq^{α} − a^{ν}dt.

For the sake of brevity, we will express them as follows (t = q^{0}):

ω^{ν} = dq^{ν} − aν ; α̂dqα̂, a^{ν} = a_{0}^{ν}, α̂ = 0, 1, …, m.

2010 Mathematics Subject Classification: Primary 37J60; Secondary 70F25.

Dedicated to the memory of Professor Antun Bilimovich, whose books and works in the area of Rational Mechanics are a part of the foundation of my professional education and serve as a model for my work. His textbook [1] is the source of my initial knowledge of the dynamics of nonholonomic systems.

The linear space N^{∗} of the 1-forms on the manifold, which represents a module over the ring of smooth functions, will be called the differential system. If dim N^{∗} is equal at all points, then it is taken as the dimension of the differential system. The differential system may also be given with (m + 1) linearly independent vector fields

ξα̂ = aν ; α̂ \frac{∂}{∂q^{ν}} + \frac{∂}{∂qα̂}

which annul the 1-forms ω^{ν}. The linearly independent vectors ξα̂ in each tangent space T_{(q,t)} (Q^{n} × R) define the (m + 1)-dimensional vector subspace D_{(q,t)} . The differentiable field of these subspaces D : (q, t) ↦ D_{q,t} is called the distribution.

For the linearly independent vector fields ξα̂ we say that they span the distribution D, and that the corresponding fields of 1-forms ω^{ν} span the codistribution or annulator D^{⊥} .

The differential equations of motion of the mechanical system determined by the Lagrangian L : T Q^{n}×R ∋ (q̇, q, t) ↦ L(q̇, q, t) ∈ R, whose motion is constrained by nonholonomic constraints (1), are derived from the basic dynamic equation

( \frac{d}{dt} \frac{∂L}{∂q̇} − \frac{∂L}{∂q} , δq) = 0

where δq is a random vector from T_{q}Q^{n} (virtual displacement). Bearing in mind the constraints imposed on the coordinates of vector δq by nonholonomic connections

δq^{ν} = a_{α}^{ν} δq^{α}

the basic dynamic equation has the following form

( \frac{d}{dt} \frac{∂L}{∂q̇^{α}} − \frac{∂L}{∂q^{α}} ) δq^{α} + a_{α}^{ν} ( \frac{d}{dt} \frac{∂L}{∂q̇^{ν}} − \frac{∂L}{∂q^{ν}} ) δq^{α} = 0.

Let us introduce a “connected" Lagrangian L̃ by using the equality (1) and excluding the coordinates q̇^{ν}:

L(q̇, q, t) → L̃(q̇^{1}, …, q̇^{m}, q^{1}, …, q^{n}, t).

The equations of motion may now be written in the following form (in scientific literature, these equations are known as Voronets’s equations ^{1} [3]:

(2)  \frac{d}{dt} \frac{∂L̃}{∂q̇^{α}} − \frac{∂L̃}{∂q^{α}} − \frac{∂L̃}{∂q^{ν}} a_{α}^{ν} − ( ˜ \frac{∂L}{∂q̇^{ν}} ) (B_{αβ}^{ν} q̇^{β} + B_{α}^{ν}) = 0,

where

B_{αβ}^{ν} = \frac{∂a_{α}^{ν}}{∂q^{β}} − \frac{∂a_{β}^{ν}}{∂q^{α}} + \frac{∂a_{α}^{ν}}{∂q^{μ}} a_{β}^{μ} − \frac{∂a_{β}^{ν}}{∂q^{μ}} a_{α}^{μ} (3)

B_{α}^{ν} = \frac{∂a_{α}^{ν}}{∂t} − \frac{∂a^{ν}}{∂q^{α}} + \frac{∂a_{α}^{ν}}{∂q^{μ}} a^{μ} − \frac{∂a^{ν}}{∂q^{μ}} a_{α}^{μ} (4)

1P. V. Voronets, 1871–1923

Along with the initially set conditions, equations (1) and (2) completely define the motion of the system. Obviously, the system of coefficient (3) is skew-symmetric. In further we shall denote

B_{αβ}^{ν} q̇^{β} + B_{α}^{ν} = B_{αβ̂}^{ν} q̇β̂, α = 1, …, m, β̂ = 0, …, m, q̇^{0} = 1.

where t = q^{0}, a^{μ} = a_{0}^{μ}, B_{α}^{ν} = B_{α0}^{ν} .

2. Ehresmann Connection

In order to examine the geometrical properties of the nonholonomic connections, we need to analyse the fibre bundle (Q^{n}, R^{m}, S^{(n−m)} , π), where Q^{n} is the total space, R^{m} is the base space, S^{(n−m)} is the fiber, π : Q^{n} → R^{(n−m)} projection (surjective submersion).

The submersion of π induces the submersion of π_{∗} : T Q → T R. The kernel of the submersion is the vertical subspace Ker(π_{∗}) = V ∈ T Q. The complement of the vertical subspace is the horizontal subspace H. At any point q ∈ Q is V_{q} ⊕ H_{q} = T_{q}Q, V_{q} ∩ H_{q} = {0}.

The separation of the horizontal subspace is linked with the structure of the linear connection at the fiber bundle, i.e., with the parallel transport. The connection can be defined in several ways and we will introduce it via the vector-valued 1-form.

Definition. The Ehresmann^{2} connection is a vector-valued differential 1-form Ω on Q^{n} such that

(1) Ω : T_{q}Q^{n} → V_{q} (vertical valued), (2) Ω(v) = v, v ∈ V (projection).

We will introduce the Ehresmann connection on Q^{n} × R via the differential form

Ω := ω^{ν} \frac{∂}{∂q^{ν}} , ω^{ν} = dq^{ν} − a_{α}^{ν} dq^{α} − a^{ν}dt.

The vertical vector component ξ (we will mark it as ξ_{v}) is

ξ_{v} = Ω(ξ) = (ξ^{ν} − aν ; α̂ξα̂)\frac{∂}{∂q^{ν}} , α̂ = 0, 1, …, m.

The map Ω is obviously vectorial and it can be easily seen that the projection is vectorial as well.

The horizontal vectors have the following form

ξ_{h} = ξ − ξ_{v} ⇒ ξ_{h} = ξα̂( aν ; α̂ \frac{∂}{∂q^{ν}} + \frac{∂}{∂qα̂} ) .

It should be noted that these vectors extend the distribution. We can see that the horizontal lift is

hlift( \frac{∂}{∂qα̂} ) = \frac{∂}{∂qα̂} + aν ; α̂ \frac{∂}{∂q^{ν}} .

The collection of all the horizontal vectors at point q ∈ Q^{n} forms a horizontal subspace. The separation of the horizontal subspace basically represents the definition of the Ehresmann connection.

2Charles Ehresmann, 1905–1979

One of the important characteristics of the connection is its curvature. The curvature of the connection Ω is the two-form on Q^{n} with values in T Q^{n}, [4], [7]

T (X, Y) = Ω[X_{h}, Y_{h}].

By using a known identity valid for a 1-form differential θ

dθ(X, Y) = X(θ(Y)) − Y (θ(X)) + θ([X, Y])

we get

T (X, Y) = dω^{ν}(X_{h}, Y_{h})\frac{∂}{∂q^{ν}} .

In our case it is

dω^{ν} = −\frac{∂aν ; α̂}{∂qβ̂} dqβ̂ ∧ dqα̂ − \frac{∂aν ; α̂}{∂q^{μ}} dq^{μ} ∧ dqα̂.

The value of the differential form dω on vectors ξ_{h} ∈ T Q, η_{h} ∈ T (Q^{n} × R) is

dω(ξ_{h}, η_{h}) = B_{αβ}^{ν} ξ^{α} η^{β} + B_{α}^{ν}ξ^{α} η^{0}.

This proves the following

Proposition. Coefficients B_{αβ}^{ν} + B_{α}^{ν} in Voronets’s equations represent the curvature of the Ehresmann connection on the vectors ξ_{h} ∈ H ⊂ T Q^{n}, η_{h} ∈ H ⊂ T (Q^{n} × R).

Example. Let us consider the rolling of a heavy homogenous ball on a horizontal plane, revolving around its axis [4]. Let us assume that the radius of the ball r(·) and the angular speed of the plane Ω(·) are given smooth functions of time. We will set the inertial (immovable) coordinate system Oxyz in such a manner that the z-axis is vertical, pointed upwards. We will determine the position of the ball by the coordinates (x, y) of the point at which the ball A touches the plane, and by the Euler angles (ψ, θ, φ). The configuration manifold of the system is R^{2} × SO(3), which can be considered as the main stratification with the base R^{2} and the fiber SO(3), at which the structural group acts on itself from the right. Based on the rolling conditions (without sliding), we get the nonholonomic constraints in the following form

(5)  ω_{x}(t) = −\frac{1}{a(t)}ẏ+ \frac{Ω(t)}{a(t)} x, ω_{y}(t) = \frac{1}{a(t)}ẋ+ \frac{Ω(t)}{a(t)} y,

t ∈ (α, β) ⊂ R, where

ω_{x} = θ̇cosψ + φ̇sinψ sinθ
(6)  ω_{y} = θ̇sinψ − φ̇cosψ sinθ
ω_{z} = φ̇cosθ + ψ̇.

Substituting (6) into (5) gives

φ̇ = \frac{cosψ}{a sinθ} ẋ− \frac{sinψ}{a sinθ} ẏ+ \frac{Ω(x sinψ − y cosψ)}{a sinθ}
θ̇= \frac{sinψ}{a} ẋ− \frac{cosψ}{a} ẏ+ \frac{Ω(x cosψ + y sinψ)}{a}

The Lagrangian is

L = \frac{m}{2}(ẋ^{2} + ẏ^{2}) + \frac{mk^{2}}{2} (ω_{x}^{2} + ω_{y}^{2} + ω_{z}^{2})

= \frac{m}{2}(ẋ^{2} + ẏ^{2}) + \frac{mk^{2}}{2} [ψ̇^{2} + φ̇^{2} + 2φ̇ψ̇cosθ + θ̇^{2}]

(mk^{2} is inertia about any axis).

From Lagrange’s equations with multipliers [1]

\frac{d}{dt} \frac{∂L}{∂q̇^{i}} − \frac{∂L}{∂q^{i}} = ∑ λ_{ν} \frac{∂Φ_{ν}}{∂q^{i}}

we get

\frac{∂L}{∂ψ̇} = mk^{2}(ψ̇+ φ̇cosθ) = mk^{2}ω_{z} = const.

The reduced Lagrangian L̃, without an additive constant, is

L̃(x, y, ẋ, ẏ) = \frac{m}{2} [( 1 + \frac{k^{2}}{a^{2}} ) (ẋ^{2} + ẏ^{2}) + \frac{2k^{2}Ω}{a^{2}} (yẋ− xẏ) + \frac{k^{2}Ω^{2}}{a^{2}} (x^{2} + y^{2})] .

The functions that determine the curvature are (q^{1} = x, q^{2} = y, q^{3} = ψ, q^{4} = φ, q^{5} = θ)

B_{12}^{4} = −\frac{cosθ}{a^{2} sin^{2} θ}, B_{13}^{4} = \frac{sinψ}{a sinθ}, B_{23}^{4} = −\frac{cosψ}{a sinθ}

B_{1}^{4} = \frac{ȧcosψ}{a^{2} sinθ} − \frac{Ω sinψ}{a cosθ} + \frac{Ω cosθ}{a^{2} sin^{2} θ}x, B_{2}^{4} = \frac{ȧsinψ}{a^{2} sinθ} + \frac{Ω cosψ}{a sinθ} + \frac{Ω cosθ}{a^{2} sin^{2} θ}y,

B_{3}^{4} = −\frac{Ω(x cosψ + y sinψ)}{a sinθ} , B_{12}^{5} = 0, B_{13}^{5} = \frac{cosψ}{a} , B_{23}^{5} = \frac{sinψ}{a} ,

B_{1}^{5} = −\frac{1}{a^{2}} (ȧsinψ+a cosψ), B_{2}^{5} = \frac{1}{a^{2}} (ȧcosψ−a sinψ) ; B_{3}^{5} = \frac{Ω}{a} (x sinψ−y cosψ)

The distribution D is determined by (horizontal vectors)

ξ_{1} = \frac{∂}{∂x} − \frac{cosψ}{a sinθ} \frac{∂}{∂φ} + \frac{sinψ}{a} \frac{∂}{∂θ},

ξ_{2} = \frac{∂}{∂y} − \frac{sinψ}{a sinθ} \frac{∂}{∂φ} − \frac{cosψ}{a} \frac{∂}{∂θ},

ξ_{0} = \frac{∂}{∂t} + \frac{Ω(x sinψ + y cosψ)}{a sinθ} \frac{∂}{∂φ} − \frac{Ω(x cosψ + y sinψ)}{a} \frac{∂}{∂θ}.

References

1. Anton Bilimovich, Rational Mechanics II, Belgrade, 1951 (in Serbian).

2. Anton Bilimovch, Sur les équations intrinsiques du mouvement d’un systéme matériel, the

Herald of Serbian Royal Academy, 138, Belgrade, 1927.

3. P. V. Voronets, Sur les equations du mouvement pour les systemes non holonomes, Mat. Sb.

22(4), (1901), 659–686 (in Russian).

4. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, R. M. Murrey, Nonholonomic mechanical

system with symmetry, Arch. Ration. Mech. Anal. 136(1) (1996), 21–99.

5. Vladimir Dragović, Darko Milinković, Analysis of Manifolds, Belgrade 2003 (in Serbian).

6. W. S. Koon, J. E. Marsden, The Hamiltonian and lagrangian approaches to the dynamics of nonholonomic systems, Rep. Math. Phys. 40(4) (1997),21–62.

7. Ivan Kolar, Peter W. Michor, Jan Slovak; Natural Operators in Differential Geometry, Springer-Verlag, 1993.

Belgrade (Received 08 04 2011) Serbia (Revised 19 10 2011)