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Suppose we have a non-holonomic mechanical system, say Lagrangian, for example the Chaplygin sleigh is a model of a knife in the plane. Its configuation space is $Q = S^1\times \mathbb{R}^2$ with local coordinates $q = (\theta,x,y)$. The non-holonomic constraint is "no admissible velocities are perpendicular to the blade" which is specified by a one-form

\begin{equation} \omega_q = \sin\theta dx + \cos\theta dy. \end{equation} When evaulating on a velocity $\dot{q}$ we specify the velocity constraint

$$ v = \omega_q\cdot \dot{q} = -\dot{x}\sin\theta + \dot{y}\cos\theta = 0. $$

What is the mathematical/physical meaning of the exterior derivative of the constraint one-form? Since $\omega_q$ is a one-form, in physics, we typically interpret it as a force. Mathematically, we integrate this form over a 'line' to get the work due to the force over the 'line'. The exterior derivative is a 2-form, (does this have any physical significance??)

$$ \boldsymbol{d}\omega = -\cos\theta d\theta \wedge dx - \sin\theta d\theta \wedge dy $$

If we evaluate this 2-form along a tangent (velocity) vector $\dot{q}$, we find the 'force'

$$ \alpha_q = \boldsymbol{d}\omega(\dot{q},.) = \left(\dot{x}\cos\theta + \dot{y}\sin\theta \right)d\theta - \cos\theta\dot{\theta}dx - \sin\theta \dot{\theta}dy. $$

Does this 'force', $\alpha_q \in T^*Q$, have any direct/obvious physical/mathematical physics/differential geometric interpretation?

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2 Answers 2

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Let us work more generally, suppose that on the $m$-dimensional configuration space $Q$, we have a set of $r$ pointwise-independent $1$-forms $\theta^\alpha$ ($\alpha=1,\cdots,r$) giving a non-holonomic constraint through the Pfaffian equation $\theta^\alpha=0$. For simplicity I am assuming the constraint is scleronomic, so it does not involve the time variable.

The Frobenius integrability theorem tells us that the constraint is equivalent to a holonomic constraint if and only if $d\theta^\alpha=0$ $\mod \theta^1,\cdots,\theta^r$, or in other words $$ d\theta^\alpha=\xi^\alpha_{\ \beta}\wedge\theta^\beta, $$ where the $\xi^\alpha_{\ \beta}$ is any matrix of $1$-forms.

Let $\Delta$ be the associated distribution on $Q$ given by $$ \Delta_p=\ker\theta^1_p\cap\cdots\cap\ker\theta^r_p. $$

An equivalent statement of Frobenius' theorm is then that the Pfaffian system is integrable if and only if $$ d\theta^\alpha(u,v)=0 $$ for any pair of vectors $u,v\in\Delta$ that belong to the distribution.

Now the $1$-forms $u\lrcorner\ d\theta^\alpha$ then have the intepretation that if they are orthogonal to $\Delta$ for all $u$, then the system is Frobenius-integrable and is thus equivalent to a holonomic constraint.

In OP's example this meaning is more pronounced for the following reason. OP's configuration manifold is $3$-dimensional, and OP's Pfaffian system is generated by a single $1$-form. Thus the distribution corresponding to the Pfaffian is two dimensional, and in the neighborhood of each point there is a local frame for the distribution that consists of a pair of independent vector fields.

Accordingly suppose that $\dot q$ is a velocity field compatible with the constraint that does not vanish anywhere. Then $d\omega(\dot q,\cdot)=\dot q\lrcorner\ d\omega$ is orthogonal to the distribution if and only if the distribution is integrable. Thus this expression essentially measures how much the constraint fails to be holonomic.

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  1. The Chaplygin sleigh is modelled by Lagrange equations $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^j}-\frac{\partial L}{\partial q^j}~=~\underbrace{\lambda a_j}_{\text{gen. constr. force}}, \qquad j~\in \{1,\ldots, n\}, \tag{A}$$ where $L$ is the kinetic energy of the sleigh; $\lambda$ is a Lagrange multiplier; and $$\omega~\equiv~\sum_{j=1}^na_j(q,t)~\mathrm{d}q^j+a_0(q,t)\mathrm{d}t~=~0\tag{B}$$ is the semi-holonomic constraint mentioned by OP. See e.g. Ref. 1 for details.

  2. The right-hand side of Lagrange equations (A), i.e. the components of $\omega$ [multiplied by $\lambda$] has an interpretation as a generalized constraint force.

  3. The exterior derivative $\mathrm{d}\omega$ is related to integrability conditions for the semi-holonomic constraint (B), cf. Bence Racsko's answer. The Chaplygin sleigh constraint is not integrable, i.e. not secretly holonomic.

References:

  1. A.M. Bloch, Nonholonomic Mechanics and Control; p. 27, eq. (1.7.3).
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