# What is the physical meaning of the exterior derivative of a non-holonomic constraint form?

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

$$$$\omega_q = \sin\theta dx + \cos\theta dy.$$$$ When evaulating on a velocity $$\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?

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.

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).