5.4. Constraints classification

5.4.1. Holonomic vs. non-holonomic constraints

Definition 5.1 (Holonomic constraint)

A holonomic constraints can be written in a form

\[g(q^k(t), t) = 0 \ .\]

Definition 5.2 (Non-holonomic constraint)

Every constraint that is not holonomic, is non-holonomic. (wow!)

todo Add some examples…; even some constraints that looks like a non-holonomic constraint that are holonomic: “integrable constraints”, Pfaffian method?…

5.4.2. Ideal constraints

Definition 5.3 (Ideal constraint)

An ideal constraint produces no net power.

Given the generalized actions \(\mathbf{f}_c\) introduced in the dynamical systems by constraints,

\[\mathbf{M} \ddot{\mathbf{q}} = \mathbf{f} + \mathbf{f}_c \ ,\]

power of ideal constraints reads

\[\dot{\mathbf{q}}^T \mathbf{f}_c = 0 \ .\]

5.4.2.1. Ideal holonomic constraints

If constraints don’t have explicit dependence (what happens if there’s explicit time dependence? Treat in the proper section…) from time \(t\)

\[\begin{split}\begin{aligned} & \mathbf{M} \ddot{\mathbf{q}} = \mathbf{f} + \mathbf{f}_c \\ & \mathbf{g}(\mathbf{q}(t)) = \mathbf{0} \end{aligned}\end{split}\]

with \(\mathbf{q} \in \mathbb{R}^N\), \(\mathbf{g} \in \mathbb{R}^C\).

Power of ideal constraint. Power of ideal constraints is zero, and this condition provides the most general form of constraint reactions \(\mathbf{f}_c\) as a linear combination of the gradient of the constraints, and a set of well-defined (extra conditions?) and determined DAEs, with equal number of equations and unknowns. Power of contraint reactions of ideal constrains is zero,

\[0 = \dot{\mathbf{q}}^T \mathbf{f}_c\]

Time derivative of the constraint equation reads \(0 = g_i (q^k) = \dot{q}^k \frac{\partial g_i}{\partial q^k}\), and thus for every \(\mathbf{s} \in \mathbb{R}^C\),

(5.7)\[0 = \dot{\mathbf{q}}^T \nabla_{\mathbf{q}} \mathbf{g} \, \mathbf{s} \ ,\]

and thus constraint reactions can be written as a linear combination of the (columns of the) gradient of the constraint equation w.r.t. the generalized coordiantes,

\[\mathbf{f}_c = \nabla_\mathbf{q} \mathbf{g} \, \mathbf{s}\]

or

\[\begin{aligned} \mathbf{f}_c = \nabla_{\mathbf{q}} g_i \, s_i \qquad , \qquad f_{c,k} = \dfrac{\partial\mathbf{g}^T}{\partial q^k} \mathbf{s} = \dfrac{\partial g^i}{\partial q^k} \, s_i \ . \end{aligned}\]

Determined set of DAEs. Introducing the expression (5.7) of the constraint reactions in the original set of DAEs, the set of equations governing the constrained system reads

\[\begin{split}\begin{aligned} & \mathbf{M} \ddot{\mathbf{q}} = \mathbf{f} + \nabla_{\mathbf{q}} \mathbf{g} \, \mathbf{s} \\ & \mathbf{g}(\mathbf{q}) = \mathbf{0} \end{aligned}\end{split}\]

5.4.2.2. Ideal non-holonomic constraints

The equations of a constrained system with non-holonomic constraints in semi-linear form and no explicit time dependence reads

\[\begin{split}\begin{aligned} & \mathbf{M} \ddot{\mathbf{q}} = \mathbf{f} + \mathbf{f}_c \\ & \mathbf{a}(\mathbf{q}(t)) \, \dot{\mathbf{q}}(t) = \mathbf{0} \end{aligned}\end{split}\]

Power of ideal constraints. As done in the section about holonomic constraints, the condition of zero power of ideal constraints provides the most general form of constraint reactions \(\mathbf{f}_c\) as a linear combination of the gradient of the constraints, and thus a determined set of DAEs.

\[0 = \dot{\mathbf{q}}^T \, \mathbf{f}_c\]

compared with the (transpose of the) non-holonomic constraint,

\[0 = \dot{\mathbf{q}}^T \mathbf{a}^T(\mathbf{q}) \mathbf{s} \ , \qquad \forall \mathbf{s} \in \mathbb{R}^C\]

and thus constraint reactions can be written as the linear combination of the columns of the transpose of matrix \(\mathbf{a}(\mathbf{q})\),

(5.8)\[\mathbf{f} = \mathbf{a}^T(\mathbf{q}) \mathbf{s} \ .\]

Determined set of DAEs. Introducing the expression (5.8) of the constraint reactions in the original set of DAEs, the determined set of equations governing the constrained system reads

\[\begin{split}\begin{aligned} & \mathbf{M} \ddot{\mathbf{q}} = \mathbf{f} + \mathbf{a}^T(\mathbf{q}) \, \mathbf{s} \\ & \mathbf{a}(\mathbf{q}) \, \dot{\mathbf{q}}(t) = \mathbf{0} \end{aligned}\end{split}\]