Section 13.6 Lagrange Equations with Multipliers
We saw in
Section 13.4 that it is always possible to derive Lagrange equations for systems which satisfy holonomic or non-holonomic constraint equations by simply ignoring these constraints and thereby guaranteeing that the chosen coordinates
\(\bq\) satisfy assumption
(13.1.2). The corresponding virtual velocity field is then defined as
\begin{equation*}
\vel_P^* = \sum_{i=1}^n \frac{\partial \br_{OP}}{\partial q_i} \dq^*_i =
\sum_{i=1}^n \frac{\partial \vel_{P}}{\partial \dq_i} \dq^*_i
\end{equation*}
where the virtual speeds \(\dbq^* = (\dq_1^*, \dq_2^*, \ldots, \dq_n^*)\) take arbitrary values.
We examine here how to derive Lagrange equations which take into account non-holonomic constraints: suppose that the coordinates \(\bq\) are known to satisfy \(l \lt n\) constraint (time-independent) equations of the type
\begin{equation}
\sum_{j=1}^n a_{ij} (\bq) \dq_j + b_i(\bq) = 0, \qquad (i=1,2,\ldots, l) \tag{13.6.1}
\end{equation}
These equations, linear in the
\(\dq_j\text{,}\) are assumed non-integrable, that is, it is not possible to express
\(l\) coordinates, say,
\((q_{m+1}, q_{m+2} , \ldots , q_n)\) as an explicit function of the
\(m=n-l\) remaining independent coordinates
\((q_1, q_2, \ldots, q_m)\text{.}\) However we assume that the linear system of equations
(13.6.1) in the unknowns
\((\dq_{m+1}, \dq_{m+2} , \ldots , \dq_n)\) is invertible to give
\begin{equation}
\dq_{m+i} = \sum_{j=1}^m \tilde{a}_{ij} (\bq) \dq_j + \tilde{b}_i (\bq), \qquad (i=1,2,\ldots, l) \tag{13.6.2}
\end{equation}
We then ask if it is possible to derive
\(m\) Lagrange equations w.r.t. the independent coordinates
\((q_1, q_2, \ldots, q_m)\text{.}\) We start with equation
(13.1.1) and substitute each
\(\dq_{m+i}\) for
\(i=1,2,\ldots, l\) according to
(13.6.2): we find
\begin{equation}
\vel_{P}= \sum_{i=1}^m \pvel_P^{q_i} \,\dq_i + \pvel_P^0 \tag{13.6.3}
\end{equation}
with
\begin{equation*}
\pvel_P^{q_i} = \frac{\partial \br_{OP}}{\partial q_i} + \sum_{j=1}^l \tilde{a}_{ji}
\frac{\partial \br_{OP}}{\partial q_{m+j}}
\end{equation*}
and
\begin{equation*}
\pvel_P^0 = \sum_{j=1}^l \tilde{b}_{j} \frac{\partial \br_{OP}}{\partial q_{m+j}}
\end{equation*}
\begin{equation}
\pvel_P^{q_i} = \frac{\partial}{\partial \dq_i} \vel_{P} \qquad (i=1,\ldots,m) \tag{13.6.4}
\end{equation}
Note however that
\begin{equation}
\pvel_P^{q_i} \neq \frac{\partial \br_{OP}}{\partial q_i} \qquad (i=1,\ldots,m)\tag{13.6.5}
\end{equation}
We now introduce the following virtual velocity field
\begin{equation}
\vel_P^* = \sum_{i=1}^m \dq^*_i \, \pvel_P^{q_i} \tag{13.6.6}
\end{equation}
where
\(\dbq^* = (\dq_1^*, \dq_2^*, \ldots, \dq_m^*)\) is an arbitrary element of
\(\mathbb{R}^m\text{.}\) It is easy to see, with equation
(13.6.4), that this virtual velocity field defines a screw. The derivation of Lagrange equations from the Principle of Virtual Power hinges on the Lagrange kinematic formula
(13.2.1). Hence, we seek to find a generalization of Lagrange kinematic formula compatible with the virtual velocity field
(13.6.6):
\begin{equation}
\ba_P \cdot \pvel_P^{q_i} = \frac{d}{dt}\left(\vel_P \cdot \frac{\partial}{\partial \dq_i} \vel_{P}
\right) - \vel_P \cdot \frac{d}{dt} \pvel_P^{q_i} \tag{13.6.7}
\end{equation}
On one hand, the first term of
(13.6.7) can be put in the form
\(\frac{d}{dt}\frac{\partial}{\partial \dq_i} \tfrac{1}{2} \vel_P^2\) found in Lagrange kinematic formula
(13.2.1). On the other hand, the second term
\(\vel_P \cdot \frac{d}{dt} \pvel_P^{q_i}\) is problematic since
\(\pvel_P^{q_i} \neq \frac{\partial \br_{OP}}{\partial q_i}\text{.}\) In fact it is easy to show that
\(\frac{d}{dt} \pvel_P^{q_i}\) cannot possibly be equal to
\(\frac{\partial}{\partial q_i} \vel_P\text{.}\) This shows that it is
not possible to evaluate the integral
\begin{equation*}
\int_\cB \ba_P \cdot \frac{\partial\vel_P}{\partial \dq_i} dm
\end{equation*}
as
\begin{equation*}
\frac{d}{dt}\left(\frac{\partial \kin_{\cB/\cE}}{\partial \dq_i}\right) - \frac{\partial \kin_{\cB/\cE}}{\partial q_i}
\qquad (i=1,\ldots,m)
\end{equation*}
\(\danger\) This makes non-holonomic systems fundamentally different from holonomic systems in the Lagrangian formalism: it is not possible to derive Lagrange equations for non-holonomic systems with respect to the reduced set of independent coordinates \((q_1, q_2, \cdots , q_m)\text{.}\)
We verify this important fact by re-examining the solution of
Example 13.4.3.
Example 13.6.1.
Derive the three Lagrange equations
\(\cL^q_{1/0}\) governing the motion of the sphere of
Example 13.4.3 for
\(q=\psi,\theta,\phi\) by taking into account the no-slip constraint equations. Show that these equations are meaningless.
Solution.
We can express \((\dx, \dy)\) in terms of \((\dpsi, \dte , \dphi)\) by relating \(\vel_{G/0}\) to \(\vel_{I\in 1/0} = \bze\text{:}\)
\begin{equation*}
\vel_{G/0} = \dx\bx_0+\dy\by_0 = -r (\dte \bv + \dphi \sin\te \bu)
\end{equation*}
leading to the non-holonomic equations
\begin{gather*}
\dx = r\dte \sin\psi - r\dphi\sin\te\cos\psi\\
\dy = -r\dte \cos\psi - r\dphi\sin\te \sin\psi
\end{gather*}
The kinetic energy \(\kin_{1/0}\) can now be expressed in terms of \(q=\psi, \te, \psi\) and their time-derivative
\begin{align*}
2\kin_{1/0}\amp
= m r^2 (\dte \bv + \dphi \sin\te \bu)^2 + \tfrac{2}{5} mr^2 ( \dpsi \bz_0 + \dte \bu + \dphi \bz_1)^2\\
\amp = \tfrac{2}{5}mr^2 \left(\dpsi^2 + \tfrac{7}{2}\dte^2 + \dphi^2 (1+ \tfrac{5}{2} \sin^2 \te)
+ 2 \dpsi\dphi \cos\te \right)
\end{align*}
Furthermore, if we take into account the no-slip velocity at \(I\text{,}\) the power coefficients \(\qQ^q_{\bar{1}\to 1/0}\) vanish for \(q= \psi, \te, \psi\text{.}\) The 3 Lagrange equations would then take the expression
\begin{align*}
\cL^{\psi}_{1/0}:\amp \qquad \dpsi + \dphi \cos\te = const. \tag{1}\\
\cL^{\te}_{1/0}:\amp \qquad \tfrac{7}{2} \ddte -\tfrac{5}{2} \dphi^2 \sin\te\cos\te+ \dpsi\dphi \sin\te = 0
\tag{2}\\
\cL^{\phi}_{1/0}:\amp \qquad \dphi (1+\tfrac{5}{2}\sin^2\te) + \dpsi \cos\te = const. \tag{3}
\end{align*}
Although equation
\(\cL^\psi\) is correct, equations
\(\cL^\te\) and
\(\cL^\phi\) do not match the results found in
Example 13.4.3. For instance, for equation (2) to match equation (4) of
Example 13.4.3 one would have to equate
\(F_v\) to
\(mr(\dphi^2 \sin\te\cos\te - \ddte)\text{.}\) It is possible to find the expression of
\(F_v\) from equations (1-7) of
Example 13.4.3:
\begin{equation*}
F_v = -mr (\ddte + \dphi\dpsi \sin\te) = mr ( \dphi^2\sin\te\cos\te-\ddte -C_1 \dphi\sin\te)
\end{equation*}
where \(C_1\) denotes the constant \(\dpsi + \dphi \cos\te\text{.}\) This last extra term leads to a discrepancy. This procedure is incorrect, as expected.
The correct way to proceed is to assume that the coordinates
\((q_1, q_2, \ldots, q_n)\) satisfy assumption
(13.1.2): hence the coordinates
\((\bq, \dbq)\) are assumed independent. However the virtual speeds
\((q_1^*, q_2^*, \ldots , q_n^*)\) which define the virtual velocity field
\(P\mapsto \vel_P^*\) as
\begin{equation}
\{ \cV^* \} = \dq_1^* \{\cV^{q_1}_{\cB/\cE} \} + \dq_2^* \{\cV^{q_2}_{\cB/\cE} \} + \cdots +
\dq_n^* \{\cV^{q_n}_{\cB/\cE} \}\tag{13.6.8}
\end{equation}
are no longer arbitrary elements of \(\mathbb{R}^n\text{,}\) but rather, are assumed to satisfy the homogeneous equations
\begin{equation}
\sum_{j=1}^n a_{ij} (\bq) \dq_j^* = 0, \qquad (i=1,2,\ldots, l)\tag{13.6.9}
\end{equation}
The virtual speeds
\((q_1^*, q_2^*, \ldots , q_n^*)\) are then said to be compatible with the non-holonomic constraints. Application of the Principle of Virtual Power with the choice
(13.6.8) for virtual velocity gives
\begin{equation}
\sum_{i=1}^n \dq_i^* \{ \cD_{\cB/\cE} \} \cdot\{\cV_{\cB/\cE}^{q_i}\}
=
\sum_{i=1}^n \dq_i^* \{\cA_{\coB \to \cB}\} \cdot\{\cV_{\cB/\cE}^{q_i}\}\tag{13.6.10}
\end{equation}
Given that the coordinates satisfy assumption
(13.1.2), the coefficients in the left-hand-side are unchanged, and are determined from the kinetic energy of
\(\cB\) by using Lagrange kinematic formula:
\begin{equation*}
\{ \cD_{\cB/\cE} \} \cdot\{\cV_{\cB/\cE}^{q_i}\} = \int_\cB \ba_P \cdot
\frac{\partial \vel_P}{\partial \dq_i} dm
= \left[\frac{d}{dt}\left(\frac{\partial}{\partial \dq_i}\right) - \frac{\partial}{\partial q_i}
\right] \kin_{\cB/\cE}
\end{equation*}
The right-hand-side of
(13.6.10) is the virtual power of external action
\(\{\cA_{\coB \to \cB}\}\)
\begin{equation}
\Pow^*_{\coB\to\cB/\cE} = \{\cA_{\coB \to \cB}\} \cdot\{\cV^*\} = \sum_{i=1}^n \dq^*_i \, \overline{\qQ}_{\coB\to\cB /\cE} ^{q_i} \tag{13.6.11}
\end{equation}
and gives the expression of power coefficients
\(\overline{\qQ}_{\coB\to\cB /\cE} ^{q_i}\text{.}\) Note that these power coefficients are in general not equal to the standard power coefficients
\({\qQ}_{\coB\to\cB /\cE} ^{q_i}\) found by assuming independent virtual speeds, since now the virtual speeds satisfy the equations
(13.6.9).
\begin{equation}
\sum_{i=1}^n \dq_i^* \left\{ \frac{d}{dt}\frac{\partial \kin_{\cB/\cE}}{\partial \dq_i}
- \frac{\partial \kin_{\cB/\cE}}{\partial q_i}
-\overline{\qQ}_{\coB\to\cB /\cE} ^{q_i} \right\} = 0\tag{13.6.12}
\end{equation}
To solve equation
(13.6.12) with the virtual speeds satisfying
(13.6.9), we use a theorem of linear algebra: a
linear form 1 is a linear combination of a finite number of linear forms
\(f_1\text{,}\) \(f_2\text{,...,}\) \(f_l\) if and only if its
kernel contains the intersection of the kernels of the
\(f_i\)’s. This implies that the linear form defined by
(13.6.12) is necessarily a linear combination of the
\(l\) linear forms defined by
(13.6.9): there must exist
\(l\) scalars
\(\la_1\text{,}\) \(\la_2\text{,}\) \(\ldots\text{,}\) \(\la_l\) such that
\begin{equation}
\sum_{i=1}^n \dq_i^* \left\{ \frac{d}{dt}\frac{\partial \kin_{\cB/\cE}}{\partial \dq_i}
- \frac{\partial \kin_{\cB/\cE}}{\partial q_i}
-\overline{\qQ}_{\coB\to\cB /\cE} ^{q_i} \right\} =
\sum_{k=1}^l \la_k \sum_{j=1}^n a_{ij} (\bq) \dq_j^*\tag{13.6.13}
\end{equation}
for all \(\dbq^* \in \mathbb{R}^n\text{.}\) This gives \(n\) Lagrange equations for rigid body \(\cB\) in the following form:
Theorem 13.6.2. Lagrange Equations with Multipliers.
The motion of a rigid body \(\cB\) relative to a Newtonian referential \(\cE\) defined by \(n\) coordinates \((q_1, \ldots, q_n)\) is governed by the \(n\) Lagrange equations
\begin{equation*}
\cL_{\cB/\cE}^{q_i}: \qquad \frac{d}{dt}\frac{\partial \kin_{\cB/\cE}}{\partial \dq_i}
- \frac{\partial \kin_{\cB/\cE}}{\partial q_i} =
\overline{\qQ}_{\coB\to\cB /\cE} ^{q_i} + \sum_{j=1}^l \la_j a_{ji} (\bq)
\end{equation*}
where the coordinates
\((q_1, \ldots, q_n)\) satisfy assumption
(13.1.2). The unknown real parameters
\(\la_1, \la_2, \ldots, \la_l\text{,}\) called
Lagrange multipliers, are associated with the
\(l\) non-holonomic constraint equations:
\begin{equation*}
\sum_{j=1}^n a_{ij} (\bq) \dq_j + b_i(\bq) = 0, \qquad (i=1,2,\ldots, l)
\end{equation*}
These equations are readily extended to the case of a system \(\Si\) of \(p\) rigid bodies by accounting for the power coefficients of interactions \(\overline{\qQ}_{j\leftrightarrow h}\) between pairs of rigid bodies:
\begin{equation*}
\cL_{\Si/0}^{q_i}: \qquad \frac{d}{dt}\frac{\partial \kin_{\Si/0}}{\partial \dq_i}
- \frac{\partial \kin_{\Si/0}}{\partial q_i} =
\overline{\qQ}_{\bSi\to\Si /\cE} ^{q_i}+ \sum_{1\leq j \lt h\leq p} \overline{\qQ}_{j\leftrightarrow h}^{q_i}
+ \sum_{j=1}^l \la_j a_{ji} (\bq)
\end{equation*}
In practice, Lagrange equations with multipliers are implemented as follows:
Identify the \(n\) coordinates \((q_1, q_2, \ldots, q_n)\) which parametrize the configuration of system \(\Si\text{.}\)
Identify all holonomic and non-holonomic constraint equations to be taken into account by the virtual velocity field.
2
Determine the kinetic energy of
\(\Si\) in a manner consistent with assumption
(13.1.2) satisfied by the coordinates: hence this must be done by ignoring all holonomic and non-holonomic equations.
Define the virtual velocity field \(\{\cV^*\}\) compatible with all holonomic and non-holonomic constraint equations.
For all external actions or interactions, find the corresponding virtual powers \(\Pow^*_{\bSi\to \Si/\cE}\) or \(\Pow^*_{j\leftrightarrow h}\) in terms of the virtual speeds \((\dq_1^*, \dq_2^*, \ldots, \dq_n^*)\) compatible with the constraints. The corresponding power coefficients are the coefficients in the virtual speeds of the virtual power.
Derive Lagrange equations by including \(l\) Lagrange multipliers to account for \(l\) constraint equations.
Once the Lagrange equations are determined, we retrieve all holonomic and non-holonomic constraint equations.
We first illustrate this implementation on the holonomic system examined in
Example 13.4.2.
Example 13.6.3.
Derive Lagrange equations
\(\cL^{x}_{1/0}\) and
\(\cL^{\te}_{1/0}\) for
Example 13.4.2 by considering virtual speeds compatible with the holonomic constraint equation governing coordinates
\((x,\te)\text{.}\) See
Figure 13.1.5.
Solution.
We know that coordinates \((x,\te)\) satisfy the geometric constraint
\begin{equation*}
x + h \cot\theta = \text{const.}
\end{equation*}
which can be differentiated w.r.t. time to give
\begin{equation*}
- \dx\sin\te + h \frac{\dte}{\sin \te} =0 \quad{(1)}
\end{equation*}
We choose virtual speeds \((\dx^*, \dte^*)\) compatible with this constraint, that is, such that
\begin{equation*}
- \dx^* \sin\te + h \frac{\dte^*}{\sin \te} =0 \quad{(2)}
\end{equation*}
Assuming that
\((x, \te)\) satisfy assumption
(13.1.2), the virtual velocity field takes the form
\begin{align*}
\{\cV^*\} \amp = \dx^* \{\cV^x_{1/0}\} + \dte^* \{\cV^\te_{1/0}\}\\
\amp =
\dx^* \begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_A + \dte^* \begin{Bmatrix} \bz_0 \\ \bze \end{Bmatrix}_A\\
\amp =
\dx^* \begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_Q + \dte^* \begin{Bmatrix} \bz_0 \\ \frac{h}{\sin\te}\by_1
\end{Bmatrix}_Q
\end{align*}
The kinetic energy
\(\kin_{1/0}\) takes the same expression found in
Example 13.4.2:
\begin{equation*}
2\kin_{1/0}= m \dx^2 - 2ml \dx\dte \sin\te+
\tfrac{4}{3}ml^2 \dte^2
\end{equation*}
We then find the virtual power of all external actions:
\begin{align*}
\Pow_{\bar{1}\to 1/0}^* \amp =
\begin{Bmatrix}F_A \bx_0 + N_A \by_0 \\\\ \bze \end{Bmatrix}_A
\cdot
\begin{Bmatrix} \dte^* \bz_0 \\\\ \dx^* \bx_0 \end{Bmatrix}_A
+
\begin{Bmatrix}-mg \by_0 \\\\ \bze \end{Bmatrix}_G
\cdot
\begin{Bmatrix} \dte^* \bz_0 \\\\ \dx^* \bx_0 + l \dte^* \by_1 \end{Bmatrix}_G\\
\amp \qquad + \begin{Bmatrix}N_Q \by_1 +F_Q \bx_1) \\\\ \bze \end{Bmatrix}_Q
\cdot
\begin{Bmatrix} \dte^* \bz_0 \\\\ \dx^* \bx_0+ \frac{h}{\sin\te} \dte^* \by_1 \end{Bmatrix}_Q \\\\
\amp = F_A \dx^* - mg l\dte^* \cos\te + N_Q (-\dx^*\sin\te + \frac{h\dte^*}{\sin\te}) +F_Q \dx^*\cos\te\\
\amp = F_A \dx^* - mg l\dte^* \cos\te + \dx^* F_Q \cos\te
\end{align*}
where the virtual power of normal force \(N_Q \by_1\) is seen to vanish due to equation (2) satisfied by the virtual speeds. We can now obtain the power coefficients from the expression of the virtual power:
\begin{align*}
\overline{\qQ}^x_{\bar{1}\to 1/0} \amp = F_A + F_Q \cos\te\\
\overline{\qQ}_{\bar{1}\to 1/0} ^\te \amp = -mgl \cos\theta
\end{align*}
The two equations \(\cL^{x}_{1/0}\) and \(\cL^{\te}_{1/0}\) are then found by introducing the Lagrange multiplier \(\la\) associated with constraint (2):
\begin{align*}
\cL^{x}_{1/0}:\amp \qquad m \ddx -ml \ddte \sin\te -ml \dte^2 \cos\te = F_A + F_Q \cos\te -\la\sin\te
\tag{3}\\
\cL^{\te}_{1/0}:\amp \qquad \tfrac{4}{3}ml^2 \ddte- ml \ddx \sin\te = -mgl \cos\theta +\la \frac{h}{\sin \te}
\tag{4}
\end{align*}
To equations (3) and (4) we now add equation (1): we have 3 equations to solve for 4 unknowns
\((x,\te,F_Q, \la)\text{.}\) We cannot use the relationship
\(|F_Q| = \mu N_Q\) without knowledge of
\(N_Q\text{.}\) Comparison with equations (1) and (2) of
Example 13.4.2 shows that the multiplier
\(\la\) can be identified as
\begin{equation*}
\la = N_Q
\end{equation*}
We can see with this example that Lagrange equations with multipliers are less than satisfying.
Although the multipliers can be determined, their physical interpretation is problematic in practice. The following examples illustrate Lagrange equations with multipliers for non-holonomic systems. These examples will show that the method has no advantage over the Lagrange equations without multipliers.
Example 13.6.4.
Derive the five Lagrange equations
\(\cL^q_{1/0}\) of
Example 13.4.3 for
\(q=x,y,\psi,\theta,\phi\) by taking into account the no-slip constraint equations with Lagrange multipliers. Interpret the physical meaning of the multipliers.
Solution.
The non-holonomic equations expressing the no-slip condition at
\(I\) were found in
Example 13.4.3:
\begin{gather*}
r \dphi \sin\te +\dx\cos\psi + \dy \sin\psi =0 \tag{1}\\
r\dte - \dx \sin\psi+ \dy\cos\psi = 0 \tag{2}
\end{gather*}
The virtual speeds \((\dx^*,\dy^*,\dpsi^*, \dte^*, \dphi^*)\) compatible with constraint equations (1-2) satisfy the equations
\begin{gather*}
r \dphi^* \sin\te + \dx^*\cos\psi + \dy^* \sin\psi =0 \tag{3}\\
r\dte^* - \dx^* \sin\psi + \dy^*\cos\psi =0 \tag{4}
\end{gather*}
The virtual velocity field \(\{\cV^*\}\) is then chosen as
\begin{align*}
\{\cV^*\} \amp = \dx^* \{\cV^x_{1/0}\} +\dy^* \{\cV^y_{1/0}\}+\dpsi^* \{\cV^\psi_{1/0}\} +\dte^* \{\cV^\te_{1/0}\}
+\dphi^* \{\cV^\phi_{1/0}\}\\
\amp = \begin{Bmatrix}\dpsi^* \bz_0 + \dte^* \bu + \dphi^* \bz_1 \\ \dx^* \bx_0 + \dy^* \by_0\end{Bmatrix}_G
=\begin{Bmatrix}\dpsi^* \bz_0 + \dte^* \bu + \dphi^* \bz_1 \\ \bze
\end{Bmatrix}_I
\end{align*}
where the last expression of \(\{\cV^*\}\) has been found by taking into account (3-4). We can now find the virtual power of all external actions:
\begin{equation*}
\Pow_{\bar{1}\to 1/0}^* =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg)\bz_0 \\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix}\dpsi^* \bz_0 + \dte^* \bu + \dphi^* \bz_1 \\ \bze
\end{Bmatrix}_I
= 0
\end{equation*}
This shows that all power coefficients
\(\overline{\qQ}^q_{\bar{1}\to 1/0}\) are zero. The kinetic energy
\(\kin_{1/0}\) takes the same expression as found in
Example 13.4.3
\begin{equation*}
2\kin_{1/0}
= m r^2 (\dx^2 + \dy^2) + \tfrac{2}{5} mr^2 ( \dpsi^2 + \dte^2 + \dphi^2 + 2 \dpsi\dphi \cos\te)
\end{equation*}
We can now write five Lagrange equations with two multipliers \((\la_1, \la_2)\) associated with constraint equations (3-4):
\begin{align*}
\cL^{x}_{1/0}: \amp \qquad m \ddx = \la_1 \cos\psi - \la_2 \sin\psi \tag{5}\\
\cL^{y}_{1/0}: \amp \qquad m \ddy = \la_1 \sin\psi + \la_2 \cos\psi \tag{6}\\
\cL^{\psi}_{1/0}: \amp \qquad \tfrac{2}{5} mr^2 \frac{d}{dt} (\dpsi + \dphi \cos\te) =0 \tag{7}\\
\cL^{\te}_{1/0}:\amp \qquad \tfrac{2}{5} mr^2 (\ddte + \dpsi\dphi \sin\te) = r\la_2 \tag{8}\\
\cL^{\phi}_{1/0}:\amp \qquad \tfrac{2}{5} mr^2 \frac{d}{dt} (\dphi + \dpsi \cos\te)= r \la_1 \sin\te
\tag{9}
\end{align*}
The seven equations (1-2) and (5-9) solves for seven unknowns
\((x,y,\psi,\te,\phi,\la_1,\la_2)\text{.}\) Comparison with equations (1-5) of
Example 13.4.3 shows that
\(\la_1 = F_u\) and
\(\la_2 = F_v\text{.}\) With this identification, Lagrange equations with and without multipliers then become identical: however, identifying the physical meaning of multipliers is an issue.
Example 13.6.5.
Derive the Lagrange equations with multiplier for Chaplygin sleigh of
Example 13.4.6 (
\(q = x,y,\theta)\text{.}\)
Solution.
The (non-integrable) non-holonomic constraint equation was found in
Example 13.4.6 to be given by
\begin{equation*}
\dy = \dx \tan\te \tag{1}
\end{equation*}
The virtual speeds \((\dx^*,\dy^*, \dte^*)\) compatible with constraint equation (1) must satisfy the equation
\begin{equation*}
\dy^*= \dx^* \tan\te \tag{2}
\end{equation*}
The virtual velocity field \(\{\cV^*\}\) is then chosen as
\begin{align*}
\{\cV^*\} \amp = \dx^* \{\cV^x_{1/0}\} +\dy^* \{\cV^y_{1/0}\} +\dte^* \{\cV^\te_{1/0}\} \\
\amp = \begin{Bmatrix} \dte^* \bz_0\\ \dx^* \bx_0 + \dy^* \by_0\end{Bmatrix}_C \\
\amp = \begin{Bmatrix} \dte^* \bz_0\\ (\dx^* \cos\te + \dy^* \sin\te) \bx_1\end{Bmatrix}_C
\end{align*}
where we have taken into account the constraint equation (2). The virtual power of all external actions is then given by:
\begin{align*}
\Pow_{\bar{1}\to 1/0}^* \amp = \begin{Bmatrix} \dte^* \bz_0\\ (\dx^* \cos\te + \dy^* \sin\te) \bx_1\end{Bmatrix}_C
\cdot
\begin{Bmatrix} X\bx_0 +Y \by_0 +Z \bz_0 \\ L\bx_0 +M\by_0 +N \bz_0 \end{Bmatrix}_C\\
\amp = N \dte^*+ (\dx^* \cos\te + \dy^* \sin\te) (X \cos\te + Y \sin\te) =0
\end{align*}
since \(N=0\) and \(X\cos\te + Y \sin\te =0\text{.}\)
Hence all power coefficients
\(\overline{\qQ}^q_{\bar{1}\to 1/0}\) are zero. Assuming that the variables
\((x,y,\te,\dx,\dy,\dte)\) are independent, the kinetic energy of the body takes the same expression as found in
Example 13.4.6:
\begin{equation*}
2 \kin_{1/0} =
m (\dx^2 + \dy^2) + 2m b \dte (-\dx\sin\te + \dy \cos\te) + I_C \dte^2
\end{equation*}
Hence we can now state that for all \(\dq^* = \dx^*, \dy^*, \dte^*\) satisfying (2)
\begin{equation*}
\sum_{i=1}^3 \dq_i^* \left[ \frac{d}{dt}\frac{\partial \kin_{1/0}}{\partial \dq_i}
- \frac{\partial \kin_{1/0}}{\partial q_i} \right]
= 0
\end{equation*}
or equivalently, that there exists a scalar \(\lambda\) such that
\begin{equation*}
\sum_{i=1}^3 \dq_i^* \left[ \frac{d}{dt}\frac{\partial \kin_{1/0}}{\partial \dq_i}
- \frac{\partial \kin_{1/0}}{\partial q_i} \right]
= \lambda (\dy^*- \dx^* \tan\te)
\end{equation*}
for arbitrary values of \(\dq^* = \dx^*, \dy^*, \dte^*\text{.}\) This leads to 3 equations:
\begin{align*}
\cL_{1/0}^x: \amp \qquad m \frac{d}{dt} (\dx- b\dte \sin\te) = - \lambda \tan\te \amp \tag{3}\\
\cL_{1/0}^y: \amp \qquad m \frac{d}{dt} (\dy+ b\dte \cos\te) = \lambda \amp \tag{4}\\
\cL_{1/0}^\te: \amp \quad I_C \ddte + mb(-\ddx\sin\te+\ddy\cos\te) = 0 \amp \tag{5}
\end{align*}
Equations (1) and (3-5) solve for the unknowns
\((x,y,\te,\lambda)\text{.}\) Comparison of equations (3-5) with equations (5-6) of
Example 13.4.6 shows that multiplier
\(\lambda\) can be identified with
\begin{equation*}
\lambda = Y = - X\cot\te
\end{equation*}
Once again, the physical identification of the multiplier can be safely made only by comparison with Lagrange equations without multipliers.
