Lagrange equations for non-holonomic monogenic system

Question

For monogenic and a special case of non-holonomic constraints where we have$$ \sum_{k} a_{l k} d q_{k}+a_{t t} d t=0 \tag{2-20} $$ we use lagrange multipliers and hamiltons principle to reach the below equation:

Goldstein, 2 edition

$$\frac{d}{d t} \frac{\partial L}{\partial \dot{q}_{k}}-\frac{\partial L}{\partial q_{k}}=\sum_{l} \lambda_{l} a_{l k}, \quad k=1,2, \ldots, n.\tag{2-29}$$

And he says further

What is the physical significance of the $\lambda_{i}$ 's? Suppose one removed the constraints on the system, but instead applied external forces $Q_{k}^{\prime}$ in such a manner as to keep the motion of the system unchanged. The equations of motion would likewise remain the same. Clearly these extra applied forces must be equal to the forces of constraint, for they are the forces applied to the system so as to satisfy the condition of constraint. Under the influence of these forces $Q_{k}^{\prime}$, the equations of motion are $$ \frac{d}{d t} \frac{\partial L}{\partial \dot{q}_{k}}-\frac{\partial L}{\partial q_{k}}=Q_{k}^{\prime}.\tag{2-31} $$ But these must be identical with Eqs. (2-29). Hence we can identify $\sum \lambda_{1} a_{l k}$ with $Q_{k}^{\prime}$, the generalized forces of constraint.

Question: How does it follow that "Under the influence of these forces $Q_{k}^{\prime}$, the equations of motion are eq. (2-31)?"

Since we've a monogenic system, potential maybe a function of velocities $V=V(r,\dot r,t)$ therefore replacing constraints by equivalent forces and using d'Alemberts principle we might get $$\frac{d}{d t}\left(\frac{\partial T}{\partial \dot{q}_{j}}\right)-\frac{\partial T}{\partial q_{j}}=Q_{j}$$ and not eq. (2-31).

Can anyone shed some light please.

Answer 1

1. Goldstein is talking about semi-holonomic constraints; not general non-holonomic constraints, such as, e.g. inequalities.

2. Goldstein's treatment of semi-holonomic constraints via a stationary action principle has already been criticize in e.g. Ref. 3. and this & this related Phys.SE posts.

3. To be specific, let us from now on only consider the 2nd edition of Goldstein.

4. We should stress that the result (2-29) is correct. For a generalization, see e.g. this Phys.SE post.

5. One may consider the constraint forces $\sum_{l} \lambda_{l} a_{l k}$ on the RHS of eq. (2-29) as examples of generalized forces, cf. eq. (2-31). This point is possibly one of OP's doubts.

6. It seems Goldstein implicitly and unnecessarily assumes that all the non-constraint forces are monogenic, i.e. that they have generalized (possibly velocity-dependent) potentials, cf. eq. (1-58).

7. OP seems to mistakenly assume that the constraint forces are also monogenic, and/or that all forces are constraint forces. Both are not the case.

8. To prove eq. (2-29) we need to establish eq. (2-23), or more generally, eq. (1-52).

9. Goldstein is now criticized for using Hamilton's principle (2-2). The issue is that one cannot just enforce the semi-holonomic constraints via Lagrange multipliers in an extended stationary action principle, cf. point 2.

10. We will instead rely on d'Alembert's principle (1-45). A minor modification of eqs. (2-24)-(2-28) and the surrounding arguments then leads to the result (2-29).

References:

1. H. Goldstein, Classical Mechanics, 2nd edition; Section 2.4.

2. H. Goldstein, Classical Mechanics, 3rd edition; Section 2.4.

3. M.R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. 73 (2005) 265.

Answer 2

EL with non holonomic constraint equations \begin{align*} &\text{you have $~n_w~$non holonomic constraint equations}\\ &F_{w}=F_w({\dot{q}}_1~, q_1~,\ldots~,{\dot{q}}_s~, q_s)=0~,w=1..n_w\tag 1\\\\ &\text{EL equations with Langrange multiplier $~\lambda _w$ }\\ &\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=\underbrace{\sum_{w=1}^{n_w} a_{(i,w)}\,\lambda_w}_{Q_i}~,i=1..s\tag 2\\ &\text{where}\\ &a_{(w,i)}=\frac{\partial F_w}{\partial \dot q_i} \end{align*} Instead of Eq. (1) and (2) you applied the EL with external forces

\begin{align*} &\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=\underbrace{\sum_{j=1}^{s}\frac{\partial r_i}{\partial q_j}f_j}_{\bar Q_{i}}~,i=1..s\tag 3\\ &\text{where $~f_j~$ are the non conservative external force components}~f_j=f_j(\mathbf q~,\mathbf{\dot{q}})\\ &\text{and $~r_i~$ the position vector components} \end{align*} notice that the conservative forces are equal to $-\frac{\partial U}{\partial q_i}$ where U is the potential enegry

Eq. (3) is equal to Eq. (1) and (2) if $~\bar Q_i=Q_i~$ and the constraint equation (1) is fulfilled, this is what Prof. Golstein wrote ?