# Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 \dots m$. Then it also holds that $\sum \lambda_\alpha f_\alpha = 0$. Using Hamilton's principle (i.e. that the action must be stationary), we get that

$$\delta \int_1^2 L\ dt = \int_1^2 dt\ \sum_{k=1}^n \left(\frac{\partial L}{\partial q_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q_k}}\right)\delta q_k = 0.$$

But we can't get Lagrange's equations from this because the $\delta q_k$ aren't independent. However, if we add this with $\sum \lambda_\alpha f_\alpha = 0$, it follows that

$$\delta \int_{t_1}^{t_2} \left(L +\sum_{\alpha=1}^m \lambda_\alpha f_\alpha\right)\ dt = 0.$$

And then Goldstein says that

The variation can now be performed with the $n\, \delta q_i$ and $m\, \lambda_\alpha$ for $m+n$ independent variables.

Why have the variables suddenly become independent? First we had $n$ dependent variables, why do we now have $m+n$ independent ones?

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Let there be $n$ coordinates $q^j$. Ref. 1 is discussing in Section 2.4 a type of non-holonomic constraints that is known as semi-holonomic constraints. However we interpret OP's question (v2) as mostly being about counting independent degrees of freedom in constrained systems, and not so much about semi-holonomic constraints per se. Therefore, to gain intuition, let us for simplicity just consider $m$ holonomic constraints

$$\tag{A}f_{\alpha}(q)~=~0,$$

where $m\leq n$ (and where we have suppress possible explicit time dependence in the notation). Granted some regularity assumptions, we may in principle solve the $m$ constraints (A) locally so that the coordinates

$$\tag{B}q^j~=~g^j(\xi, \varphi)$$

become functions of $n-m$ independent physical coordinates $\xi^a$, and $m$ coordinates $\varphi^{\alpha}$, in such a way that locally the $n-m$ dimensional constraint surface

$$\tag{C}\{q\in\mathbb{R}^n|f(q)=0\}$$

is parametrized as

$$\tag{D}\{g(\xi, \varphi=0)\in\mathbb{R}^n| \xi\in \mathbb{R}^{n-m}\}.$$

Thus we have at least two equivalent variational formulations:

1. Reduced formalism: Replace $q^j$ with $g^j(\xi, \varphi=0)$ in the action $S[q]$. Vary the corresponding action $S[\xi]$ wrt. the $n-m$ independent variables $\xi^a$.

2. Extended formalism: Replace the action $S[q]$ with $$\tag{E} S[q,\lambda]=S[q]+\int\!dt~\lambda^{\alpha}f_{\alpha}(q).$$ Vary the corresponding action $S[\xi,\lambda]$ wrt. the $n+m$ independent variables $q^j$ and $\lambda^{\alpha}$.

The role of the $m$ Lagrange multipliers $\lambda^{\alpha}$ can be view as putting the $m$ variables $\varphi^{\alpha}=0$, so that only the $n-m$ physical variables $\xi^a$ remains, and formulation (2) reduces to (1).

References:

1. H. Goldstein, Classical Mechanics, Section 2.4.
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