Problems that Lagranges equations of the 1st kind can solve whereas the 2nd kind can't?

Question

Can anyone give examples of mechanics problems which can be solved by Lagrange equations of the first kind, but not the second kind?

Answer 1

1. Let us call the equations

$$ \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial L}{\partial \dot{q}^j}\right)-\frac{\partial L}{\partial q^j}~=~0 \tag{s}\label{eq:s}$$ for Lagrange equations of second kind in strong sense, and let

$$ \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial L}{\partial \dot{q}^j}\right)-\frac{\partial L}{\partial q^j}~=~Q_j\tag{w}\label{eq:w}$$

be Lagrange equations of second kind in weak sense, where $Q_j$ are the generalized forces that don't have generalized potentials.

2. A Frictional force like

$${\bf F}_f ~=~- k{\bf v}$$

can be modeled in Lagrangian mechanics using Rayleigh's dissipation function

$${\cal F}~=~ \frac{k}{2} {\bf v}^2.$$

It will not be Lagrange equations of second kind in the strong sense $\eqref{eq:s}$, but only in the weak sense

$$ \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial L}{\partial \dot{q}^j}\right)-\frac{\partial L}{\partial q^j}~=~ -\frac{\partial{\cal F} }{\partial \dot{q}^j}. $$

3. Another source of examples are non-holonomic constraints. This is, in particular, true if one insists on the strong form $\eqref{eq:s}$.

It is a bit more tricky if one allows the weak form $\eqref{eq:w}$, and the $m$ non-holonomic constraints are on semi-holonomic form, say for simplicity of the form

$$ \sum_i a_i(q,t)~{\rm d}q^i + a_t(q,t)~{\rm d}t ~=~ 0, $$

where the functions $a_i(q,t)$ and $a_t(q,t)$ do not depend on the generalized velocities $\dot{q}^j$.

Then, it is possible to introduce $m$ Lagrange multipliers $\lambda^\alpha$ so that the $n$ Lagrange equations become of second kind in the weak form $\eqref{eq:w}$. However, that is not the full set of equations. Together with the $m$ constraints

$$ \sum_i a_i(q,t)~\dot{q}^i + a_t(q,t) ~=~ 0, $$

the full system will have $m+n$ equations, corresponding to $n$ $q$'s and $m$ $\lambda$'s.

Reference:

1. Herbert Goldstein, Classical Mechanics, Chapter 1 and 2.