Classical Mechanics – Validity of Lagrange’s Equations According to Thornton and Marion

Question

I am a bit confused about the 2nd statement below from Thornton and Marion 7.4:

It is important to realize that the validity of Lagrange's equations requires
the following two conditions:

1. The forces acting on the system (apart from any forces of constraint) must
   be derivable from a potential (or several potentials).
2. The equations of constraint must be relations that connect the coordinates of
   the particles and may be functions of the time—that is, we must have
   constraint relations of the form given by $$f_k(x_{\alpha,i},t)=0.\tag{7.9}$$

If the constraints can be expressed as in condition 2, they are termed holonomic
constraints. If the equations do not explicitly contain the time, the constraints
are said to be fixed or scleronomic; moving constraints are rheonomic.

Do they mean only under this condition the method of lagrange multiplier can be applied?

How can Lagrange's equations be "invalid"?

Answer 1

• It should be stressed that Thornton & Marion take the stationary action principle/Hamilton's principle as a first principle.

  They therefore need that:

  1. all forces are implemented via potentials.

  2. The position coordinates ${\bf r}_{\alpha}={\bf r}_{\alpha}(q,t)$ are expressed via the generalized coordinates, or equivalently, the position coordinates ${\bf r}_{\alpha}$ satisfy holonomic constraints.$^1$

• However, there are more general first principles where 1 & 2 are violated. One may e.g. derive Lagrange equations from d'Alembert's principle, cf. e.g. this Phys.SE post.

  Note that even d'Alembert's principle may fail.

--

$^1$ There are various issues with non-holonomic constraints, cf. e.g. this Phys.SE post.