Meirovitch says in his "Principles and Techniques of Vibrations" (1997) on p.85:

In the case of holonomic systems, the variation and integration processes are interchangeable (...)

which means that

$\delta \int_{t_1}^{t_2} L(q, \dot{q}) dt = \int_{t_1}^{t_2} \delta L(q,\dot{q}) dt$

subject to

$f(r_1(q), r_2(q),...,r_N(q),t) = 0.$

where $L$ is the Lagrangian, $q$ are generalized coordinates, $r_i$ are the coordinates of points of mass in the configuration space and $f$ is the constraint.

Can anybody tell me why?

  • $\begingroup$ If you like this question, you may also enjoy reading this Phys.SE post. $\endgroup$
    – Qmechanic
    Nov 6, 2014 at 12:51
  • $\begingroup$ Thanks a lot for the link to the discussion about commutativity of the $d/dt$ and $\delta$ operator! It does not (seem to) answer my question, though. How come that $\delta \int L dt = \int \delta L dt$ holds and what has holonomy got to do with it? $\endgroup$ Nov 6, 2014 at 13:55
  • $\begingroup$ interchange has to do with uniform convergence (having to do with path-independence, thus holonomic vs non-holonomic systems which are path-dependent) $\endgroup$
    – Nikos M.
    Nov 6, 2014 at 17:29
  • $\begingroup$ @NikosM.: Thanks a lot for the hint! I'll investigate into that.. $\endgroup$ Nov 6, 2014 at 22:33


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