Given a mechanical system with degrees of freedom $q=(q_1,\cdots,q_n)$, whose dynamics is given by

$$\ddot{q}_i = f_i(q,\dot{q},t),\tag{1}$$

for $i=1,\cdots,n$, are there criteria on the $f_i$'s which ensure the existence of a Lagrangian? I mean, of course, that the system (1) would be the Euler-Lagrange equations for that Lagrangian. iirc for $n=1$, it has been known since the 19th century, that there is always a Lagrangian. But what is known for more than one degree of freedom?

Actually, it comes to think of it that the system (1) might not be the Euler-Lagrange equations in general. Instead, we could have functions $a_{ij}(q, \dot{q},t)$ such that

$$a_{ij}(\ddot{q}_j-f_j)=\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i},$$

with a summation on repeated indices.


This is known as the inverse problem for Lagrangian mechanics. There exist necessary and sufficient Helmholtz conditions (for details, see Wikipedia), which are usually very difficult to work with.

  • $\begingroup$ Thanks but I am puzzled by this Wikipedia page: at face value, what they refer to as Douglas theorem is presented as valid for any $n$ but I checked Douglas' paper and he proves it only for $n=2$. Is there nothing proved for higher values of $n$? $\endgroup$ – user154997 Sep 18 '17 at 22:49
  • $\begingroup$ I haven't had time to read Douglas' paper in detail. As for Helmholtz conditions, they seem to be discussed in Part II, $\S4+\S5$, p. 81-84, for general $n$. $\endgroup$ – Qmechanic Sep 24 '17 at 13:07
  • $\begingroup$ Afaiu §4 and §5 are concerned with proving the equivalence between Euler-Lagrange equations and the system of equations $\mathfrak{S}$ i.e. (4.7)-(4.10), which is only an intermediate step to the kind of result I sought, i.e. eqn. (1.3) in Douglas paper. $\endgroup$ – user154997 Sep 25 '17 at 8:09
  • $\begingroup$ As the Wikipedia page indicated, the Davis solution is on manifold and the most difficult part is proving the differential $1$-form $g_{i}$ in dimensions $n>2$ is closed. For $n>2$, you should try posting on math.stackexchange.com - they may know some tricks for proving higher dimensional differential $1$-forms are closed - assuming $H(1)$ and $H(2)$ are doable. In short, it's a math question. $\endgroup$ – Cinaed Simson Oct 24 '19 at 18:29

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