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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.

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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.

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  • $\begingroup$ 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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