A linear differential form $\sum_{i}\mathcal{E}_{i}(q)\, dq_{i}$ is an exact differential if the conditions $\partial\mathcal{E}_{i}(q)/\partial q_{j}=$ $\partial\mathcal{E}_{j}(q)/\partial q_{i}$ are met for any $i, j$.

Corresponding closed exact differential form is equivalent to a linear first order homogeneous differential equation: $$\sum_{i}\mathcal{E}_{i}(q)\frac{dq_{i}}{d\tau}=0;$$ for which, in case when the potential function $E(q)$ for the corresponding form, such that: $$\mathcal{E}_{i}(q)=\partial E(q)/\partial q_{i};$$ is known, the general one-parametric solution is simply: $$E(q_{1},...q_{n})=Const;$$

What can we say about the solution of this differential equation when the explicit potential function is not known, can not be given in a closed explicit form (does not exist)?
Can we still expect that there exist some implicit form of a general solution, as a one-parametric family of relations between variables $q_{i}$?

If so, than how to find / build those solutions?

I appreciate your suggestions, specific references.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.