There is another way around, starting with Lagrange, but bypassing his variational construction to reach immediately the presymplectic construction. Considering the motion of a point submitted to a force $F$, you can define the following $2$-form(${}^*$) $\omega$ on
the space $Y$ of initial conditions $(x,v,t) \in {\bf R}^3 \times{\bf R}^3 \times {\bf R}$
$$
\omega(\delta y)(\delta'y) = \langle\delta v - F\delta t , \delta' x - v\delta' t \rangle - \langle \delta' v - F\delta' t , \delta x - v\delta t \rangle,
$$
where $\delta y = (\delta x, \delta v, \delta t) \in T_yY$, idem for $\delta'y$. You can notice that the kernel of this 2-form is generated by the equations of motion:
$$
\ker(\omega_y) = \{ \delta y \mid \delta x = v \delta t \quad \mbox{and} \quad \delta v = F \delta t\}.
$$
In other words, the motions of the point submitted to the force $F$ are the integral curves of the vectorial distribution $y \mapsto \ker(\omega_y)$.

You can check that if $F$ does not depend on $v$ and if $F$ is derived from a potential $F = - dU$ then $\omega$ is closed, therefore presymplectic, and the space of characteristics of $\omega$, that is, the solutions of your dynamical system, is then equipped with a symplectic structure.

Now we can rephrase your question in the presymplectic framework: *what are the condition on $F$, depending a priori from $y=(x,v,t)$, for $\omega$ to be closed?*

First of all you can notice that the form $\omega$ writes
$$
\omega = \omega_0 \oplus F \cdot dx \wedge dt,
$$
where $\omega_0$ is the standard symplectic form on ${\bf R}^3 \times {\bf R}^3$. The condition for $\omega$ to be closed writes then $d[F\cdot dx] \wedge dt = 0$. In other words we need $d[F \cdot dx]$ to be proportional to $dt$. If you make the computation you'll find that the partial derivative $\partial F_i / \partial v^j$ must vanish and for all $t$, $d[x \mapsto F_t(x)] = 0$, that is, $F_t = -dU$, where $U$ depends on $(x,t)$.

Thus if you add a dissipative term in your force, you lose the symplectic structure on the space of solutions, and therefore all the conservation theorems associated.

(*) This construction is due to Cartan, Galissot, Souriau.

Research-level? – Dimensio1n0 Oct 28 '13 at 11:14