Lagrangian dynamics by nonlocal constants of motion (Q827474)

In this paper, a method to find first integrals for the extended Euler-Lagrange equation \[\frac{d}{dt}\partial_{\dot q}L-\partial_{q}L=Q,\] where \(L=L(t,q(t),\dot{q}(t))\) is the Lagrangian and \(Q=Q(t,q(t),\dot{q}(t))\), is given. This method is used in some applications in the variational case (\(Q=0\)) and also in the non-variational case. In Section 2, the case of potentials with simple symmetries is considered. In Section 3, first integrals for a variational mechanical system of a point moving in a potential field, which is positively homogeneous of degree \(-2\), are obtained. In Section 4, the method is used to prove the global existence of solutions of the equation of motion of a particle under a bounded from below potential and viscous fluid resistance in the past. In Section 5, a non-variational case is considered, namely the equation for hydraulic resistance in a bounded potential field. It is proved that, under some conditions, all the solutions explode in the past in finite time. In Section 6, the Maxwell-Bloch equations, which describe laser dynamics for a system of two-level atoms in a cavity resonator, are considered.