The author solves the following problems:
i) Suppose one has a positional, but not conservative force field acting on a point P, further one imposes a perfect constraint to P; then the question arises how to find conditions for the conservability of the force field acting on such a point P. It is well known that if the constraint is formed by a regular curve \(\gamma\), the force field becomes conservative automatically on this curve. If the constraint is formed by a regular surface \(\sigma\), the author give the following necessary and sufficient condition in order that the force field becomes conservative on \(\sigma\) : rot \(\vec f\cdot {\vec \nu}=0\), where \({\vec \nu}\) is the normal to \(\sigma\).
ii) Suppose to have a positional, but not conservative force field, acting on a point P subjected to the anholomic perfect constraint: \(a(x,y,z)dx+b(x,y,z)dy+c(x,y,z)dz=0\), where the first member is a non- integrable form. The author gives the necessary and sufficient condition in order that the force field on P subjected to the above constraint becomes conservative.
iii) Suppose to have a discrete system S of N material points, subjected to a positional but not conservative global collection of forces \(\{\vec f_ h\}\) \(h=1,...,N\) with anholonomic, perfect constraints. The author gives the necessary and sufficient condition in order that the global force field acting on the system S becomes conservative. If in particular each single force \(\vec f_ h\) only depends on position of its application point, the above necessary and sufficient condition is strongly simplified.
For these three cases many applications are given for the existence of the first integral of the energy.