I've been doing a bit more searching on this in my spare time and came across a comment regarding Jacobi's integral in

**Classical Dynamics**, by Donald T. Greenwood,

*Dover Pub.* (1977). On page 73 he defines a

*natural system* as a

*conservative* system which has the additional properties

(1) it is described by the standard holonomic form of Lagrange's equations

(2) the kinetic energy is expressed as a homogeneous quadratic function of the generalized velocities.

He then states that under these circumstances Jacobi's integal, aka the energy function, is the total mechanical energy of the system and is an integral of motion, i.e. constant.

Nice!