A question regarding the resonant circuits
Should the power source negative be connected to the circuit or not.

As far as resonance is concerned it doesn't matter. Either of your two circuits will exhibit resonance.
Resonance requires that energy continually moves backwards and forwards between the coil (where the energy is stored in a magnetic field) and the capacitor (where it's stored in an electric field). The LC circuit may or may not be connected to earth, as in your two diagrams. (In turn earth may or may not be connected to the circuit negative power rail.)
A practical circuit will need some way of getting energy into and out of the LC circuit (also known as a tank circuit, because it stores energy). Many different circuits have been devised to do this, some of which connect earth or a power rail to different points of the tuned circuit. However none of these variations change the basic resonance principle of energy moving between the coil and the capacitor at a rate determined by the values of their inductance and capacitance.