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I think the consensus is Lagrangian methods alone (or even at all) will not help you design a filter until you have desired properties somewhat defined by some category like the shape of amplitude, phase, group delay for the passband and stopband.

You need to research more about filter design from Zverev and coupled oscillators to understand your question is unanswerable.

Now keep your hand raised for the Professor until the oscillation stops from this mechanical version of a coupled dual resonator.

Please read my tagline below which only applies to people who have had bad experiences from forums getting succinct accurate responses.