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I mention this because atwith stuff like this ... stuff that doesn't pass the giggle test.
http://tabitha.phas.ubc.ca/wiki/index.php/Phase_Space
we have a detailed calculation using canonical mechanics to "prove" that
E/É÷ is invariant for a pendulum where the mass is changing because the
bob is a box of sand with a slow leak. It also "proves" that you can't
increase the energy of a swingset (at constant É÷) by slow parametric
pumping.
And that page is just one example among many. The literature is crawling