Hi Ashwin. Sorry for the late reply. It may be a problem of definitions. Mathematically, canonical coordinates are those for which the canonical one-form can be written as p_i dq^i. See, for example, https://en.wikipedia.org/wiki/Canonical_coordinates . So, yes pdx-Hdt is preserved, but then the canonical one-form is not, since it’s only the first part. The other issue here is that t is a parameter, and not a coordinate. So p dx – H dt, from the perspective of differential geometry, is not really well defined. If you do, then note how you have (p, -H) \cdot (dx, dt), which is quasi-relativistic.

As for invariance of Lagrangian, I don’t follow. We are talking about coordinate transformations only, so q to Q. And you are saying that a Lagrangian is invariant over all of those… What is the issue?

And, yes: relativity could have been discovered through particle mechanics instead of field theories. People were not looking there, though: they had no reason to.

]]>Here’s the first thing on google I found with an example, http://www.srl.caltech.edu/phys106/p106b01/topic2.pdf at p 9 ish. They have a generating function for the galileian boost.

You also write “What they do is start from the Lagrangian, which is invariant under all transformations…” but this isn’t true. Lagrangians are invariant under all point transformations, which is a subset of all possible transformations in phase space. In a point transformation, you change q to Q, and dQ/dt is derived from this transformation. A general phase space transformation is less demanding, allowing the momenta p to change independently of the Q. All point transformations are canonical, and the Galileian boost should be no exception.

Maybe there’s something I’m missing here, but it strikes me as extremely strange that Galileian boosts could be non canonical – people surely would have thought of relativity much sooner!

]]>Yes! See: Can Hamiltonian mechanics describe dissipative systems?

]]>Does this post exist already?

]]>I think the cat is still white. You are blindfolded and the cat is in the box, so you simply poured the dye on the box.

Did I get it right? 😛

]]>– put a white cat in a box with the usual stuff (a Geiger counter, a tiny bit of radioactive substance, a small flask of hydrocyanic acid…)

– blindfold myself

– then randomly pick a dye out of a 2-piece set that includes a blonde dye and a black dye

– pour the dye over the cat

…what are the chances that the cat is blonde AND alive? ;p

Keep up with the good work, always a pleasure to read your posts!

]]>Hi Peter,

Time travel inconsistencies are with thermodynamics, which is not time symmetric. Conservation of energy is just one aspect (i.e. isolated systems conserve energy).

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