$ All sub-SYSTEMs have "surroundings", duh.
Sub-SYSTEMs are "submerged" in SYSTEM "working fluid" AMBiENT.
Sub-SYSTEMs ONLY EXCHANGE energy with "working fluid" AMBiENT.

Go-go Google GROUP SEARCH: My BiGGER bang ..My REAL STUFF.

TOTAL LaGrangian h*f Dis-iNFORMATiON follows with Tom Roberts.

brian a m stuckless

Tom Roberts wrote: Koobee Wublee wrote:
"Tom Roberts" wrote in message
...
L = \integral g_ij (dx^i/d\tau) (dx^j/d\tau) d\tau

Yes, the Lagrangian is buried inside there; the integral itself is
called the action.

we can discuss logically and clearly how Noether's Theorem applies to
indicate a universal phenomenon to the conservation of observed energy.

Except, of course, that Noether's theorem states quite explicitly that
energy is conserved ONLY for systems with a Lagrangian that is invariant
under time translations. shrug

And you never responded to the fact that for two interacting subsystems,
if conservation of energy is "universal", why isn't energy conserved
in each subsystem? The answer is, of course, that as long as the
interactions between a subsystem and its surroundings are independent of
time then energy will be conserved in the subsystem; but if the
interactions vary over time then the subsystem's Lagrangian won't be
time independent and energy in the subsystem won't be conserved. That
_IS_ what Noether's theorem says about such a case. shrug

As I said before, before Noether this was a puzzle[#], and
ad hoc rules had to be established to save energy
conservation. So "... for an isolated system that does
not interact with its surroundings" had to be added to
the "principle of conservation of energy".
Not very "universal" is it?

[#] Admitedly one that not many people worried about. It
was considered self-evident that energy won't be conserved
in a subsystem that interacts with its surroundings.

[...]

Tom Roberts