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The Hamilton action $S$ is defined as following: $$S=\int^T_0 L(q,\dot{q})dt$$ the integral along any actual or virtual (conceivable or trial) space-time trajectory q(t) connects two specified space-time events, initial event t=0 and final event t=T , And the principle of least action is $$ \delta S=0$$ There are questions:

Can any procedure satisified by the principle be implemented in real physics world?

Is any process in real physics world able to be discribed with the principle?

Is any procedure satisified by the principle able to be simulated by Turing Machine ( computed by Turing Machine or appoximated numerically to arbitary precision by Turing Machine) ?

Any reference is appreciated

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    $\begingroup$ The second question is out of scope, the first one is likely not in scope but it might be. The third question requires clarification as to what is your model of computability. In any case, you might be interested in worldscientific.com/worldscibooks/10.1142/8306#t=aboutBook $\endgroup$
    – Andrej Bauer
    Commented Sep 4 at 15:39
  • $\begingroup$ Yes and thanks, those questions have to be posted on mathoverlfow, physics, but they will refuse or have refused to accept the post. :D $\endgroup$
    – XL _At_Here_There
    Commented Sep 5 at 3:35

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Concerning my limited understanding of your last question and my interpretation of Alan Turing's outlook: the answer leans towards 'no'.

More generally, we live in an uncountable physical world (according to many but not all of us) and there are only countably many Turing machines.

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  • $\begingroup$ I don't know who votedown your answer and my question, does the votdowner understand the question and the answer? $\endgroup$
    – XL _At_Here_There
    Commented Sep 5 at 3:29
  • $\begingroup$ Hao Wang's book Reflections on Godel has discussed some aritcles and works that do some related researches. But those works and articles have not answered the questions. $\endgroup$
    – XL _At_Here_There
    Commented Sep 5 at 3:32
  • $\begingroup$ The inconvenient truth for some theorists is that not everything is a Turing machine. Suggesting that logic is limited w.r.t. the physical world, as you do in your questions, makes them vote down your contribution and follow-ups. They want to dictate the do's and don'ts of engineers and physicists without listening to counter-arguments coming from people versed in both logic and engineering. It so happens that Alan Turing would not have endorsed a "one size fits all" mentality in theoretical discourse. $\endgroup$
    – Edgar Daylight
    Commented Sep 5 at 7:09
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    $\begingroup$ Uncountability does not imply non-computability. $\endgroup$
    – Andrej Bauer
    Commented 2 days ago
  • $\begingroup$ Please read the sources (which takes days, if not weeks) to avoid misquoting Edward A. Lee's book on uncountability and computability. $\endgroup$
    – Edgar Daylight
    Commented 2 days ago

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