Is a Lagrangian unique in the same field? Is Hamiltonian unique?
If it is unique then please explain why is it so and if it is not then please explain why is it not so.
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$\begingroup$ Also please specify a time parameterized function $\endgroup$– Debarghya AdhikariCommented Sep 18, 2019 at 6:58
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$\begingroup$ Related: physics.stackexchange.com/q/174137/2451 , physics.stackexchange.com/q/87628/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Sep 18, 2019 at 8:38
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1 Answer
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The system is described by the Euler-Lagrange equations. So basically the Lagrangian is the description of the system. However the Euler-Lagrange equations can be the same for different Lagrangians. The field will be the solution of the Euler-Lagrange equations. They are differential equations so the boundary conditions can change the solution.
Now you can go from a Lagrangian description to a Hamiltonian via Laplace transform which is an one to one process.
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2$\begingroup$ I downvotd this answer, because it contains a wrong statement. Lagrangians that differ only by a "total derivative of time" (gauge-term) yield the same equations of motion. $\endgroup$ Commented Sep 18, 2019 at 8:17
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$\begingroup$ Ok, my statement was that the ELeqs may be the same for different fields. What is wrong about that? Sure you added an important detail. But it doesn't seem like a said something wrong. $\endgroup$– jmstf94Commented Sep 18, 2019 at 8:20
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$\begingroup$ You wrote "Different Lagrangian means different field" (I assume you also meant different equations, since as you pointed out, even one equation of motion can yield different fields due to different boundary condicitons). However (as I wrote), Lagrangians that differ by a gauge-term yield the same equations, and can thus also produce the same fields. In this case the said information is the most important part of your answer, because OP explicitly asked for the uniqueness of the lagrangian. $\endgroup$ Commented Sep 18, 2019 at 8:34
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$\begingroup$ OK. I see your point. I didn't assume different equations. All I said is: You can take the same ELequations from different Lagrangians. Thank you I will edit $\endgroup$– jmstf94Commented Sep 18, 2019 at 8:45