I Hamilton’s principle maximises potential energy?

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Hamilton's principle minimizes the difference between kinetic energy and potential energy, effectively maximizing potential energy when kinetic energy is fixed. In scenarios where kinetic energy or mass is negligible, the Lagrangian simplifies to the negative of potential energy. The discussion references Feynman's explanation in his lectures, particularly around Fig 19-6, to illustrate these concepts. The limit of kinetic energy approaching zero raises questions about the meaningfulness of maximizing potential energy in this context. Ultimately, the focus remains on finding the path where the integral of kinetic and potential energy is extreme.
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Hamilton’s principle minimises kinetic energy minus potential energy, that is, with a fixed kinetic energy, Hamilton's principle maximises potential energy. What if we consider the limit that the kinetic energy or the mass/the inertia can be ignored then the lagrangian is solely the negative of potential energy. How to understand the potential energy needs to be maximised?
 
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anuttarasammyak said:
Feynmann does a good explanation around Fig 19.3 in https://www.feynmanlectures.caltech.edu/II_19.html.
Thank you, however, I don't think he said anything about maximization of PE and its meaning?
 
Sorry, Fig 19-6 and its around explains it.
 
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anuttarasammyak said:
Sorry, Fig 19-6 and its around explains it.
Thanks for pointing it out. Then how should we understand the limit of KE->0, then min(L)=-max(PE)?
 
As Feynman stated we are looking for the path KE-PE integral on which should be extreme.
KE=0 takes place at the top of trajectory in Fig 19-6 but I do not think considering such "limit of KE->0" is meaningful.
 

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