With Lagrangian, is there any way to intuitively grasp why total energy equals the difference between the kinetic and potential energy? Seems counter-intuitive - whereas Hamiltonian calculation (sum of KE and PE) makes more sense to me.
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6The Lagrangian is not the energy, but rather its Legendre transform. That's why there's a minus sign. – Javier Jun 05 '16 at 02:46
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Related: http://physics.stackexchange.com/q/78138/2451 , http://physics.stackexchange.com/q/41138/2451 , http://physics.stackexchange.com/q/207736/2451 , http://physics.stackexchange.com/q/9686/2451 , http://physics.stackexchange.com/q/50075/2451 and links therein. – Qmechanic Jun 05 '16 at 03:33
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See for example: Why is the Hamiltonian the Legendre transform of the Lagrangian? – John Rennie Jun 05 '16 at 07:29
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There is no intuitive way to grasp this because it is not true! The Lagrangian is NOT the energy of the system. The energy of the system is $(KE+PE)$, of course. I can define lots of quantities with the units of energy: $KE$, $PE$, $(KE+PE)$, $(KE-4PE)$, $({KE}^2/PE)$, $(KEk_bT/PE)$. Of course, only one of them is the total energy. Some of them are still useful for calculation, such as $KE$, and some of them are totally useless, like ${KE}^2/PE$. The Lagrangian happens to be one that is useful.
Now, what's the intuition? Well, the claim of Lagrangian mechanics is that the path the particle actually takes should minimize the integral of $L$ over time. So in general, when a particle goes along a path, Lagrangian mechanics tells us the particle will spend lots of time at places where $L$ is small, and not a lot of time at places where $L$ is big. This makes sense: when L is big, we have LOTS of kinetic energy, so you expect the particle to zoom right by those spots; when $L$ is small, the particle has very little kinetic energy, so it will stay there longer.
Jahan Claes
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@DanielSank It's a mathematical claim. I.e. Claim: there are infinite prime numbers. Proof: etc. Claim: If you do this, you get Newtonian mechanics. – Jahan Claes Jun 05 '16 at 04:08
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It's not always "minimised", may be "extremized" would be a better word – Courage Jun 05 '16 at 05:16