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Physvillain
Brief Review of Undergraduate Classical Mechanics (4) - How to construct a Lagrangian? 본문
Brief Review of Undergraduate Classical Mechanics (4) - How to construct a Lagrangian?
Physvillain 2020. 10. 30. 11:12이 글은 2020.10.14에 작성됨.
This post was written on Oct 14, 2020.
Let's next think about why our Lagrangian is the form of
(1) Consider a freely moving particle in an inertial frame. There is no preferred point in the space (nor the time), thus there is no dependence on
Of course, this implies
(2) Let's do a Galilei's transformation. Take two inertial frame
The last term is a total derivative of something.
From the above, we can notice that
(3) Now consider the case with a potential. The exact differential of
Suppose
From the fact that
(4) Then what if
[ Example : Constructing EM Lagrangian ]
We already know in this case Lorentz force is
Expanding
Then rearranging this, we obtain
Suppose RHS is
Thus in this non-conservative potential, we can find Lagrangian from pre-known Lerentz force law. (In fact, vector potential is just a differential one-form, magnetic field is two-form, and electric field is one-form. Think about field strength tensor