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Brief Review of Undergraduate Classical Mechanics (4) - How to construct a Lagrangian? 본문

Classical Mechanics

Brief Review of Undergraduate Classical Mechanics (4) - How to construct a Lagrangian?

Physvillain 2020. 10. 30. 11:12

이 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 TU in Classical Mechanics.

 

(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 x or t. In this case we can obtain L=L(x˙2). It follows 

Lxi=0ddtLx˙i=0

Of course, this implies v is a constant.

 

(2) Let's do a Galilei's transformation. Take two inertial frame K and K, moving one respect to other with a small velocity w. Then our physics must be same, thus Lagrangian must be up to total time derivative of some function.

L(v2)=L(v2+2wv+w2)=L(v2)+2Lv2wv

The last term is a total derivative of something.

Lv2wv=ddtg(x,t)

From the above, we can notice that L/v2 is just a constant. So we can conclude Lv2.

 

(3) Now consider the case with a potential. The exact differential of L is

dL=Lqdq+Lq˙dq˙

Suppose 2Lqq˙=0. (i.e. Lq˙=Lq˙(q˙)) Then the first term should be a function of only q. (Lq=Lq(q)) From (2), we know Lq˙=2aq˙. Take a=m/2, the half mass of the particle. Then we know Lq=F(q).

dL=p(q˙)dq˙+F(q)dq

From the fact that F=dU/dq, we can construct L(q,q˙)=T(q˙)U(q).

 

(4) Then what if 2Lqq˙0? Lets consider an example of particle in a scalar field (ϕ(x,t)) and a vector field (A(x,t)). You can also think this situation as an analogue of electromagentics.

 

[ Example : Constructing EM Lagrangian ]

We already know in this case Lorentz force is

mx¨=qE+qcv×B

Expanding i th component,

mx¨i=qEi+qcϵijkx˙jBk=q(iϕ1ctAi)+qcϵijkx˙jϵklmlAm=qiϕqcddtAi+qcx˙jjAi+qc(x˙jiAjx˙jjAi)

Then rearranging this, we obtain

ddt(mx˙i+qcAi)=qiϕ+qcx˙jiAj

Suppose RHS is L/xi and LHS is d/dt(L/x˙i). Lets check exactness. (if not exact, we should multiply some factor)

2Lx˙jxi=qciAj=2Lxix˙j

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 Fμν. Not vectors strictly.)

L(x,x˙)=12mx˙2+qcx˙Aqϕ

 

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