How Does Landau Justify the Additivity of Lagrangians in Isolated Systems?

[gionole]

TL;DR Summary

Lagrangian analysis by Landau

Trying to grasp the Landau's book and struggling here. (Attaching the image).If you multiply L by some constant and put it in in the Euler-Lagrange equation, motion equation won't be changed.

Q1: Though, what does he base his logic to say $Lim L = L_A + L_B$. If we got 2 separated system, he says total lagrangian of the whole thing is the sum of each part's lagrangian. What's the proof of this and how does he know it ? At that point, he doesn't even know what form Lagrangian has.

Q2: He proceeds to mention: "the Lagrangians of different isolated mechanical systems may be multiplied by different arbitrary constants. The additive property, however, removes this indefiniteness, since it admits only the simultaneous multiplication of the Lagrangians of all the systems by the same constant. " which is also unclear.

 

[wrobel]

Take it as two different Lagrangian systems which are considered simultaneously. These systems are described by different sets of different generalized coordinates. In the theory of dynamical systems, this is called the cross product of the dynamical systems.
$$L_A=L_A(t,q,\dot q),\quad L_B=L_B(t,x,\dot x).$$ The variables $x,q$ are separated in the differential equations with the Lagrangian $L=L_A+L_B$

 

[gionole]

Take it as two different Lagrangian systems in different generalized coordinates. In the theory of dynamical systems, this is called the cross product of the dynamical systems.
$$L_A=L_A(t,q,\dot q),\quad L_B=L_B(t,x,\dot x).$$ The variables $x,q$ are separated in the differential equations with the Lagrangian $L=L_A+L_B$

1. why does he use Lim ?
2. When wouldn't we have $L = L_A + L_B$ all the time ? it simply means adding 2 things give the sum of these together which should always hold true. It's interesting that I asked a question why it would hold true at all and now I'm asking why wouldn't it hold true all the time ?

 

[wrobel]

why does he use Lim ?

He said that he moves away the systems from each other to the infinite distance

When wouldn't we have L=LA+LB all the time ?

we have it all the time

UPD: consider for example two pendulums that do not interact

 

[gionole]

He said that he moves away the systems from each other to the infinite distance

we have it all the time

1. So he just uses Lim to say that even for the infinite distance, it would hold true. He basically states mathematically that it would hold true for any distance. right ?
2. Isn't it pointless to say this at all ? how can summing 2 things ever give something which is different from the sum ? it's like 2+2 != 4

 

[wrobel]

So he just uses Lim to say that even for the infinite distance, it would hold true.

the lim means "system $L_A$ does not interact with system $L_B$"

 

[gionole]

the lim means "system $L_A$ does not interact with system $L_B$"

Well, I get that but even if they are interacting, would not Lim L = L_A+L_B hold true ? If not why ?

 

[gionole]

@wrobel

1. Do you also agree that Landau is unneccesarily complicated ? but what I like about the book is it goes point by point why Lagrangian should contain 1/2mv^2 and doesn't directly say that it's K-V. Do you happen to know a book that figures out why principle of least action should contain K and V - kinetic and potential ?

2. I get you point about pendulum, but remember, that at the point where landau explains all it, it doesn't know that L is a function of kinetic and potential energies in which case in connected pendulum by a spring, why would you assume total lagrangian wouldn't be the sum of each part ? For the total lagrangian, you say spring also gets included, but does it get included as potential energy or just separate Lagrangian(equation of motion of the spring) ? if separate lagrangian, it might be clearer now. Thoughts. ?

 

[wrobel]

Do you also agree that Landau is unneccesarily complicated

He is not complicated, he is messy.

Do you happen to know a book that figures out why principle of least action should contain K and V - kinetic and potential ?

Arnold Math methods of classical mechanics
Greenwood Classical Dynamics.
There are a lot of good books.

I get you point about pendulum, but remember, that at the point where landau explains all it, it doesn't know that L

I remember that. I just try to give an acceptable explanation.

For the total lagrangian, you say spring also gets included, but does it get included as potential energy or just separate Lagrangian(equation of motion of the spring) ?

It depends on the statement of the problem. If the spring is massless, then the Lagrangian is written in one way, if not then in another way.

 

[PeroK]

TL;DR Summary: Lagrangian analysis by Landau

Trying to grasp the Landau's book and struggling here. (Attaching the image).If you multiply L by some constant and put it in in the Euler-Lagrange equation, motion equation won't be changed.

Q1: Though, what does he base his logic to say $Lim L = L_A + L_B$. If we got 2 separated system, he says total lagrangian of the whole thing is the sum of each part's lagrangian. What's the proof of this and how does he know it ? At that point, he doesn't even know what form Lagrangian has.

Q2: He proceeds to mention: "the Lagrangians of different isolated mechanical systems may be multiplied by different arbitrary constants. The additive property, however, removes this indefiniteness, since it admits only the simultaneous multiplication of the Lagrangians of all the systems by the same constant. " which is also unclear.

Here's a question for you. If $L(q, t)$ is considered a function of two independent variables $q$ and $t$, then how is the total derivative defined?

Note that, usually, the total derivative applies only where all variables are functions of the common variable wrt which the total derivative applies. E.g. $\frac d {dt} f(x(t), y(t), z(t), t)$.

In other words, how is the total derivative wrt $t$ reconciled with the Lagrangian as a function of indepedent variables; rather then a quantity defined along a defined time-dependent path?