This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to answer the titular question on my own, I was recently led to think that scale transformations and addition by a total time derivative are the only transformations of the Lagrangian which give the same Equation of motion according to this answer by Qmechanic with kind help from respected users in the hbar. I will ignore topological obstructions from now on in this question. The reasoning was that if we have that
Two Lagrangians $L$ and $L'$ which satisfy the same EOM then we have that the EL equations are trivially satisfied for some $L-\lambda L'$. The answer "yes" to (II) in Qmechanic's answer then implies that $L-\lambda L'$ is a total time derivative.
However I am confused upon reading the Reference 2 i.e. "J.V. Jose and E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998"; (referred to as JS in the following) mentioned in the answer. I have several closely related questions---
JS (section 2.2.2 pg. 67) says
For example, let $L_1$ and $L_2$ be two Lagrangian functions such that the equations of motion obtained from them are exactly the same. Then it can be shown that there exists a function $\Phi$ on $\mathbb{Q}$ such that $L_1-L_2 = d\Phi/dt$.
(i) Why doesn't this assertion not violate Qmechanic's answer to (I) here, namely why should two lagrangians having same EOM necessarily deviate by a total derivative of time, when Qmechanic explicitly mentioned scale transformation to be such a transformation? Infact on the next page(pg. 68) JS also says
It does not mean, however, that Lagrangians that yield the same dynamics must necessarily differ by a total time derivative.
exactly what Qmechanic asserts with example.
But then JS gives an example
$L_a= (\dot{q_1}\dot{q_2} -q_1q_2)~~\text{and}~~L_b = \frac{1}{2}\left[\dot{q_1})^2 +(\dot{q_2})^2 - (q_1)^2- (q_2)^2\right]$ yield the same dynamics, but they do not differ by a total time derivative.
Also, this is (atleast manifestly) not equivalent to a scale transformation of the Lagrangian. So we found a transformation which is not a scale transformation or adding a total derivative but gives the same dynamics contrary to what I was led to think from Qmechanic's answer by the reasoning mentioned in the first paragraph(actually first block-quote) of this question.
(ii) So where was the reasoning in the first para wrong /How is the assertion in the first para consistent with this particular example?
Surprisingly JS pg 68 also says
What we have just proven (or rather what is proven in Problem 4) means that if a Lagrangian is changed by adding the time derivative of a function, the equations of motion will not be changed.
(iii) But didn't they just state and prove the converse on pg. 67?
- If "Yes" then
$~~~(a)$ As asked in part (A) of this question how is this consistent with part (I) of the previous answer by Qmechanic.
$~~~(b)$ Did JS make a mistake on pg. 68 then?
- If "No" i.e. what is stated on pg 68 is what is proved on pg 67 then
$~~~(a)$ Did JS make a mistake on pg. 67 then? i.e. How are the statement on pg 67 and the proof consistent with the statement on pg 68?
$~~~(b)$ Why did Qmechanic cite this as a reference to support his answer to (II) which is almost(*) the converse of what is given in JS?
(*) "Almost" because I don't see how the statement on pg. 67 uses the word "trivially" from Qmechanic's statement
If EL equations are trivially satisfied for all field configurations, is the Lagrangian density $\mathcal{L}$ necessarily a total divergence?
Assuming, the answer to the above question is somehow "yes", what I understood from pg. 67 of JS is that they are simply taking two Lagrangians which satisfy the same EOM as the assumptions of the theorem that they move on to prove and what Qmechanic is taking as assumption in his answer to (II) is a single Lagrangian which satisfy the EL equations for any arbitrary choice of $q,\dot{q}$ and $t$ on which the Lagrangian depends. So how are these two assumptions are supposed to be the same? Or more simply put,
(iv) How is Qmechanic's statement in answer to (II) in the previous question equivalent to what is stated in JS?
(v) Finally, is the titular question answerable with or without using Qmechanic's previous answer? If so, How?
If someone is not interested in answering all these questions (labelled $(i)-(v)$ with ($iii$) containing multiple subquestions), an overall explanation/discussion in new words, of the answer by Qmechanic and what's written on JS and their comparison might also help.