Lagrangian Mechanics - Non Commutativity rule

[muzialis]

Hi there,

I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and possible non-commutative rule.
I do not get it: given a path q(t) and its variation $$\deltaq(t)$$, the equivalence between the variation of a derivative and the derivative of a variation seems to me a fact, not an arbitrary choice.
Could maybe anyone shed some light?
Thanks a lot

 

[Khashishi]

What book are you reading? In Lagrangian mechanics, ##q## and ##\dot q## are treated as independent parameters, so variation of time derivative of q and time derivative of variation in q don't mean the same thing.
Variation of derivative: ##\delta \dot q##
Derivative of variation: ##\frac{\partial}{\partial t} \delta q##

 

[muzialis]

I am reading Cornelius Lanczos' "The Variational principles of Mechanics", and Vujanovic, Atackanovic "Introduction to Modern Variational tecniques in Mechanics and Engineering".

I understand that in some derivations q and its time derivative are treated as independent, that they are to be viewed as independent while partially differentiating the Lagrangian, but I am struggling to understand your reply in full, could you please expand?
How can an object and its time derivative be independent?

Many thanks for your help

 

[haael]

The notation physicists use has been specifically designed to scare new students. Srsly, 90% of problems would be mitigated if physics adopted a better notation for derivatives, differentials, variations and integrals.

 

[muzialis]

Haeel,
your comment is interesting. It would be certianly the most welcome shouldyou expnad upon it: for example, how would a better notation shed light on the topic of non-commutativity?
thanks a lot

 

[voko]

I am reading Cornelius Lanczos' "The Variational principles of Mechanics", and Vujanovic, Atackanovic "Introduction to Modern Variational tecniques in Mechanics and Engineering".

The Lanczos' book is in agreement with your interpretation. I have not read the other one.

Some texts have two sorts of variations.

1. Isochronous variation. This is the variation of the kind ##\delta q(t) = \epsilon s(t) ##. With this sort of variation, ## {d \over dt} \delta q = \delta \dot q ##. This is the most commonly encountered kind of variations.

2. Non-isochronous variation. This chief idea behind this variation is that not only do we morph the function into something else, but we also mess with time, so the varied function is both changed and evaluated at a different time. So ##\delta q = \epsilon s + \dot q \delta t##. This sort of variation is less common, and ## {d \over dt} \delta q \ne \delta \dot q ##.

I would have to agree that the notation in the calculus of variations is not particularly illuminating. It's both a blessing and a curse.