Orthogonality from infinitesimal small rotation

AI Thread Summary
The discussion focuses on understanding the mathematical principles behind vector rotations and their properties. The first equation highlights that vector length remains constant during rotations, while the second equality derives from this using the chain rule, assuming the variation of gik is zero. Participants seek clarification on the chain rule's application and the time dependence of vik, questioning whether the infinitesimal matrix leads to a specific identity. The conversation emphasizes the need for detailed explanations of these mathematical concepts. Overall, the thread aims to clarify the intricacies of orthogonality in the context of infinitesimal rotations.
Warlord_
Messages
2
Reaction score
0
Hello buddies,

Could someone please help me to understand where the second and the third equalities came from?
Thanks,

Page-question-tensor.jpg
 
Physics news on Phys.org
Um... Ok, it's a little difficult to know what level of detail you want.

The first equation essentially says that the length of a vector does not change under rotations. The second one follows from the first one using the chain rule, and assuming delta of gik is zero.
 
Thanks for answering,

gik is basically the inner product of the base set vector, i.e., <ei,ek>.
- Could you please explicit the chain rule?
- is vik time dependent? And since it is an infinitely small matrix, the ##\delta v^k_i = I##?

Thanks
 

Thread '1:1 versus 2:1 Mechanical Advantage debate'

The rope is tied into the person (the load of 200 pounds) and the rope goes up from the person to a fixed pulley and back down to his hands. He hauls the rope to suspend himself in the air. What is the mechanical advantage of the system? The person will indeed only have to lift half of his body weight (roughly 100 pounds) because he now lessened the load by that same amount. This APPEARS to be a 2:1 because he can hold himself with half the force, but my question is: is that mechanical...
View full post »

Thread 'Newton's first law?'

Some physics textbook writer told me that Newton's first law applies only on bodies that feel no interactions at all. He said that if a body is on rest or moves in constant velocity, there is no external force acting on it. But I have heard another form of the law that says the net force acting on a body must be zero. This means there is interactions involved after all. So which one is correct?
View full post »
Thread 'Beam on an inclined plane'

Thread 'Beam on an inclined plane'

Hello! I have a question regarding a beam on an inclined plane. I was considering a beam resting on two supports attached to an inclined plane. I was almost sure that the lower support must be more loaded. My imagination about this problem is shown in the picture below. Here is how I wrote the condition of equilibrium forces: $$ \begin{cases} F_{g\parallel}=F_{t1}+F_{t2}, \\ F_{g\perp}=F_{r1}+F_{r2} \end{cases}. $$ On the other hand...
View full post »

Similar threads

A Exploring Infinitesimal Rotations in Classical Mechanics
Replies
8
Views
2K
I Confusion about adding angular velocities
Replies
4
Views
2K
Observation about the rotation of a disc
Replies
4
Views
1K
Angular Velocity Vector; Relationship to Angular Momentum
Replies
1
Views
2K
I Mechanics of an inertial balance
Replies
6
Views
1K
B How to find the infinitesimal coordinate transform along a hyperbola?
Replies
1
Views
1K
Classical Rotation: Referencing Rotational Motion in Mechanics
Replies
5
Views
1K
A Prove that points which are indistinguishable from 0 exist (using logic)
Replies
17
Views
1K
I Does a rotating accelerometer moving linearly measure linear velocity?
Replies
47
Views
4K
Angular veolcity in rotating frame
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top