Scalar and vector defined by transformation properties

Question

In Classical Mechanics, we are defining scalars as objects that are invariant under any coordinate transformation. Vectors are defined as objects that can be transformed by some transformation matrix $ \lambda $ .

Why is this important? I get that it leads to other properties such as the invariance of the dot product under coordinate rotations but how does this relate to physics? This is supposed to lead to another question but I will refrain from posting so that I may think a little about it.

I also have seen Noether's Theorem explaining that symmetries pop out conservation laws, such as the time independence of the Lagrangian gives you the Hamiltonian equating to the total energy of the system.

Answer 1

Just as a quick example: say that the dot product were not invariant under transformations. Then let's say that we have two reference frames, A and B, where reference frame B is rotated and displaced with respect to A and which moves at a constant speed w.r.t. A (where $v\ll c$).

Then the researcher in A wants to calculate the gravitational attraction between two known masses. For this he seperates the masses by a rod which seperates the masses and which extends from the origin of his reference frame to some point $\vec r={(x,y,z)}$. The gravitational strength is proportional to the length of this rod. How does he measure the length? He takes the dot product of $\vec r$ and takes the square root of the resulting number.

Now the researcher in B wants to know the length and the resulting gravitational attraction from the known masses, but since he is far away from the rod he can't measure the length directly. So he says to researcher A: ok, I can't see the rod, but there is a set of rules which tell me how to calculate where your origin is and where the point $\vec r$ is viewed from my frame, from this I can calculate the length. These rules are of course the galilei transformations.

After he has done the calculation he finds out that the rod is shorter in his frame since the dot product is not invariant, and because the rod has a different length the gravitational attraction between the masses is different. This is nonsensical! The gravitational attraction between two masses should not depend on the reference frame.

This is why vectors and transformations and tensors and what not exist: such that the laws which govern one frame of reference are also valid in any other frame which is obtained from the first by a transformation.

Answer 2

Draw an arrow to represent a vector, with its length representing the vector magnitude. Draw a coordinate system and get the components of the vector. Now draw another coordinate basis, rotated with respect to the first, and get the components with respect to the new basis. The length of the arrow is the same in both systems - i.e length is invariant - and so there must be a relationship between the components in each system. That relationship is the vector transformation relation.

If it wasn't for invariants, mechanics would be impossible. The maximum displacement of a bridge, for example (and whether it falls down or not), would depend, quite literally, on how you look at it. Which is, of course, nonsense.

Answer 3

All these mathematical objects (scalars, vectors, tensors) are carefully selected to represent real world phenomena and their observable properties.
For example, invariance under transformation, means that the variable (or physical object) represented via the transformed, let's say vector or tensor, does not change ,i.e. it is conserved.
In general, theories could be either:

1. Deduced experimentally: meaning, observed through experimental results and developed as a model according to it.
2. Created from insight or thought experiment and/or based on existent mathematical theories.