groups and physics coadjoint action momentum
| 3 |
|---|
Group of translations:
...Consider the 2D space (x,y). In this space, a translation corresponds to the pair of scalars (Dx, Dy), and it is customary to write
x' = x + Dx
y' = y + Dy
We then use addition. Could we manage to encode the translation using a ... multiplication?
Consider the matrices:
and the group action:
Note that this is no longer simple matrix multiplication
g × r
but rather group action:
In passing, we can consider translations in three, four, or more dimensions:
The corresponding group action is then:
...Incidentally, the group of translations is commutative, and the identity element is "the zero translation." In 3D, the group's dimension is three; in 4D, it is four.
The usefulness of matrix groups. Example: the Euclidean group.
...The advantage of a matrix group is that we can simultaneously handle multiple things that previously seemed fundamentally different, such as rotation and translation. All we need to do is consider matrices:
and apply the group element matrix to the column vector to observe that this is equivalent to the combination of a rotation by angle a and a translation along the vector (Dx, Dy).
...As we see, the matrix g does not act "directly" on the points (x,y) in this 2D space, but rather through what is called a "group action," which obeys certain axioms.
...Thus, a group "acts" and "transports," in this case points. This is the Euclidean group. Linked to a 2D space (x,y), this group is defined by three parameters. It is g(a, Dx, Dy): the dimension of this group is 3. In particular:
g(0, Dx, Dy) represents the subgroup of translations.
g(a, 0, 0) represents the subgroup of rotations about the origin.
g(0, Dx, 0) represents the subgroup of translations parallel to a line (the OX axis).
...The Euclidean group transports points that, by themselves, have no intrinsic properties (whereas dynamical groups assign to a simple "material point" properties called mass, energy, momentum, spin).
...With the Euclidean group, we are forced to consider sets of points. As if, in chemistry, atoms were indistinguishable from one another and only the geometry of molecular assemblies carried meaningful information.
...A geometric figure—such as a triangle (viewed as a set of three points or three segments), or a square (viewed as a set of four points or four segments)—can be transported by the group. This is where the fundamental idea of species comes into play. Two "objects" are said to belong to the same species if there exists a group element that can map one onto the other.
With respect to the Euclidean group, squares with the same side length a form a species:
Squares of the same species.
...If the sides a and b are different, these objects do not belong to the same species. There is no group element that can transform one into the other. With respect to the Euclidean group
these squares do not form the same species.
Euclidean geometry does not allow dilations (homotheties). To handle this, we would need to move to another group—the Cartesian group:
a four-parameter group g(λ, a, Dx, Dy), where λ is a dilation coefficient. Thus, the dimension of this group is 4.
From here, we can easily imagine a 3D Euclidean group acting on objects in three dimensions.
...We are not aiming to launch into a full course on groups, but rather to grasp a few key ideas. What is zoology? A science devoted to studying animals and classifying them. If we limit ourselves to shape alone, the Euclidean group allows us to classify adult rabbits. To classify rabbits of different sizes into the same species, we would need to resort to the Cartesian group, since no element of the 3D Euclidean group can transform a small rabbit into a large one.
...You're smiling? You're mistaken. Perhaps in your apartment or house, you have a baby learning to play in a corner. You gave them a classic toy: a shape sorter with holes for cylinders, cubes, or triangular prisms.
...What is the baby doing? They are becoming familiar with the 3D Euclidean group. They are classifying objects by species, which will later allow them to recognize shapes—i.e., perform "shape recognition."
...Even though the cylinders are different colors, the baby checks whether there exist group actions (movements of these objects in 3D space) that can align cylinder A and cylinder B perfectly, using the hollow shape of the cylinder or prism as a guide—the entry slot into the sorting compartment. In this way, the baby learns that cylinders A and B, with respect to the shape criterion (Euclidean group), belong to the same species.
