V. I. Arnold’s Mathematical Methods of Classical Mechanics (Springer, 1989) provides a contemporary approach to classical mechanics. We follow the presentation here but modify it to six dimensions of space-time.
1 The principles of relativity and determinacy
A series of experimental facts is at the basis of classical mechanics. We list some of them.
A Geometry and order
Space and time are both three-dimensional and Euclidean.
B Galileo’s principle of relativity
There exist basic coordinate systems possessing the following two properties:
- All the laws of motion are in all cases the same in all basic coordinate systems.
- All coordinate systems in uniform rectilinear motion with respect to a basic one are themselves basic coordinate systems.
In other words, if a coordinate system attached to the earth is basic, then an experimenter on a train which is moving uniformly in a straight line with respect to the earth cannot detect the motion of the train by experiments conducted entirely inside their car.
In reality, the coordinate system associated with the earth is only approximately basic. Coordinate systems associated with the sun, the stars, etc. are more nearly basic.
C Newton’s principle of determinacy
The initial state of a mechanical system (the totality of positions and motions of its points at some index of events) uniquely determines all of its motion.
It is hard to doubt this fact, since we learn it very early. One can imagine a world in which to determine the future of a system one must also know the change of motion of the initial state, but experience shows us that our world is not like this.
2 The Galilean group and Newton’s equations
A Notation
We denote the set of all real numbers by ℜ. We denote by ℜn an n-dimensional real vector space.
Affine n-dimensional space An is distinguished from ℜn in that there is “no fixed origin.” The group ℜn acts on An as the group of parallel displacements:
a → a + b, a ∈ An, b ∈ ℜn, a + b ∈ An.
[Thus the sum of two points of An is not defined, but their difference is defined and is a vector in ℜn.]
A Euclidean structure on the vector space ℜn is a positive definite symmetric bilinear form called a scalar product. The scalar product enables one to define the metric
ρ(x, y) = || x − y || = √(x − y, x − y)
between points of the corresponding affine space An. An affine space with this metric function is called a Euclidean space and is denoted by En.
B Galilean structure
The Galilean space-time structure consists of the following three elements:
1. The universe—a six-dimensional affine space A6. The points of A6 are called world points or events. The parallel displacements of the universe A6 constitute a vector space ℜ6.
2. Order—a linear mapping r: ℜ6 → ℜ from the vector space of parallel displacements of the universe to the real “order axis.” The order interval from event a ∈ A6 to event b ∈ A6 is the number from the mapping r(b − a). If r(b − a) = 0, then the events a and b are called simulordinate.
The set of events simulordinate with a given event forms a three-dimensional affine subspace in A6. It is called a space of simulordinate events A3.
The kernel of the mapping r consists of those parallel displacements of A6 which take some (and therefore every) event into an event simulordinate with it. This kernel is a three-dimensional linear subspace ℜ3 of the vector space ℜ6.
The Galilean structure includes one further element.
3. The metric between simulordinate events
ρ(a, b) = || a − b || = √(a − b, a − b) a, b ∈ A3
is given by a scalar product on the space ℜ3. This metric makes every space of simulordinate events into a three-dimensional Euclidean space E3.
A space A6, equipped with a Galilean space-time structure, is called a Galilean space.
One can speak of two event occurring simulordinately in different zones, but the expression “two non-simulordinate events a, b ∈ A6 occurring at one and the same zone in three-dimensional space” has no meaning as long as we have not chosen a coordinate system.
The Galilean group is the group of all transformations of a Galilean space which preserve its structure. The elements of this group are called Galilean transformations. Thus, Galilean transformations are affine transformations of A6 which preserve intervals of order and the metric between simulordinate events.
Let M be a set. A one-to-one correspondence φ1 : M → ℜ3 × ℜ3 is called a Galilean coordinate system on the set M. A coordinate system φ2 moves uniformly with respect to φ1 if φ1 • φ2−1 : ℜ3 × ℜ3 → ℜ3 × ℜ3 is a Galilean transformation. The Galilean coordinate systems φ1 and φ2 give M the same Galilean structure.