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 ℜ^{n} an *n*-dimensional real vector space.

Affine n-dimensional space *A*^{n} is distinguished from ℜ^{n} in that there is “no fixed origin.” The group ℜ^{n} acts on *A*^{n} as the *group of parallel displacements*:

*a* → *a* + **b**, *a* ∈ *A*^{n}, **b** ∈ ℜ^{n}, a + b ∈ *A*^{n}.

[Thus the sum of two points of *A*^{n} is not defined, but their difference is defined and is a vector in ℜ^{n}.]

A *Euclidean structure* on the vector space ℜ^{n} 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 *A*^{n}. An affine space with this metric function is called a Euclidean space and is denoted by *E*^{n}.

B *Galilean structure*

The Galilean space-time structure consists of the following three elements:

1. The universe—a six-dimensional affine space *A*^{6}. The points of *A*^{6} are called *world points* or *events*. The parallel displacements of the universe *A*^{6} constitute a vector space ℜ^{6}.

2. Order—a linear mapping r: ℜ^{6} → ℜ from the vector space of parallel displacements of the universe to the real “order axis.” The *order interval* from event *a* ∈ *A*^{6} to event *b* ∈ *A*^{6} 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 *A*^{6}. It is called a *space of simulordinate events* *A*^{3}.

The kernel of the mapping *r* consists of those parallel displacements of *A*^{6} which take some (and therefore every) event into an event simulordinate with it. This kernel is a three-dimensional linear subspace ℜ^{3} of the vector space ℜ^{6}.

The Galilean structure includes one further element.

3. The *metric between simulordinate event*s

*ρ*(*a, b*) = || *a − b* || = √(*a − b, a − b*) *a, b* ∈ *A*^{3}

is given by a scalar product on the space ℜ^{3}. This metric makes every space of simulordinate events into a three-dimensional Euclidean space *E*^{3}.

A space *A*^{6}, 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 ∈ *A*^{6} 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 *A*^{6} which preserve intervals of order and the metric between simulordinate events.

Let *M* be a set. A one-to-one correspondence φ_{1} : M → ℜ^{3} × ℜ^{3} is called a *Galilean coordinate system on the set M*. A coordinate system φ_{2} *moves uniformly* with respect to φ_{1} if φ_{1} • φ_{2}^{−1} : ℜ^{3} × ℜ^{3} → ℜ^{3} × ℜ^{3} is a *Galilean transformation*. The Galilean coordinate systems φ_{1} and φ_{2} give *M* the same Galilean structure.