Is this a Valid Axiomatization for a System of Particles in the formalism of Newtonian Mechanics?

Question

I read an article proposing four axioms for a system of particles in the formalism of Newtonian Mechanics:

1. Inertial frames of reference
2. Conservation of Momentuum
3. Superposition of forces
4. Instantaneous action of force

I don't really know what to make of inertial frames of reference. I haven't understood very well this concept.

I know that temporal, translational, and rotational symmetries - via Noether's Theorem - gives conservation of energy, linear momentuum and angular momentuum.

This would modify the axioms to the following:

1. Inertial frames of reference (?)
2. Temporal symmetry
3. Translational symmetry
4. Rotational symmetry
5. Superposition of forces
6. Instantaneous action of force

As a symplifying assumption, I will suppose only contact forces, no fields of force.

I know that Heaviside and Gibbs' scalar and vectorial formulation does not accounts for certain objects only described by pseudo-scalars (bivectors?) and pseudo-vectors (trivectors?). I suppose pseudo-scalars are not needed for the very simple case I am talking about. I suppose that pseudo-vectors would not be needed if I constrain the degrees of freedom of the particles, not allowing for rotations.This way it would suffice a 3 dimensional euclidean space, plus a scalar parameter for absolute time, plus a scalar parameter for inertial mass.

I was wondering if superposition of forces and instantaneous action of forces follows mathematically from the linearity and ortogonality of a 3 spatial dimensions like an euclidean space, plus a scalar parameter for absolute time, plus a scalar parameter for inertial mass.

I was told that in special and general relativity what we observe as forces don't obey superposition of forces and the instantaneous action of forces. And that these behaviors are mathematical consequences of the nature and geometry of the space and time and mass the theories pressupose.

I was wondering if there would be a similar reasoning for the simpler case I am reffering to (a description of a System of Particles in the formalism of Newtonian Mechanics, preferably using vector analysis and Newtonian mechanics, not calculus of variations and Lagragian.)

This would make the axioms:

1. Inertial frames of reference (?)
2. Temporal symmetry
3. Translational symmetry
4. 3 dimensional euclidean space
5. Scalar parameter for absolute time
6. Scalar parameter for inertial mass

Related:

The foundations of geometric formulation of Newton's axioms

Answer 1

Neither Newtonian spacetime nor relativistic spacetime are four-dimensional Euclidean spaces. In a four-dimensional Euclidean space, the metric would look like $(\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$. In relativistic spacetime, the metric is $-(\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$. In Newtonian spacetime, there is no unified metric that applies to both time and space.

Answer 2

Newtonian mechanics cannot be described by just 3-dimensional Euclidean space.

You need the space that includes all 3D vectors, but also scalar values (like time). In fact, you need all 3D pseudo-vectors, and might as well include pseudo-scalars to complete the Grassmann Algebra.

So 1 scalar, 1 3D vector, 1 3D pseudo-vector and 1 pseudo-scalar creates a space with 8 dimensions. Only then you can start using the exterior algebra between these objects to describe mechanics fully.

A lot of research in this area has been done already (example), and it sounds really promising to untie the oddities and quirks of 3D vector mechanics with simpler more coherent 8D laws.

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I am a student of Screw Theory which tackles mechanics from a 6D perspective using vectors and pseudo-vectors and there are many benefits to doing so such as combining all translational and rotational quantities into one formulation.

But I have understood recently that 6D isn't sufficient, and a more complete description is required, and Grassmann Algbera offers the best tools to do so. The description of dual quaternions was a big step in this direction. And its application to mechanics makes a compelling case for GA (geometric algebra), as mechanics is fundamentally linked to geometry.