Describing quantities independently from frames in Newtonian mechanics

[burakumin]

Hello

I know that it's possible to reformulate Newtonian mechanics in such a manner that absolute velocities of objects can be defined. By absolute I mean defined without any reference to a specific frame of reference (just as in the article Notes on Mathematical Physics for Mathematicians). I wondered if in a similar manner an object representing kinetic energy could be defined too (after a few attemps, it appears to me that yes) and if all Newtonian physics could be expounded that way.

I'd like to go further in this approach but the previous article only uses absolute objects in its beginning and does not contain any bibliography. I'm pretty sure books or documents on that topic have already been written.

Does anybody have any reference to something similar ?

Thank you

 

[Simon Bridge]

I think you are talking about Hamiltonian Mechanics.

 

[burakumin]

Hi simon Bridge

No I'm not talking specifically about Hamiltonian Mechanics. This is an independent concern. I'm talking about the ability to describe a non-einsteinian spacetime (the 4 dimensions "galilean spacetime") and to expound classical mechanics in it without frames of reference (using Hamiltonian formulation, Lagrangian formulation or none, I don't care)

 

[Simon Bridge]

Therefore I don't understand your problem - there are several formulations that do not rely on a specific frame of reference. You have just named two of them. There are, indeed, many books and papers written about them.

To my knowledge, absolute velocities cannot be defined in Newtonian Mechanics though - do you have an example?

 

[burakumin]

Therefore I don't understand your problem - there are several formulations that do not rely on a specific frame of reference. You have just named two of them. There are, indeed, many books and papers written about them.

I have named two ? We're not talking about the same thing. Hamiltonian and Lagrangian formulation both use kinetic ernergy. And kinetic energy is generally defined within a specific frame. Of course considering a different frame will lead us to the same physical result. But this result will be formulated in a peculiar frame.

To my knowledge, absolute velocities cannot be defined in Newtonian Mechanics though - do you have an example?

One can think about galilean spacetime as a 4 dimensions affine space $\mathcal{A}$ with a linear form $\tau : \mathcal{E} \rightarrow \mathbb{R}$ where $\mathcal{E}$ is the vector space under $\mathcal{A}$. If $\epsilon_1$ and $\epsilon_2$ are events (elements of $\mathcal{A}$), the quantity $\tau(\epsilon_1 - \epsilon_2)$ represent the delay elapsed between them. Finally one needs a scalar product defined on the kernel of $\tau$.

Within this framework, absolute velocities of bodies can be defined as vectors of $\{ v \in \mathcal{E} : \tau(v) = 1 \}$. And I guess kinetic energy could be defined as a tensor of $\mathcal{E} \otimes \mathcal{E}$ with specific properties.

This way you can get rid of frames, or describe them as objects with their own absolute velocities.

I'm looking for books that describes completely Newtonian physics with this kind of formulation. To me using Hamiltonian or Lagrangian is just a way of describing a physical problem using state spaces and to find the good trajectory in them, but these state spaces can be based on any spacetime you prefere.

 

[robphy]

I suspect there are two related but distinct uses of the word "absolute" here.

One usually says there are "no absolute speed [i.e. magnitude of a 3-d velocity vector] in Galilean relativity" to mean that observers generally disagree on the speed of an object... in particular, there is nothing "absolutely at rest" (as there would be in an Aristotelian spacetime).

One can also talk about "absolute structures [i.e. tensorial or "geometrical" structures in 4-d]", like the 4-velocity of an object, which has different 3-dimensional components depending on the reference frame of the observer studying that object. Malament's papers might be a good starting place. Follow the references (to Ehlers, Kunzle, Trautman, etc...)
www.socsci.uci.edu/~dmalamen/bio/papers/GRSurvey.pdf#page=36
http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf ("Newtonian Gravitation Theory)

[I am actively working on aspects of formulating classical physics in a Galilean spacetime framework... with an eye to clarifying the connection to its special relativistic analogoues.]

 

[burakumin]

Hi Robphy

I'm reading the second reference you've given and it's definitely the kind of approach I'm looking for. Thank you.

I'm a bit surprised that the chosen structure is a manifold and thus is weaker that the affine space I was thinking about, but I guess the reason will become more obvious when I have finished reading.