An ab initio Hilbert space formulation of Lagrangian mechanics

[andresB]

I want to share my recent results on the foundation of classical mechanics. Te abstract readWe construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, several concepts from analytical mechanics will be rediscovered from an entirely new perspective. We start by expressing the basic concepts of the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of Holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position-velocity to a position-momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman-von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized
to include holonomic constraints.

On the other hand, the biggest flaw of the work is the absence of an operational formulation of the D'Alembert principle. I have no idea how to even define virtual work using operators.Commentaries and suggestions are welcomed

https://arxiv.org/abs/2204.02955

 

[wrobel]

Before reading this I have a question: which problems that have not been solved before are now solved by means of the theory developing here ?

 

[andresB]

None.

While operational methods do allow the use of new methods to solve problems in classical mechanics, like here https://www.nature.com/articles/s41598-018-26759-w, my work doesn't aim in that direction.

The paper deal with obtaining some results from analytical mechanics in new ways. From a practical side, Using the Heisenberg equations to find Lagrange equations in generalized coordinates is quite laborious, the standard ways are much more suited for the task.

 

[coquelicot]

What about non holonomic constraints? e.g. rolling, like a ball rolling on a plane.

 

[andresB]

What about non holonomic constraints? e.g. rolling, like a ball rolling on a plane.

That's still on my to-do list.

 

[andresB]

That's still on my to-do list.

To give an actual answer:
There is a Hilbert space for the unconstrained dynamics. One of the central results of the paper is that holonomic scleronomous constraints restrict the dynamics to a subspace of the unconstrained Hilbert space. When evaluated in that subspace, we can ignore the constraining forces in the time evolution operator.

I don't know if that is the case for non-holonomic constraints.

 

[coquelicot]

thx for your answer. I saw a lot a physicists/mathematicians have produced theories according to similar ideas. What originality do you think your theory have with respect to similar theories?

 

[andresB]

thx for your answer. I saw a lot a physicists/mathematicians have produced theories according to similar ideas. What originality do you think your theory have with respect to similar theories?

To my knowledge, the points (1) to (5) described in the abstract are original.

For example, I have never seen the transformation from velocity to momentum being done without the use of a Legendre transformation.

 

[coquelicot]

To my knowledge, the points (1) to (5) described in the abstract are original.

For example, I have never seen the transformation from velocity to momentum being done without the use of a Legendre transformation.

OK, I looked at the the link you provided (their bibliography), but only very superficially, so, I have probably not understood the essential points.

 

[coquelicot]

This seems beautiful. A pity that I have not the time and energy to study your paper.
I have a last question though. Do you think that your theory could unify classic mechanics and quantum mechanics?

 

[andresB]

This seems beautiful. A pity that I have not the time and energy to study your paper.
I have a last question though. Do you think that your theory could unify classic mechanics and quantum mechanics?

Well, define "unify".

If you mean, putting them in the same mathematical formalism, then yes, that's the point.

There are also quantization and dequantization rules from QM to operational classical mechanics, if that is what you are looking for. They are very close to geometric quantization.

Also, one of the main interest I have recently, are quantum-classical hybrid theories. When two systems are interacting, and one of them is big enough and it behaves classically, but not big enough so the backreaction has to be taken into consideration https://arxiv.org/abs/2107.03623