# An ab initio Hilbert space formulation of Lagrangian mechanics

• A
• andresB
In summary, an ab initio Hilbert space formulation of Lagrangian mechanics is a mathematical framework for studying the dynamics of mechanical systems. It uses the concept of a Hilbert space, a mathematical space where vectors represent the states of the system, to describe the motion of particles and the forces acting on them. This approach provides a more general and elegant way to derive the equations of motion and allows for the inclusion of quantum mechanical effects. It has applications in a wide range of fields, including quantum mechanics, statistical mechanics, and quantum field theory.
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

coquelicot and dextercioby
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 ?

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vanhees71
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.

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

coquelicot said:
What about non holonomic constraints? e.g. rolling, like a ball rolling on a plane.
That's still on my to-do list.

andresB said:
That's still on my to-do list.

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.

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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?

coquelicot said:
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.

andresB said:
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.

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?

coquelicot said:
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

coquelicot

## 1. What is an ab initio Hilbert space formulation?

An ab initio Hilbert space formulation is a mathematical framework used to describe the dynamics of a physical system in terms of Hilbert spaces, which are mathematical spaces that are used to represent vectors and operators. This formulation is based on the principles of quantum mechanics and allows for a more fundamental understanding of the behavior of a system.

## 2. How does this formulation relate to Lagrangian mechanics?

This formulation is an alternative way of expressing the equations of motion in Lagrangian mechanics. It provides a more rigorous and fundamental approach, as it is based on the underlying principles of quantum mechanics rather than classical mechanics.

## 3. What are the advantages of using an ab initio Hilbert space formulation?

One of the main advantages is that it allows for a more fundamental understanding of the dynamics of a system. It also provides a more elegant and concise way of expressing the equations of motion, as well as the ability to incorporate quantum effects into the analysis.

## 4. Are there any limitations to this formulation?

One limitation is that it may be more difficult to apply in certain situations, such as systems with a large number of particles or systems with strong interactions. It also may not be as intuitive as other formulations, making it more challenging for some to understand.

## 5. How is this formulation used in practical applications?

This formulation is commonly used in theoretical physics and quantum mechanics research, as it provides a more fundamental understanding of the dynamics of physical systems. It can also be applied in fields such as quantum chemistry and materials science to study the behavior of atoms and molecules.

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