[Eternal Student (OP)]

Hi everyone.

   What proof or good reason exists for suggesting that all physical systems that we might want to study can be modeled with a Lagrangian and a least action principle?

  I honestly don't know.  Why is Nature lazy and always trying to minimise an action?

Biologists and Chemists welcome - please check this and comment if it's wrong:
    Chemists and Biologists don't seem to assume that every system they want to study and model can be represented with a Lagrangian.   I'm fairly sure that when a vaccine for Covid was developed no one started by proposing a new Lagrangian for the system.

Best Wishes.

[evan_au]

Lagrangian

Lagrange was a very productive mathematician. Which Lagrangian do you mean?

Lagrangian may refer to:
Mathematics
- Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
- Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set
- Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem
- Lagrangian, a functional whose extrema are to be determined in the calculus of variations
- Lagrangian submanifold, a class of submanifolds in symplectic geometry
- Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density
Physics
- Lagrangian mechanics, a reformulation of classical mechanics
- Lagrangian (field theory), a formalism in classical field theory
- Lagrangian point, a position in an orbital configuration of two large bodies
- Lagrangian coordinates, a way of describing the motions of particles of a solid or fluid in continuum mechanics
- Lagrangian coherent structure, distinguished surfaces of trajectories in a dynamical system

Maybe we can reach it this way? https://en.wikipedia.org/wiki/Stationary-action_principle

[hamdani yusuf]

Which Lagrangian do you mean?

When I first read th OP,

Chemists and Biologists don't seem to assume that every system they want to study and model can be represented with a Lagrangian.

I assumed that he referred to Lagrangian mechanics.
https://en.wikipedia.org/wiki/Lagrangian_mechanics

Lagrangian mechanics defines a mechanical system to be a pair (M,L) of a configuration space M and a smooth function L=L(q,v,t) called Lagrangian. By convention, L=T-V, where T and V are the kinetic and potential energy of the system, respectively. Here q ∈ M, and v is the velocity vector at q (v is tangential to M).

CMIIW.

Describing a system using its kinetic and potential energy is also the idea behind Schrodinger equation, which is used by chemists, as shown in this 60 seconds video.

[TommyJ]

For me, the ‘Stationary-action principle’ (principle of least action) also looks applicable as the answer.

The article explains all details, history, disputes and the ‘See also’ part is very useful.
https://en.wikipedia.org/wiki/Stationary-action_principle

As for vaccines, the principle of least action might be explained like this.

Any approach from state or point A to B takes a path that minimizes ‘something’: curves, spirals, anything.
‘Something’ is the action (A), which is described with:
- how fast is it going (K)
- position, where it is now (P)
A = K - P of the object in the entire path.
A= ∫ (K - T) dt

[Eternal Student (OP)]

Hi all and thanks for your time.
   

Which Lagrangian do you mean?

    I probably should have said more in the OP but the answer was always in the title:
"Lagrangian Mechanics -  Why is nature Lazy?"

    For a simple mechanical system,  the Lagrangian,  L   =    K   -   V   
where  K = kinetic energy of the system;   V = potential energy of the system.   ....etc..
  Hamdani correctly provided this reference:

I assumed that he referred to Lagrangian mechanics.
https://en.wikipedia.org/wiki/Lagrangian_mechanics

  Anyway, you (all) have the right idea.  That is the Lagrangian and the least action principle that I am asking about.
However, Lagrangian mechanics is not limited to considering just simple mechanical systems.   The Einstein Field Equations can be derived from a least action principle and a suitable Lagrangian (See Hilbert-Einstein Action in Wiki if you're interested,  https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action).   Most field theories involve a suitable Lagrangian [ https://en.wikipedia.org/wiki/Lagrangian_(field_theory) ].   In general, it seems like most physical systems can be described with a suitable Lagrangian and then a least action principle will describe the evolution of the system from an initial state to a later state.

    Hamdani mentioned quantum mechanics which is perfectly sensible since it has a Hamiltonian in it - but it is a little bit of side-step from the main topic.  Hamiltonian mechanics and Lagrangian mechanics are very similar and we can often derive one from the other.  However, in general Lagrangian mechanics is considered as the parent technique and identifying a Lagrangian to describe a system is preferred rather than identifying a Hamiltonian.
    If you want more information, see https://en.wikipedia.org/wiki/Hamiltonian_mechanics  and note that to define quantities like the generalised momenta  p_i we often assume the Lagrangian, L is already known,   since    p_i =       However, none of this really needs to concern us too much - I think it's fair to say that Lagrangian mechanics is the parent technique but the differences are not worth worrying about.  Both techniques assume there is an action, S(x), and a system evolves from one state to anther so that the action functional is stationary.

    Also, yes, I acknowledge that Chemists use Lagrangian mechanics when they are actually doing Physics.  However, I think that most Chemistry is done with orbital and bonding theory and this is an "effective theory" that was only loosely based on solutions from Quantum Mechanics.   For example, when developing the vaccine for Covid someone probably did model a new molecule with the ideas of p orbitals and π and σ bonds.   However, it seems unlikely that they solved the Schrodinger equation to find an exact solution for that particular molecule.

   Anyway, back to the original question:
Can all Physical Systems be modeled with a Lagrangian?
Why would a system evolve in such a way as to minimise (or at least find a stationary value of) the action?

Best Wishes.

[Eternal Student (OP)]

Hi @TommyJ
   Sorry I didn't directly take a quote or acknowledge your reply.  Anyway it's perfectly good apart from one minor typing error:

A = K - P of the object in the entire path.
A= ∫ (K - T) dt

In the top line that is L,  the Lagrangian  not   A.   But obviously you knew what you were talking about.

Anyway let's take your answer directly:

Any approach from state or point A to B takes a path that minimizes ‘something’: curves, spirals, anything.
‘Something’ is the action (A)

   Why is Nature lazy?   Why does it always want to minimise something?
   What actually makes you so sure there is always a thing, the action, that we can define and Nature would want to minimise that?

[Bored chemist]

Chemists and Biologists don't seem to assume that every system they want to study and model can be represented with a Lagrangian.   I'm fairly sure that when a vaccine for Covid was developed no one started by proposing a new Lagrangian for the system.

As far as I can tell, that statement is correct for any of the given uses of the word.
My understanding is that Lagrangians are only useful for simple "toy" systems like physicists use. (Statisticians do a similar thing with Normal Distributions)

I seem to recall using Lagrangian multipliers for something once. Specifically, I used them to pass an exam in 1985, and then I forgot about them.

[Bored chemist]

For example, when developing the vaccine for Covid someone probably did model a new molecule with the ideas of p orbitals and π and σ bonds. 

I wouldn't bet on that.
With big molecules you might as well model them as balls on springs.

[TommyJ]

Why is Nature lazy?   Why does it always want to minimise something?
   What actually makes you so sure there is always a thing, the action, that we can define and Nature would want to minimise that?

Thank you, for understanding and clarifying the question.
My first explanation was rather 'project management' approach: where we are, were we need to get (result), and what time we have to get the result?
Nice that many things in other areas come to differential equations.

The question on the level of biology/chemistry/physics is deeper. Probably they went with experience, knowledge and trials, and the nature of viruses just follows the nature.
1. Do the biologists and chemists use similar equations? (I was always trying to escape with math from that) - needs answer.
2. Why Nature evolution goes like this 'lazy' way and if it does it all the time? - needs answer (probably or approximate hypothesis comes to some of 'energy conservation' and entropy answers?).

[Bored chemist]

Do the biologists and chemists use similar equations?

Please pardon my ignorance, but similar to what?

[alancalverd]

From any state B a system could evolve into two states A or C where B→A requires the input of energy or results in a decrease of entropy, and B→C is the opposite. A either presumes the existence of an external agent  (so B was not a complete description of the system) or that evolution will lead to a less probable state of the system, which is clearly less likely than C.

We have discussed elsewhere that energy is simply "that which is conserved"  so in a fully described (i.e. closed) system, ΔK = -ΔV by definition, so the most probable path is B→C with ΔS > 0.

[evan_au]

Do the biologists ... use similar equations?

The Least Action Principle refers to "the action is stationary (no change) to first order".
- This implies that there is some differentiable function describing all the alternative ways the system could evolve
- And the path taken is the path which has a stationary point (minimum, maximum or saddleback in some multiple-dimensional space).

However, at its heart, biology is based  on genetics, which is digital rather than analog.
- Differentiating digital functions rarely produces stationary points (unless the digital function is a constant)
- So genetic changes in the offspring can produce no visible change, or a total change (death), or somewhere in between.
- And it can be the same change (eg changing a DNA letter from A to C) that can produce any of these wide variety of outcomes
- One change (A to C) might change the color of your hair, while the same change in a different part of your DNA may provoke early-onset diabetes.
- Both changes have the same action. But because it is digital rather than analog, the effort to make the change is non-zero, so the code is (in some sense) preserved; the system does not slide smoothly from one state to another state, as implied by the Least Action principle, guided by a Lagrangian.

[hamdani yusuf]

   Anyway, back to the original question:
Can all Physical Systems be modeled with a Lagrangian?
Why would a system evolve in such a way as to minimise (or at least find a stationary value of) the action?

AFAIK,  any Physical Systems which can be modeled with Newtonian mechanics can also be modeled with Lagrangian mechanics.

Why would a system evolve in such a way as to maximize entropy?

[Eternal Student (OP)]

Hi all.

We have discussed elsewhere that energy is simply "that which is conserved"  so in a fully described (i.e. closed) system, ΔK = -ΔV by definition, so the most probable path is B→C with ΔS > 0.

   This sounds perfectly sensible to me.   It sounds like statistical mechanics with ΔS as the usual symbol for the change in entropy.  It's obviously a (perfectly reasonable) way of predicting how a system may evolve.  It doesn't seem to be easily formulated as a least action principle.  So were you proposing that the evolution of some systems cannot be described by a Lagrangian and a least action principle?
- - - - - - - - - -

However, at its heart, biology is based  on genetics, which is digital rather than analog............

  (and so derivatives and a least action principle don't necessarily apply)
   Yes, thanks for that evan_au.   This also seems sensible to me.
I just needed to check that Biologists and chemists aren't unhappy with the idea that they do not assume every system of interest to them can be modeled with a Lagrangian and a least action principle.
Anyway, it's just what we need as fuel for the main thrust of the question:   Why do Physicists assume that every system of interest to them can be modeled this way?

Best Wishes.

[Eternal Student (OP)]

Hi again,

AFAIK,  any Physical Systems which can be modeled with Newtonian mechanics can also be modeled with Lagrangian mechanics.

   Yes.
....and for Newtonian systems, it's not too difficult to explain why a least action principle would apply when the Lagrangian is constructed in the usual way as  K - V    (Kinetic  - Potential).

   I should probably make it clear that some scientific dictionaries will state that "Lagrangain mechanics" means only the Lagrangian formulation of what could otherwise be a Newtonian system.    However, I am using the term  "Lagrangian Mechanics" in a much broader sense.  Any system that can be represented with a suitable Lagrangian and a least action principle,  i.e. so that it "looks like" Lagrangian mechanics but the Lagrangian is more elaborate or abstract and the "generalised co-ordinates" do not just refer to the properties of particles.
   See, for example, the Lagrangian formulation of classical field theory.   https://en.wikipedia.org/wiki/Lagrangian_(field_theory)   

Why would a system evolve in such a way as to maximize entropy?

  I suspect this question was aimed at Alancalverd's earlier reply.

Best Wishes.

[Eternal Student (OP)]

Hi again.

    Perhaps I should explain why I'm interested:

There has been a "fashion" or a "passion" for expressing new Physics with a Lagrangian and a least action principle.
Why is this?

Examples:

1.   General Relativity.   Einstein and Hilbert developed a suitable "Action" so that the Einstein Field Equations could be derived from a least action principle.   They did this some time AFTER General Relativity and the E.F.E.  had been published.    For some reason, it was felt important to show that you could use a Lagrangian formulation if you wanted to.  Apparently, it gave more credence to General Relativity.

2.   Noether's theorem.   As far as I can see, this says nothing about general physical systems and symmetries.  The entire paper published in 1918 only manipulates Lagrangians and describes conserved quantities that will exist and symmetries that will be present in the Lagrangian.    If you can't express a system with a Lagrangian then there is no reason to believe that Noether's theorem has any bearing on it at all.   (Don't get me wrong - this is some reason to hope that all physical systems can be represented with a Lagrangian.  There seems to be enough confidence that this is true that Noether's theorem is just assumed to apply to all physical systems).

3.   Lagrangian mechanics started as an alternative formulation for Newtonian systems.  It was very successful.  The ideas seem to have then spread far and wide.  Almost everything is now represented with a Lagrangian.   However, I have never seen any proof that every physical system can be represented with a Lagrangian or exactly why Nature should always want to minimise something (this "something" being what we can call the action).

Don't worry yourselves too much about it.   I honestly don't know why Nature is lazy and why there should always be a quantity we can identify as a Lagrangian.   Chemists and Biologist's don't seem to think that there's a Lagrangian for every problem and every system they want to examine.

Best Wishes.

Late editing:   Changed  "some years after"   to  "some time after".  I can't find a reference for exactly how much time had passed, just that it was after.

[Bored chemist]

 Chemists and Biologist's don't seem to think that there's a Lagrangian for every problem and every system they want to examine.

There may  be one (I don't know), but it doesn't seem to be the most practical way to a solution.

In principle, there's an electron wave function of every nut and bolt in a car (and for the car as a whole).
But it would be absurd for a mechanic to calculate them.

[TommyJ]

Chemists and Biologist's don't seem to think that there's a Lagrangian for every problem and every system they want to examine.

There may  be one (I don't know), but it doesn't seem to be the most practical way to a solution.

In principle, there's an electron wave function of every nut and bolt in a car (and for the car as a whole).
But it would be absurd for a mechanic to calculate them.

'Lagrangian mechanics' is a foundation to many fields. I know better of its principles to be used in economics.
I have got a consultancy from medical experts.
For medical practices there is no ready model (which can be expressed by equations), there is previous experience (that might be put as potential energy analogue) with lots of statistical data, and a vector, which is applied to ‘draw’ the path with experiments. And sometimes this path goes more or less steady, giving a trustable result. But in some cases it may have an unexpected turn, that gives new statistical data and experience to start again.
In the case of RNA vaccine, the result might really look as ‘0’ or ‘1’. But still it is probabilistic model.
RNA gets into a cell, it instructs the cell to make the SARS-CoV-2 spike protein for the immune system to respond to. It depends on a certain state of the immune system of a person.

For me, to put it to a model, this is a Machine Learning principle. Putting result to ‘0’ or ‘1’, where ‘1’ - should be vaccine applicable to a majority of people with certain features (e. g. age, blood type, etc.). Then run tests.

[alancalverd]

So were you proposing that the evolution of some systems cannot be described by a Lagrangian and a least action principle?

Quite the reverse! L = K - V applies, but you asked why nature is lazy and the reason is that by definition the most probable evolution is the one where ΔS > 0 whilst conserving K + V.

However I  do note some potential confusion  as S (not S) is often used as the integral of L!

I am slightly uneasy with the thought that folk might confuse the model L with the reality, hence the anthropomorphic concept of laziness.  If you start with the statistical mechanics and the axiom that the most likely outcome of an evolution is that which is most probable (all the best axioms are tautologies!) and the experimental observation that
K + V is conserved, you can work backwards  to your Lagrangian analysis of energy transfer.

I'll happily admit to taking an engineering approach to most subjects, but it must be remembered that thermodynamics owes more to the development of the steam engine than the steam engine owes to the development of thermodynamics! Most physics begins with product failure.

[Bored chemist]

For me, to put it to a model, this is a Machine Learning principle. Putting result to ‘0’ or ‘1’, where ‘1’ - should be vaccine applicable to a majority of people with certain features (e. g. age, blood type, etc.). Then run tests.

We already know that we need to run tests, what tests to run and how to interpret the results.
What does the extra number crunching add?