Action, Lagrangian

[Ioannis1404]

In classical mechanics to establish the Euler-Lagrange equations of motion of a particle we "minimize" the action, that is the integral of the Lagrangian, prescribing as the integral limits the initial and final positions of the particle. Usually, for a problem in mechanics we do not know the initial and final positions but the initial position q and initial velocity q'. How can we "minimize" the action under the conditions of given initial position and initial velocity (instead of initial and final positins) to get the Euler-Lagrange equations?

 

[PeroK]

In classical mechanics to establish the Euler-Lagrange equations of motion of a particle we "minimize" the action, that is the integral of the Lagrangian, prescribing as the integral limits the initial and final positions of the particle. Usually, for a problem in mechanics we do not know the initial and final positions but the initial position q and initial velocity q'. How can we "minimize" the action under the conditions of given initial position and initial velocity (instead of initial and final positins) to get the Euler-Lagrange equations?

The solutions to the Euler-Lagrange equations are general equations of motion for the system. It's only after you have the E-L equations that you specify the initial conditions.

The initial and final positions used in the derivation of E-L equations are arbitrary and thus the solution is as general as $F = ma$, where no initial conditions are assumed.

 

[Ioannis1404]

Thank you for your answer. Well, perhaps I did not make clear what I am trying to understand. What seems a contradiction to me is the following. We derive the Euler-Lagrange equations by minimising the action assuming the initial and final positions are known. But then we solve the Euler-Lagrange equations given initial position and velocity.
My question is then: Would not it be more consistent to derive the Euler-Lagrange equations from the action using the same initial conditions we use when it comes to solve these equations? In simple words take the integral of the Lagrangian (the action), assume given the initial position and initial velocity (but leave unknown the final position), and find a way which minimises the action under these conditions leading to the very same Euler-Lagrange equations which we get usually. Thanks again.

 

[PeroK]

We derive the Euler-Lagrange equations by minimising the action assuming the initial and final positions are known.

This is not what we are doing. We are not solving a dynamical problem. We are solving a functional problem. We find a family of solutions based on the principle of least action over a class of functions in an abstract function space.

Only when we have the Euler-Lagrange equations do we tackle specific dynamic problems with specific initial conditions and solutions.

 

[Ioannis1404]

Thanks again, however I think I found a paper by Chad R. Galley who, seems to me, raise a similar question as mine. I copy from the abstract '... Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data'. The title of the paper is 'The classical mechanics of non-conservative systems'. I will try to study it and if I understand it I will report here.

 

[PeroK]

Hamilton's principle has a subtle pitfall that often goes unnoticed in physics ...

This is true. There are several threads on here about this.

 

[PeroK]

E.g.

https://www.physicsforums.com/threads/independent-coordinates-are-dependent.1006998/#post-6539554

 

[vanhees71]

I think to really understand, why the Hamilton action principle works beyond the usual observation that it gives the same Lagrange equations of motion as d'Alembert's principle and can be used for more subtle general derivations (among them as the most important ever Noether's theorems about symmetries and conservation laws, the Hamilton canonical formalism, Poisson brackets, and the Hamilton-Jacobi partial differential equations, all of which paved the way to formulate quantum mechanics heuristically), is indeed quantum mechanics using Feynman's path-integral formulation to evaluate the propagator of the quantum state. In the position representation it's the transition amplitude for a particle being at position $\vec{x}'$ at time $t'$ to be found at position $\vec{x}$ at time $t>t'$. The functional integral is thus over all paths with this boundary conditions (!) at times $t'$ and $t$. There's no constraint on the (canonical) momenta in this formulation.

Now the classical limit is if the action for the "most relevant" paths is very large compared to $\hbar$, and then to evaluate the path integral you may use the "method of steepest descent", i.e., you may consider only the path which is a stationary point of the action functional. The contributions from the other paths are negligible, because the functional integrand is so rapidly oscillating that these contributions cancel nearly each other out. That's why the classical limit is given by the variational action principle with the boundary conditions on the configuration variables but not the (canonical) momenta.

 

[Ioannis1404]

Thanks vanhees71 for this informative comment but it seems to me that it is completely unrelated to my initial question. However, I am very new in this forum and perhaps I did not understand yet the way the questions and provided answers are handled.

 

[Cleonis]

Let me write some things in general about 'initial value problem' versus 'boundary value problem', in the context of mechanics, as I think that is the core of your question.


For completeness I start with the trivial case of an object moving at a uniform velocity, with one spatial degree of freedom:

Initial value problem
A train is moving at 120 kilometers per hour. How long does it take to reach a destination 150 kilometers away?

Boundary value problem:
A train travels to a train station 150 kilometers away. At what velocity must the train travel such that it arrives at the destination after 2 hours?


For the next level up we allow a known uniform acceleration, and two spatial degrees of freedom.

You have a cannon that can shoot projectiles. Firing of a projectile occurs from a level surface, so that the projectile impacting the ground is at the same height as when it was fired. Assume standard Earth gravity.

Initial value problem:
Given a nozzle velocity, and angle of the barrel with respect to the horizontal, what is the horizontal distance that the projectile will travel, and what will be the duration of the flight?

Boundary value problem:
What must the horizontal velocity component be, and what must the vertical velocity component be, such that the horizontal distance traveled is x meters, and the duration of the flight is t seconds?

From a physics point of view the initial value problem and the boundary value problem are the same case. Mathematically the order of operations is different of course, but those mathematical details are outside the scope of physics

Let's take a closer look at this projectile motion case:
If you use Newtonian mechanics then you obtain the requested horizontal velocity component from the requested distance and the requested duration of t seconds. And then the vertical velocity component needs to be such that given the Earth's gravitational acceleration the duration of the vertical motion is also the requested duration of t seconds.


Of course, in physics it is customary to express a physics problem in terms of interconversion of energy, because of the advantages that that offers.

We have what I like to call the biggest windfall in the history of phyics: the synergy of Pythagoras' theorem and kinetic energy.

There is a contrast between dealing with momentum and dealing with kinetic energy: when you have motion in, say, two spatial dimensions, then to go from momentum components (along the two spatial degrees of freedom) to the resultant momentum you have to apply Pythagoras' theorem. But for kinetic energy the velocity is squared anyway: so the resultant kinetic energy is simply the sum of the kinetic energy components.
(Also: the velocity vector is squared using the inner product, which means that you lose directional information. How is it that you can afford to lose the directional information of the velocity vector? That is because you still have the necessary directional information. The necessary directional information is available in the form of the gradient of the potential energy.)

In math notation:

$v_{ \text{resultant}}^2 = v_x^2 + v_y^2$

$E_{K, \text{resultant}} = E_{K,x} + E_{K,y}$

The connection between expressing physics taking place in terms of F=ma, and expressing the physics taking place in terms of interconversion of kinetic and potential energy is given by the Work-Energy theorem.

So: let's now look at the projectile problem in terms of interconversion of kinetic and potential energy:

Initial value problem:
Given an initial kinetic energy, and a given barrel angle (of the cannon), where will the projectile impact?

Boundary value problem:
What must the initial kinetic energy be, and what angle must the barrel be at, such that the horizontal distance traveled is x meters, and the duration of the flight is t seconds?

Just as in the case of treating the problem in terms of F=ma: you need a different order of operations to solve the problem, but the underlying physics is the same.



With the above in place:
In your question you point out that the application of the Euler-Lagrange equation addresses intial value problems, whereas the derivation of the Euler-Lagrange equation adresses the physics taking place as a boundary value problem.

My point is: such distinction as one can make between 'initial value problem' and 'boundary value problem' is mathematical only. It changes the order of mathematical operations, but not the physics taking place.
In the case of uniform velocity (train traveling from one station to another) you just casually interconvert between initial value problem and boundary value problem.

That possibility of interconversion between stating the case as initial value problem or stating the case as boundary value problem isn't just a feature of the uniform velocity case; it generalizes to all of mechanics.


(I am aware that some authors do state that treatment in terms of initial value problem is very much distinct from treatment in terms of boundary value problem, but that position is untenable.)

 

[sysprog]

@Cleonis, is the cannon embedded into the ground? If not, how does this work?

Firing of a projectile occurs from a level surface, so that the projectile impacting the ground is at the same height as when it was fired. Assume standard Earth gravity.

 

[wrobel]

Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data'

This trivial observation is not unnoticed just nobody has problems with it except the author of the article . There are two different problems.
The first one is to find a critical point of the Action functional under certain boundary conditions. This problem is reduced to the corresponding boundary value problem for the well-known ode.
The second problem is the initial value problem for these ode.
Both problems are physically reasonable. No pitfalls

https://arxiv.org/ is a very useful resource but one should understand that the majority of papers from there have not been reviewed by the scientific community

 

[sysprog]

. . . the well-known ode.

I thought this was the only well-known ode:

Ode on a Grecian Urn​

BY John Keats
Thou still unravish'd bride of quietness,
Thou foster-child of silence and slow time,
Sylvan historian, who canst thus express
A flowery tale more sweetly than our rhyme:
What leaf-fring'd legend haunts about thy shape
Of deities or mortals, or of both,
In Tempe or the dales of Arcady?
What men or gods are these? What maidens loth?
What mad pursuit? What struggle to escape?
What pipes and timbrels? What wild ecstasy?

Heard melodies are sweet, but those unheard
Are sweeter; therefore, ye soft pipes, play on;
Not to the sensual ear, but, more endear'd,
Pipe to the spirit ditties of no tone:
Fair youth, beneath the trees, thou canst not leave
Thy song, nor ever can those trees be bare;
Bold Lover, never, never canst thou kiss,
Though winning near the goal yet, do not grieve;
She cannot fade, though thou hast not thy bliss,
For ever wilt thou love, and she be fair!

Ah, happy, happy boughs! that cannot shed
Your leaves, nor ever bid the Spring adieu;
And, happy melodist, unwearied,
For ever piping songs for ever new;
More happy love! more happy, happy love!
For ever warm and still to be enjoy'd,
For ever panting, and for ever young;
All breathing human passion far above,
That leaves a heart high-sorrowful and cloy'd,
A burning forehead, and a parching tongue.

Who are these coming to the sacrifice?
To what green altar, O mysterious priest,
Lead'st thou that heifer lowing at the skies,
And all her silken flanks with garlands drest?
What little town by river or sea shore,
Or mountain-built with peaceful citadel,
Is emptied of this folk, this pious morn?
And, little town, thy streets for evermore
Will silent be; and not a soul to tell
Why thou art desolate, can e'er return.

O Attic shape! Fair attitude! with brede
Of marble men and maidens overwrought,
With forest branches and the trodden weed;
Thou, silent form, dost tease us out of thought
As doth eternity: Cold Pastoral!
When old age shall this generation waste,
Thou shalt remain, in midst of other woe
Than ours, a friend to man, to whom thou say'st,
"Beauty is truth, truth beauty,—that is all
Ye know on earth, and all ye need to know."

(Maybe a math guy might have a different idea.)