Hamilton's Variation Principle - Fundamental Questions

[Jimbone]

I read the following in Fowles & Cassiday's Mechanics:

"The correct motion that a body takes through space is that which minimizes the time integral of the difference between the kinetic and potential energies"
or
[itex]\delta[/itex]J = ∫ L dt = 0


I understand that this is describing the minimization of the "action" which allows us to find a unique path which a body must travel through space. But this statement strikes me as especially profound, why this quantity (T-V) ? and what does it mean that nature requires the path which minimizes it's time integral? Can anyone think of a physical situation that shows how a counter-intuitive path doesn't minimize the integral?

Thanks for your thoughts

 

[Ken G]

The classic example is the trajectory of light in vaccum. The Lagrangian for light in vacuum is essentially constant (fixed kinetic energy, no potential energy), so the action works just like the propagation time. Since light has a very fast frequency associated with it, neighboring paths that take a different amount of time show up with very different phases, so cancel. The path that contributes an amplitude that does not cancel is therefore the path of least time, because around an extremum, neighboring paths will also take pretty close to the same amount of time. This is also called a "stationary phase" constraint, and can be generalized to media with indices of refraction (where it gives you Snell's law). Examples of least-time paths in a vacuum are a straight line, or a broken line that bounces off a straight boundary so as to satisify angle of incidence equals angle of reflection.

This example suggests that Hamilton's principle has something to do with wave-particle duality. Of course, Hamilton never knew that, nor did anyone of his era.

 

[Erland]

Allow me to catch on here with a reflection of my own.

Although I agree of the mathematical beauty of the Hamiltonian formulation, I find it highly counterintuitive. It seems to be inconsistent with our view of causality.

Suppose a system moves from state S1 at time t1 to state S2 at time t2. The system will then follow the path through phase space that minimizes or maximizes (or more generelly, is stationary point of) the integral of the Lagrangian over the time interval.

But then, at t1:

1. How does the system know to which state it will move at t2?
2. How can the system calculate the path that is optimal in the above sense?

This just doesn't make sense if we think in terms of causality. It is much easier think that at t1, the system reacts in a way prescribed by its state and moves in accordance with that, so that the relative positions and velocities of the particles in the system determine their accelerationns in that moment. But in the Hamiltonian view, the system must be able to predict its future state. That seems strange.

 

[Bill_K]

The classic example is the trajectory of light in vaccum...The path that contributes an amplitude that does not cancel is therefore the path of least time.

What you're describing is Fermat's principle of geometrical optics, not Hamilton's principle. They're not even analogous.

 

[vanhees71]

In a way it is analogous. The answer, why the equations of motion follow Hamilton's principle is a pretty deep one. To my knowledge it can only be answered using the path-integral formulation of quantum mechanics. If a particle is observed in basically classical circumstances, the action is large compared to [itex]\hbar[/itex], and thus the path integral for the time-evolution kernel (propagator) is dominated by the path, along which the action is minimal (method of stationary phase for the approximate calculation of the path integral).

In more conventional terms, one starts from the Schrödinger equation and does an expansion in orders of [itex]\hbar[/itex] (WKB approximation), which in leading order leads to the Hamilton-Jacobi partial differential equation of classical mechanics which is equivalent to Hamilton's canonical equations, which follow from the action principle.

Schrödinger came to his version of quantum theory ("wave mechanics") by the analogy with optics. Geometrical optics follows as an approximation from Maxwell's equations of electromagnetism (eikonal approximation, valid if the typical wavelength of the em. wave is small compared to the typical measures of refracting matter around). He took the opposite path by taking up de Broglie's idea of matter waves and asked for the equation, which will lead to the Hamilton-Jacobi equation for the particle. In this way, he first came to an equation, that we nowadays call the Klein-Gordon equation. Evaluating the eigensolutions for the hydrogen atom, this gave him the wrong energy levels compared to the observed spectral lines. Since the deviations were only on the level of accuracy of the fine-structure splitting, which has been thought to be due to relativistic effects (Sommerfeld's relativistic treatment of the Bohr model), he stepped back from a relativistic equation and asked the same question for non-relativistic particles, which finally lead him to his famous non-relativistic wave equation, that is named after him. The solution for the puzzle with the fine structure was given later by Dirac, introducing the idea of spin 1/2 into relativistic quantum mechanics, leading him to the equation, that is named after him.

 

[Bill_K]

No it is not. Before launching into a discussion of path integrals and the deep analogy between optics, quantum mechanics and classical mechanics, it would help to understand classical mechanics itself first. There are several variational principles in classical mechanics, and Hamilton's principle is not to be confused with the others.

Hamilton's Principle: δ ∫_t1^t₂ L dt = 0
1) The limits t₁ and t₂ are fixed. (So any comparison to Fermat's Principle, which singles out the "path of least time", is clearly incorrect.)
2) The variation is built from a sequence of virtual displacements, in which the coordinates are varied with no change in t, subject to the constraints. Virtual displacements do not in general correspond to possible actual displacements of the system, and consequently the varied path does not have to correspond to a possible path of motion for the system.

What you're apparently thinking of is the Principle of Least Action: Δ ∫ ∑ p_iq^·_i = 0
Here the variations can involve displacements in time, and the varied path is required to be consistent with the physical motion. The time t may be varied, even at the end points.

 

[vanhees71]

That's true, there's not only Hamilton's principle but also others. As you pointed out, they differ in how the variations of the end points are treated and whether time is varied or not.

All are equivalent and all cannot be understood physically without taking some recourse to quantum theory. Within classical mechanics, Hamilton's principle is justified simply by the fact that it is a very powerful and convenient way to write down the equations of motion, which can be further analyzed within this formulation of the dynamics than in the "naive" Newtonian one. Nevertheless, physics wise they just contain Newton's equations of motion (or their relativistic generalizations).

As far as I know, a deeper reason, why the equations of motion satisfy Hamilton's variational principle, can only be given from quantum mechanics.

 

[epenguin]

Although I agree of the mathematical beauty of the Hamiltonian formulation, I find it highly counterintuitive. It seems to be inconsistent with our view of causality.

I suggest that could be because Newtonian mechanics is inconsistent with your view of causality and vice versa.

It cannot be consistent with it in one formulation and inconsistent in another when each formulation is mathematically equivalent to the other, can be derived from the other.

We do often talk of 'cause' in mechanics but we are using the word in a very loose sense. Whenever I meet the word in that context I go along with it but with this mental reservation at back of mind that the word was unnecessary.

According to many, including Bertrand Russell who also wrote an essay about what 'cause' means, Newtonian mechanics is not causal but merely descriptive of paths of material bodies in space and time.

 

[Erland]

I suggest that could be because Newtonian mechanics is inconsistent with your view of causality and vice versa.

It cannot be consistent with it in one formulation and inconsistent in another when each formulation is mathematically equivalent to the other, can be derived from the other.

If the word "inconsistent" is used in the formal sense, you are right. I had a more informal meaning in mind. It is hard to find the right words. Maybe I should have used the formulation that the Hamiltonian formulation makes no sense in terms of (our or my understanding of) causality.

 

[Ken G]

What you're describing is Fermat's principle of geometrical optics, not Hamilton's principle. They're not even analogous.

They certainly are analogous. From the Wiki on Fermat's principle, at http://en.wikipedia.org/wiki/Fermat's_principle, it states:
"Fermat's principle has the same form as Hamilton's principle and it is the basis of Hamiltonian optics. The Hamiltonian formulation of geometrical optics, similar to Winston's formulation, shares much of the mathematical formalism with Hamiltonian mechanics."
vanhees71 described the equivalences even more clearly, the nature of the allowed variations that go into the variational principle are not crucial, and different choices about independent variables can be made.

 

[Ken G]

As far as I know, a deeper reason, why the equations of motion satisfy Hamilton's variational principle, can only be given from quantum mechanics.

Yes, I believe this is the key point, and it is the reason for both Fermat's principle and Hamilton's principle, as well as for the deep connection between them.

 

[Ken G]

It cannot be consistent with it in one formulation and inconsistent in another when each formulation is mathematically equivalent to the other, can be derived from the other.

We do often talk of 'cause' in mechanics but we are using the word in a very loose sense. Whenever I meet the word in that context I go along with it but with this mental reservation at back of mind that the word was unnecessary.

I think you have hit the nail on the head on both counts. Causality is a kind of interpretation of mechanics, but it doesn't really exist in any of the postulates. None of Newton's three laws refer to it explicitly, even if they are easy to interpret that way. So we are free to interpret forces as the cause of acceleration or forces as required by the presence of acceleration (the latter being closer to how Hamiltonian mechanics looks at it). Even Newton's laws on action/reaction pairs never says which one is the action and which one is the "re"-action! Is it that when I push on the wall, the wall pushes back on me, or is the reason that I am able to push on the wall because the wall will push back on me?

But I do think Erland is onto something when asking, how does the object know to take that path? It is really the first glimpse of what becomes so much clearer in quantum mechanics-- path doesn't have meaning unless it is queried at each point along the way, in which case we have many actualizations of Hamilton's principle along the way. Even Hamilton could have gotten a glimpse of quantum mechanics by thinking along those lines.

 

[Jimbone]

Thanks everyone, this is precisely the sort of discussion I was looking for. Fortunately all I'm taking this quarter is a Goldstein Mechanics course and a Quantum course so I've plenty of time to give this some further thought.

In more conventional terms, one starts from the Schrödinger equation and does an expansion in orders of ℏ (WKB approximation), which in leading order leads to the Hamilton-Jacobi partial differential equation of classical mechanics which is equivalent to Hamilton's canonical equations, which follow from the action principle.

This is really interesting but it's difficult for me to make any sense out of the fact that the least action principle(I know this is intimately connected to the variation principle) can be derived from the Shrodinger equation.

What does it imply about the Hamiltonian(or least action principle) that it has such strong implications in both macroscopic and microscopic systems? Why is it's role in quantum so much more profound?

 

[Ken G]

Those are excellent questions, and call for a deep understanding of the connections between QM and CM. I'm going to wait for vanhees71's answer rather than diving in on that one, but at least you can see that you are asking questions at the most fundamental level at this point! About all I could say is that Feynman's path integral approach to QM is illuminating-- it basically explains how particle trajectories (and the least action principle that governs them) and wave mechanics (and the Huygens' principle that governs them) are unified by the path integral picture. In this picture, constructive interference is seen as the crux of pretty much everything that is allowed to happen, and this places even wave mechanics, which already emphasizes constructive interference, into a wider context of all the things that could be happening but aren't. The bottom line seems to be, contrary to the old single particle and the forces that act on it picture of Newton, what happens appears to be the result of a kind of gigantic choir of voices, all saying different things, but drowning each other out except for the lone few who are saying the same things. On the other hand, it is ever only the effect of this "choir" that we actually observe, so no empiricist would ever hold that the voices actually exist-- they are the voices of those proverbial "angels on the pin", if you will.