Are there any experiments that examine Hamilton's Principle directly?

Question

Or can it be examined?
I 'd glad if you can share some ideas about "principles" in general.

Answer 1

Often in theoretical physics, there can be a large gap between the logical starting point of a theory, and the actual experimental tests.

The principle of least action (aka Hamilton's principle) is such an example. The principle of least action is a mathematical prescription for how to model a wide range of systems. As a purely mathematical statement, and as one that is very general and not specific to one system, it is not really possible to test directly. However, the consequences of the principle of least action can be tested. In particular, since it can be shown that Hamilton's principle produces the same equations of motion as Newton's laws, any system that Newton's laws can be used to model, an action can be used to model as well.

Hamilton's principle is not really that valuable -- on it's own -- from a purely experimental point of view. It is a framework that produces the same results as Newton's laws. From the point of view of an experimentalist, a theory is a black box machine that takes as input the conditions of some experiment, and then outputs a prediction for one or more variables. The experimentalist doesn't care how the black box operates. But since Newton's laws and the principle of least action produce the same predictions, they only differ in how the black box is constructed. The value of Hamilton's principle (or, generally, having multiple mathematical formulations of classical mechanics) is not empirical. Instead, the benefit comes from two different points. First, there are true facts about classical mechanics that can be hard to see in one formalism, but easier in another -- for instance, Noether's theorem is relatively easy to prove starting from an action. Second, when generalizing the laws of classical mechanics to move into describing systems "beyond its boundary" (for example, when going from classical to quantum mechanics), it may be easier to generalize starting from one formulation versus another. In going from classical to quantum mechanics, one doesn't really start from Newton's laws, but usually from either a Lagrangian or Hamiltonian formalism.

Answer 2

About principles in physics:

I want to refer to a discussion by stackexchange contributor Kevin Zhou.

There is a Januari 2020 question titled: 'Why can't the Schrödinger equation be derived?'

In his answer Kevin Zhou takes the opportunity to consider the notion of derivation in a wide perspective.

A quote from that answer:

There is often confusion here because derivations in physics work very differently than proofs in mathematics.

For example, in physics, you can often run derivations in both directions: you can use X to derive Y, and also Y to derive X. That isn't circular reasoning, because the real support for X (or Y) isn't that it can be derived from Y (or X), but that it is supported by some experimental data D. This two-way derivation then tells you that if you have data D supporting X (or Y), then it also supports Y (or X).

Interestingly, Hamilton's stationary action is an instance of the above; the relation between F=ma and Hamilton's stationary action can be walked through in both directions. I will get back to that further down.

About experimental results:

As an example of what forms of calibration are required in order to obtain measurement results I take the example of tracking elementary particles with cloud chambers and/or bubble chambers.

To assess the inertial mass of a particular electrically charged particle: accelerate the particle with a calibrated electric field, and then have the particle traverse a calibrated magnetic field. The magnetic field deflects the particle. From the amount of deflection the mass-to-charge ratio can be inferred. The accuracy of the end result is dependent on how accurately the values of the electric field and the magnetic field are known.

The point is: it's very rare for a value for some property to be obtained through direct measurement; in most cases the value is inferred, the proces requiring that several contributing factors are measured/calibrated with high accuracy.

In our modern time extremely high accuracy standards for time (the second), and length (the meter) have been put into place. With those standards we have the means to obtain a very high accuracy value for Earth gravitational acceleration. That value then forms the basis for calibration of a device that measures amount of force exerted. Video (Veritasium) featuring the biggest force calibration setup in existence: World's heaviest weight

In the effort towards ever greater repeatability and accuracy a new standard for mass has been developed. That definition of mass involves a device called a Kibble Balance.

The operating principle of a Kibble balance is that it compares forces. A Kibble balance allows the process to be repeated to an accuracy within 1 part in $10^8$

The fact that force is a measurable is related to the fact that force has an intrinsic zero point.

For comparison: potential energy does not have an intrinsic zero point. For potential energy we are free to choose any point in the range as zero point. For potential energy the value that counts is difference of potential.

Of course it's not a coincidence that potential energy doesn't have an intrinsic zero point; by definition the potential energy is an integral; potential energy is defined as the negative of the integral of force wrt position coordinate. When you perform integration the value is only defined up to an additive constant.

This is also an instance of bi-directional relation. Some people may prefer to think of force as that which is obtained when you differentiate potential energy with respect to position coordinate.

Whatever the preference, what stands is the nature of the relation between the two concepts: integration/differention.

We have the measurables, such as acceleration, mass, and force.

The concepts of potential energy and kinetic energy are an abstraction level away from the measurables.

As we know: a recurring result is that we see confirmation of conservation of energy. And if it looks as if energy is not conserved then again and again a refinement of measurements eventually finds confirmation of conservation of energy.

There is no way of measuring that directly. The measurables are things like acceleration, mass, and force. To assess what the energies are doing you re-express the measurement data in terms of energy.

So: it's not possible to subject the notion of conservation of energy to experiment directly; the measurement data are not in terms of energy; the measurement data must be converted to expressions in terms of energy.

I think of Hamilton's principle as being an abstraction level away from the level of expressing energies. Abstraction upon abstraction.

Corroboration of Hamilton's principle is accordingly more indirect than corroboration of conservation of energy.

From $F=ma$ to Hamilton's stationary action

As stated as the start of this answer, the relation between $F=ma$ and Hamilton's stationary action can be walked through in both directions.

I recently submitted an answer about Hamilton's stationary action, in which I give a series of steps starting with F=ma, arriving at an expression that is equivalent to Hamilton's stationary action.

(Questions about Hamilton's stationary action have been submitted multiple times; I opted to submit that answer to the first time that a question about Hamilton's stationary action was submitted here on physics stackexchange.)