Exploring the Concept of Action: Definitions and Equivalencies

[Hetware]

What is "action"?

I've seen various definitions of "action" in the context of a principle of least action. Lemons states it as the integral of momentum over path length. https://www.amazon.com/dp/0691026637/?tag=pfamazon01-20 Others, for example, John C. Baez, (http://math.ucr.edu/home/baez/classical/#lagrangian) present it as the time integral of the Lagrangian. He also gives an alternative definition, later in his presentation, which I have not read yet.

Are there different constructs known as "action" which cannot be shown to be identical, or are all of these definitions of "action" demonstrably equivalent?

 

[Simon Bridge]

It is something most people need a run-up to understand.
I've seen the action best described as the difference between potential and kinetic energy (in classical mechanics).

So your action is the difference between your capability of motion and your actual motion.
The principle of least action says that there is no difference without some sort of constraint - in which case the motion will be such that this difference is as small as possible.

Have a look through:
http://www.youtube.com/view_play_list?p=189C0DCE90CB6D81
... start at lecture 1 ... lecture 2 describes action but you want to see lecture 1 for base concepts.

 

[Studiot]

OK action has the dimensions ML²T^-1

This is energy x time or joule-seconds in SI.

Now your path integral (momentum over distance) has dimensions momentum x distance ie (MLT^-1).(L) = ML²T^-1 (unsurprisingly)

Your second integral is energy times time directly with dimensions (ML²T^-2).(T) = ML²T^-1 as before.

So other constructs that have these dimensions are candidates eg power times time² (So are the neon adverts in NYC considered actionable? )

@Simon, That was a most interesting summary, thank you.

 

[Dickfore]

Action S is the functional
$$S[q(t)] = \int_{t_i}^{t_f}{L\left(t, q, \dot{q} \right) \, dt}$$
where L is the Larange's function of the system, that depends on the generalized velocities, generalized coordinates and (for open systems and/or time-dependent constraints) on time.

It has the property that it acquires minimal (extremal) values on the true trajectory of the system.

 

[Hetware]

OK action has the dimensions ML²T^-1

This is energy x time or joule-seconds in SI.

Now your path integral (momentum over distance) has dimensions momentum x distance ie (MLT^-1).(L) = ML²T^-1 (unsurprisingly)

Your second integral is energy times time directly with dimensions (ML²T^-2).(T) = ML²T^-1 as before.

So other constructs that have these dimensions are candidates eg power times time² (So are the neon adverts in NYC considered actionable? )

@Simon, That was a most interesting summary, thank you.

Indeed, they have the same units. But so do torque and work. I'll have to think about this some more.

Reading further, I see that Lemons rejects the designation of "action" for the integral of [T - U]dt over t. He even explicitly states that Feynman was in error to use the term action for this quantity.

No wonder I get confused when I read about this subject.

 

[Dickfore]

Who is lemons?

 

[Simon Bridge]

Don S Lemons of Perfect Form fame?

Also see:
http://www.eftaylor.com/pub/energy_to_action.html

But seriously - watch the lecture series.
If you want to know why Lemons took that position, you need to look at the reasons he used to back it up. You may also want to look at Hamilton's Paper:

http://link.aip.org/link/?&l_creator=fthtml&l_dir=REV&l_rel=CITES&from_key=AJPIAS000072000004000514000001&from_keyType=CVIPS&from_loc=AIP&to_j=AJPIAS&to_v=73&to_p=831&to_loc=AIP&to_url=http%3A%2F%2Flink.aip.org%2Flink%2F%3FAJP%2F73%2F831%2F1

 

[Studiot]

Indeed, they have the same units. But so do torque and work. I'll have to think about this some more.

I was trying to offer a simple down to Earth view.

Are you quite sure about this?

For both torque and work are the product of two vectors.
However torque is a vector and work is a scalar since the first is the vector product and the second is the scalar product.
Further the second vector is different ie the perpendicular distance in one case and the parallel distance in the other. Can you see what happens when we take the other product in each case?

 

[Hetware]

Don S Lemons of Perfect Form fame?

Also see:
http://www.eftaylor.com/pub/energy_to_action.html

But seriously - watch the lecture series.
If you want to know why Lemons took that position, you need to look at the reasons he used to back it up. You may also want to look at Hamilton's Paper:

http://link.aip.org/link/?&l_creator=fthtml&l_dir=REV&l_rel=CITES&from_key=AJPIAS000072000004000514000001&from_keyType=CVIPS&from_loc=AIP&to_j=AJPIAS&to_v=73&to_p=831&to_loc=AIP&to_url=http%3A%2F%2Flink.aip.org%2Flink%2F%3FAJP%2F73%2F831%2F1

Lemons's argument is historical.

I'm reading Hanc and Taylor's paper. I've also been watching Suskind's other classical mechanics lectures.

 

[Hetware]

I was trying to offer a simple down to Earth view.

Are you quite sure about this?

For both torque and work are the product of two vectors.
However torque is a vector and work is a scalar since the first is the vector product and the second is the scalar product.
Further the second vector is different ie the perpendicular distance in one case and the parallel distance in the other. Can you see what happens when we take the other product in each case?

That's my point. Work, energy and torque have the same units, but torque and work are clearly distinct. Whereas, work and entergy are different sides of the same coin. Having the same units doesn't necessitate two quantities are of the same nature.

I'm trying to figure out if there is some transformation that shows these two definition of action to be identical.

 

[Simon Bridge]

That's my point. Work, energy and torque have the same units, but torque and work are clearly distinct. Whereas, work and entergy are different sides of the same coin. Having the same units doesn't necessitate two quantities are of the same nature.

That is not the assertion Studiot made - go back and reread: he is saying the opposite - for two quantities to have the same nature, we would expect them to have same dimensions. Not everything with the same dimensions has the same nature... though torque is cheating a bit since it is not well defined for generalized coordinates. Can you find another example?

I'm trying to figure out if there is some transformation that shows these two definition of action to be identical.

The dimensional analysis Studiot did shows you that there certainly is.

A good place to start is by looking at the context: why did the authors choose that particular representation for the action?

BTW: $$dW=F.dx = \frac{dp}{dt}dx \Rightarrow dWdt=dpdx$$ eg. starting with the definition of Work ... end up with something that almost looks like p.dx - the line integral thingy. Work done by an additional applied force would be W=T-U wouldn't it?

Should get you started...

Lemons's argument is historical.

That the word "action" has, historically, been used to denote something different and this sort of gay misappropriation is a debasement of language ;) Something like that?

 

[Studiot]

Wasn't it used at one time for something like we now call a jerk or a boost and before that the agent that precipated a reaction?

 

[haruspex]

That's my point. Work, energy and torque have the same units, but torque and work are clearly distinct. Whereas, work and energy are different sides of the same coin. Having the same units doesn't necessitate two quantities are of the same nature.

Similarly, angular momentum and action have the same 'units', though the former is a vector and the latter a scalar. I've tended to dismiss this as coincidence, yet, intriguingly, the magnitude of angular momentum is quantized...

 

[Hetware]

That is not the assertion Studiot made - go back and reread: he is saying the opposite - for two quantities to have the same nature, we would expect them to have same dimensions. Not everything with the same dimensions has the same nature... though torque is cheating a bit since it is not well defined for generalized coordinates. Can you find another example?The dimensional analysis Studiot did shows you that there certainly is.

A good place to start is by looking at the context: why did the authors choose that particular representation for the action?

BTW: $$dW=F.dx = \frac{dp}{dt}dx \Rightarrow dWdt=dpdx$$ eg. starting with the definition of Work ... end up with something that almost looks like p.dx - the line integral thingy. Work done by an additional applied force would be W=T-U wouldn't it?

Should get you started...

OK, I have to admit to confusing the equivalence of the integrand and the integral. Tomorrow is another day, and I will try to hit this fresh, in my "spare" time. (I don't get payed for this.)

That the word "action" has, historically, been used to denote something different and this sort of gay misappropriation is a debasement of language ;) Something like that?

Yes, but understandably. Pierre Louis Maupertuis apparently used the designation to denote what amounts to the integral of momentum over distance.

 

[Hetware]

http://link.aip.org/link/?&l_creator=fthtml&l_dir=REV&l_rel=CITES&from_key=AJPIAS000072000004000514000001&from_keyType=CVIPS&from_loc=AIP&to_j=AJPIAS&to_v=73&to_p=831&to_loc=AIP&to_url=http%3A%2F%2Flink.aip.org%2Flink%2F%3FAJP%2F73%2F831%2F1

Hamilton’s principle: why is the integrated difference of kinetic and potential energy
minimized?

Alberto G. Rojo

Department of Physics, Oakland University, Rochester, MI 48309.

There are different versions of that paper. It's really good. (I'll refrain from posting links)

I can't offer a good first source on the topic of analytic dynamics. Used in conjunction with a good mentor, Lemons's book might be very good. I've had to return to it several times because I missed the subtleties the first and subsequent times.

Rojo's approach, with a bit of expansion on the mathematical details might be a good way to introduce the topic. The description of:

"Frictionless pulleys that can slide in horizontal lines
with a string passing through them a sufficient number of
times gives the trajectory of the particle if $T_{i}$ is identified
with $mv_{i}$ at each segment. Since the string can only pass
through each pulley an integer number of times, the ratios of
the velocities are approximated by the ratio of the times the
rope passes through each segment."


is a bit nebulous to me. But it doesn't appear essential to understanding the rest of the article.

I'm the kind of person who has to "get the feel" of what the mathematical expressions say. Without that, I get mental blocks. I'm still trying to catch the groove of the Euler-Lagrange equation. Susskind seems to have presented it in a way that helps. I shall review his lectures.

Years ago, I was reading Donald H. Menzel's Mathematical Physics When I realized q and q^' being treated as independent variables was contrary to my physical intuition. I felt better about my confusion when Penrose explained "Each $\dot{q}^r$ has to be treated as an independent variable (independent of ${q}^r$, in particular) in this expression. This is one of the initially baffling features of Lagrangians--but it works!"

I tried terribly hard to explain to physicists at the University of Maryland why this bothered me. None of them really caught on to my objection. It is this: in the real world, with real objects following real trajectories, ${q}^r$ and $\dot{q}^r$ are coupled. For any trajectory, other than uniform rectilinear motion, a change in ${q}^r$ changes $\dot{q}^r$.

The reason I'm so pedantic about this is because statements of QM are typically made in terms of phase space, or it's first cousin, $\Re^n\times\Re^n$ (${q}^r$, $\dot{q}^r$) configuration space. These are mathematical fantasies that only tangentially correspond with physical reality.

 

[Studiot]

Hamilton's principle and Lagrangian mechanics can be derived from the principle of virtual work which is older and in some ways more far reaching.

 

[Hetware]

Hamilton's principle and Lagrangian mechanics can be derived from the principle of virtual work which is older and in some ways more far reaching.

Will you please provide a reference to a representative development?

 

[Simon Bridge]

BTW: off the observations about dimensional analysis:
Strictly: the magnitude of the torque has units of "energy per radian" ... hence, it has dimensions of energy. There is probably a formulation of the action in terms of torque somewhere but I imagine you'll see it more in terms of angular momentum.

Torque is not the same thing as energy but compare:

Mass has units of "energy over speed-squared" and dimensions of the same. What you need to make mass the same thing as energy is some square-speed that is a constant for everybody ;)

Aside: I have associated the principle of virtual work with statics.

 

[Studiot]

Aside: I have associated the principle of virtual work with statics.

Yes this correct and the reason to find the references sought in books primarily on statics.
But don't forget that statics includes the response of deformable bodies to loads. This response includes movement so must a full treatment must include the equations of motion.

Anyway to the question Hetware asked, here is an extract form

Energy Methods in Stress Analysis by T.H. Richards

It is often necessary to compute the response of a structure to dynamic loading: when a structure is set in motion, its elements experience accelerations so that inertial forces are called into play. According to D'Alembert's Principle, a dynamics problem may be converted into a statics one by taking these intertia forces into account so that the principle of virtual work may be employed. If the displacements are considered to be prescribed at two instants of time t₁ and t₂ so that any virtual displacements must vanish then, integrating the virtual work equation with respect to time, with the inertia forces incorporated into the body force term, leads to Hamilton's Principle. This may be regarded as the basic principle of dynamics and states that:

Of all the geometriccally possible motions which a system may execute, the true one is one which renders

δA = δ∫t2t1Ldt = 0

The quantity A is called the action whilst L = T - U - Ω is the Lagrangian function. T is the kinetic energy, U the strain energy, and Ω the potential energy of the applied loads.

Of course the existence of potential implies conservative forces. One of the advantages of virtual work is that it is valid for such a general range of conditions. Linear and non linear elastic, conservative and non conservative fields, rigid and deformable bodies, particles and systems of particles and so on.

Richards devotes most of the rest of the book to expanding on this theme, in classical engineering calculus terminology.

If you want a modern physics derivation couched in modern mathematical terms you will need to read

Mathematical foundations of Elasticity by Marsden and Hughes.

However, as its name suggests it covers elasticity so is essential about conservative fields.

 

[Hetware]

Thanks. I'm glad that I was reserved in my skepticism. Elasticity and fluid mechanics (continuum mechanics) is the path to understanding all of physics.