In this post, I provide my answer to the question which I had raised last time, viz., about the differences between the , the , and the (the first two, of the usual calculus, and the last one, of the calculus of variations).
Some pre-requisite ideas:
A system is some physical object chosen (or isolated) for study. For continua, it is convenient to select a region of space for study, in which case that region of space (holding some physical continuum) may also be regarded as a system. The system boundary is an abstraction.
A state of a system denotes a physically unique and reproducible condition of that system. State properties are the properties or attributes that together uniquely and fully characterize a state of a system, for the chosen purposes. The state is an axiom, and state properties are its corollary.
State properties for continua are typically expressed as functions of space and time. For instance, pressure, temperature, volume, energy, etc. of a fluid are all state properties. Since state properties uniquely define the condition of a system, they represent definite points in an appropriate, abstract, (possibly) higher-dimensional state space. For this reason, state properties are also called point functions.
A process (synonymous to system evolution) is a succession of states. In classical physics, the succession (or progression) is taken to be continuous. In quantum mechanics, there is no notion of a process; see later in this post.
A process is often represented as a path in a state space that connects the two end-points of the staring and ending states. A parametric function defined over the length of a path is called a path function.
A cyclic process is one that has the same start and end points.
During a cyclic process, a state function returns to its initial value. However, a path function does not necessarily return to the same value over every cyclic change—it depends on which particular path is chosen. For instance, if you take a round trip from point to point and back, you may spend some amount of money if you take one route but another amount if you take another route. In both cases you do return to the same point viz. , but the amount you spend is different for each route. Your position is a state function, and the amount you spend is a path function.
[I may make the above description a bit more rigorous later on (by consulting a certain book which I don’t have handy right away (and my notes of last year are gone in the HDD crash)).]
The , the , and the :
The denotes a sufficiently small but finite, and locally existing difference in different parts of a system. Typically, since state properties are defined as (continuous) functions of space and time, what the represents is a finite change in some state property function that exists across two different but adjacent points in space (or two nearby instants in times), for a given system.
The is a local quantity, because it is defined and evaluated around a specific point of space and/or time. In other words, an instance of is evaluated at a fixed or . The simply denotes a change of position; it may or may not mean a displacement.
The (i.e. the infinitesimal) is nothing but the taken in some appropriate limiting process to the vanishingly small limit.
Since is locally defined, so is the infinitesimal (i.e. ).
The of CoV is completely different from the above two concepts.
The is a sufficiently small but global difference between the states (or paths) of two different, abstract, but otherwise identical views of the same physically existing system.
Considering the fact that an abstract view of a system is itself a system, also may be regarded as a difference between two systems.
Though differences in paths are not only possible but also routinely used in CoV, in this post, to keep matters simple, we will mostly consider differences in the states of the two systems.
In CoV, the two states (of the two systems) are so chosen as to satisfy the same Dirichlet (i.e. field) boundary conditions separately in each system.
The state function may be defined over an abstract space. In this post, we shall not pursue this line of thought. Thus, the state function will always be a function of the physical, ambient space (defined in reference to the extensions and locations of concretely existing physical objects).
Since a state of a system of nonzero size can only be defined by specifying its values for all parts of a system (of which it is a state), a difference between states (of the two systems involved in the variation ) is necessarily global.
In defining , both the systems are considered only abstractly; it is presumed that at most one of them may correspond to an actual state of a physical system (i.e. a system existing in the physical reality).
The idea of a process, i.e. the very idea of a system evolution, necessarily applies only to a single system.
What the represents is not an evolution because it does not represent a change in a system, in the first place. The variation, to repeat, represents a difference between two systems satisfying the same field boundary conditions. Hence, there is no evolution to speak of. When compressed air is passed into a rubber balloon, its size increases. This change occurs over certain time, and is an instance of an evolution. However, two rubber balloons already inflated to different sizes share no evolutionary relation with each other; there is no common physical process connecting the two; hence no change occurring over time can possibly enter their comparative description.
Thus, the “change” denoted by is incapable of representing a process or a system evolution. In fact, the word “change” itself is something of a misnomer here.
Text-books often stupidly try to capture the aforementioned idea by saying that represents a small and possibly finite change that occurs without any elapse of time. Apart from the mind-numbing idea of a finite change occurring over no time (or equally stupefying ideas which it suggests, viz., a change existing at literally the same instant of time, or, alternatively, a process of change that somehow occurs to a given system but “outside” of any time), what they, in a way, continue to suggest also is the erroneous idea that we are working with only a single, concretely physical system, here.
But that is not the idea behind at all.
To complicate the matters further, no separate symbol is used when the variation is made vanishingly small.
In the primary sense of the term variation (or ), the difference it represents is finite in nature. The variation is basically a function of space (and time), and at every value of (and ), the value of is finite, in the primary sense of the word. Yes, these values can be made vanishingly small, though the idea of the limits applied in this context is different. (Hint: Expand each of the two state functions in a power series and relate each of the corresponding power terms via a separate parameter. Then, put the difference in each parameter through a limiting process to vanish. You may also use the Fourier expansion.))
The difference represented by is between two abstract views of a system. The two systems are related only in an abstract view, i.e., only in (the mathematical) thought. In the CoV, they are supposed as connected, but the connection between them is not concretely physical because there are no two separate physical systems concretely existing, in the first place. Both the systems here are mathematical abstractions—they first have been abstracted away from the real, physical system actually existing out there (of which there is only a single instance).
But, yes, there is a sense in which we can say that does have a physical meaning: it carries the same physical units as for the state functions of the two abstract systems.
An example from biology:
Here is an example of the differences between two different paths (rather than two different states).
Plot the height of a growing sapling at different times, and connect the dots to yield a continuous graph of the height as a function of time. The difference in the heights of the sapling at two different instants is . But if you consider two different saplings planted at the same time, and assuming that they grow to the same final height at the end of some definite time period (just pick some moment where their graphs cross each other), and then, abstractly regarding them as some sort of imaginary plants, if you plot the difference between the two graphs, that is the variation or in the height-function of either. The variation itself is a function (here of time); it has the units, of course, of m.
The is a local change inside a single system, and is its limiting value, whereas the is a difference across two abstract systems differing in their global states (or global paths), and there is no separate symbol to capture this object in the vanishingly small limit.
Consider one period of the function , say over the interval ; is a small, real-valued, constant. Now, set . Is the change/difference here a or a ? Why or why not?
Now, take the derivative, i.e., , with once again. Is the change/difference here a or a ? Why or why not?
Which one of the above two is a bigger change/difference?
Also consider this angle: Taking the derivative did affect the whole function. If so, why is it that we said that was necessarily a local change?
An important and special note:
The above exercises, I am sure, many (though not all) of the Officially Approved Full Professors of Mechanical Engineering at the Savitribai Phule Pune University and COEP would be able to do correctly. But the question I posed last time was: Would it be therefore possible for them to spell out the physical meaning of the variation i.e. ? I continue to think not. And, importantly, even among those who do solve the above exercises successfully, they wouldn’t be too sure about their own answers. Upon just a little deeper probing, they would just throw up their hands. [Ditto, for many American physicists.] Even if a conceptual clarity is required in applications.
(I am ever willing and ready to change my mind about it, but doing so would need some actual evidence—just the way my (continuing) position had been derived, in the first place, from actual observations of them.)
The reason I made this special note was because I continue to go jobless, and nearly bank balance-less (and also, nearly cashless). And it all is basically because of folks like these (and the Indians like the SPPU authorities). It is their fault. (And, no, you can’t try to lift what is properly their moral responsibility off their shoulders and then, in fact, go even further, and attempt to place it on mine. Don’t attempt doing that.)
A Song I Like:
May be I have run this song before. If yes, I will replace it with some other song tomorrow or so. No I had not.]
Hindi: “Thandi hawaa, yeh chaandani suhaani…”
Music and Singer: Kishore Kumar
Lyrics: Majrooh Sultanpuri
[A quick ‘net search on plagiarism tells me that the tune of this song was lifted from Julius La Rosa’s 1955 song “Domani.” I heard that song for the first time only today. I think that the lyrics of the Hindi song are better. As to renditions, I like Kishor Kumar’s version better.]
Minor editing may be done later on and the typos may be corrected, but the essentials of my positions won’t be. Mostly done right today, i.e., on 06th January, 2017.]