The difference between Invariants and Constants
- Thread starter studiot
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I'm a EE, and I can't say I've ever had the need or desire to make the distinction. In general usage there isn't much difference, but maybe there is for math/physics context.
The only important difference I can think of is that the term invariant often applies to an entity that is unchanged by a transformation rule, and maybe constant means unchanging always, implying with respect to time?
With regards to the speed of light, I would tend to say that the speed of light is a constant, at least based on usual physics. Although there are some theories that speculate that the speed of light changes slowly over time, in which case it would not be.
I wouldn't be bothered that someone called the speed of light invariant, since we can talk about reference frame transformations and note that the speed of light remains invariant to these transformations, while the speed of any solid object changes when the reference frame changes.
I'm sure you're going to tell me I'm wrong, but I thought I'd stick my neck on the chopping block, just for the sake of learning.
As I understand it the speed of light does change, depending somewhat on its environment. Glass, for example, causes the speed to change, and is different according to frequency (hence prisms). I understand TWT do something similar, though subtler. Ditto for going through wire in antennas.
The speed we use as a constant is in a vacumn.
The speed we use as a constant is in a vacumn.
OK the ayes have it.
In mathematics we distinguish between the variables and constants of an equation.
eg in y = ax y and x are variables - a is a constant.
In physics we assign meaning to the variables and sometimes also to the constants.
The equations now describe different states of a system, where the variables do just that and the constants are, well constant.
The equations may be said to describe simple transformations.
We even formally collect bunches of these variables to form complex (complicated) objects such as the momentum vector or eelctric stress tensor or whatever.
This allows us to manipulate more difficult transformations.
A transformation represents two views of the same system either before and after some process or concurrently from different viewpoints.
Often the variables will interact during a complex transformation. We handle this by generalising our simple equation above to a matrix equation
Y = A X
or in a simple example (sorry I can't do tex matrices)
y1 \(\mapsto\) 1 0 x1 = x1
y2 \(\mapsto\) 1 1 x2= x1+x2
A = 1 0
......0 1 if someone could explain how to tidy up the tex I will replace.
Under this transformation variable x1 is Invariant, but x2 is not.
The constants involved are of course 1 and 0
The speed of light is a constant, not a variable and not manipulable by us.
Equally we physicists search for quantities (variables) like x1 that are invariant under the given transformation as they are of special interest.
One such is electric charge which is invariant under all mechanical transformations, unlike mass, length and time!
In mathematics we distinguish between the variables and constants of an equation.
eg in y = ax y and x are variables - a is a constant.
In physics we assign meaning to the variables and sometimes also to the constants.
The equations now describe different states of a system, where the variables do just that and the constants are, well constant.
The equations may be said to describe simple transformations.
We even formally collect bunches of these variables to form complex (complicated) objects such as the momentum vector or eelctric stress tensor or whatever.
This allows us to manipulate more difficult transformations.
A transformation represents two views of the same system either before and after some process or concurrently from different viewpoints.
Often the variables will interact during a complex transformation. We handle this by generalising our simple equation above to a matrix equation
Y = A X
or in a simple example (sorry I can't do tex matrices)
y1 \(\mapsto\) 1 0 x1 = x1
y2 \(\mapsto\) 1 1 x2= x1+x2
A = 1 0
......0 1 if someone could explain how to tidy up the tex I will replace.
Under this transformation variable x1 is Invariant, but x2 is not.
The constants involved are of course 1 and 0
The speed of light is a constant, not a variable and not manipulable by us.
Equally we physicists search for quantities (variables) like x1 that are invariant under the given transformation as they are of special interest.
One such is electric charge which is invariant under all mechanical transformations, unlike mass, length and time!
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someonesdad
- Joined Jul 7, 2009
- 1,583
It's largely a matter of the semantics in the particular technical area.
For example, in math, most would agree that 3 and 7.2 are constants. If one has an equation f(x) = 0, then one refers to the quantity x as a variable, as its value comes from a finite or infinite set of constants. If one is only interested in the value(s) of x that make the equation true, then one might call those constants the root(s) of the equation.
In physics, definitions change over time (meaning they're not constant). In studiot's example, the speed of light is currently defined to be a constant. Note, however, that for many decades it was an experimentally determined value that changed slightly over the years as different experimenters got slightly different values. For the last few decades it has been in vogue to consider it a constant, most likely because it being a constant is one of the two fundamental postulates of special relativity, which is a physical theory that has withstood a large variety of experimental tests. The wikipedia page also says that most of the measurement uncertainty of the speed of light was traced to the uncertainty in the length of the meter. Regardless, there is no a priori reason why it has to be a constant nor that it has been constant over time. Only experimental measurements can verify that. Similar considerations apply to the fundamental charge, which hasn't been an experimental constant historically as humorously related by Feynman.
Invariants are things that don't change when one changes one's viewpoint. They're invariantly useful (pun intended). For example, in programming, a good programmer will know the invariants of an algorithm and use them to check that his code hasn't changed the invariant (and thus it serves as a useful check against bugs). In math, often a particular function (or functional) should be invariant regardless of what coordinate system you use to evaluate it. For example, a line integral over a path should yield the same value whether you use Cartesian or spherical coordinates. The modern parlance is that the invariant doesn't depend on what chart(s) you put on the manifold.
Invariants in physics are quite important and are often deeply related to the symmetries of a system. These can be discussed using the machinery of Noether's theorem or the work Lie did on the symmetries of differential equations in the 1800's. In a nutshell, the symmetries correspond to "constants of the motion". These, in turn, get related to the integration constants you learned about when (if) you studied differential equations. An example in mechanics is that if a certain function of coordinates and momenta (the "Hamiltonian") doesn't depend explicitly on time, then the total energy of the system is a constant. There's thus a deep connection between the constants and the points of view (i.e., the transformations).
I was recently cleaning out the garage and came across some work I did when I was a student in the early 70's on this symmetry stuff. The stuff is gibberish to me now, but at the time my teacher and I thought it was pretty neat. We were able to do things like show that the n dimensional free particle and n dimensional isotropic harmonic oscillator had symmetry groups that could be related through a number of different transformations -- and even integrate some of the local infinitesimal transformation generators to get the (rather complicated) finite transformations. They were all representations of the same SL(n+2, R) transformation group, so, in some sense, these physical systems were equivalent. I look at the complicated transformations now and wonder if they've ever been of any practical use to anyone (certainly not to me, as I pretty much ignored that stuff after I got out of school).
For example, in math, most would agree that 3 and 7.2 are constants. If one has an equation f(x) = 0, then one refers to the quantity x as a variable, as its value comes from a finite or infinite set of constants. If one is only interested in the value(s) of x that make the equation true, then one might call those constants the root(s) of the equation.
In physics, definitions change over time (meaning they're not constant
). In studiot's example, the speed of light is currently defined to be a constant. Note, however, that for many decades it was an experimentally determined value that changed slightly over the years as different experimenters got slightly different values. For the last few decades it has been in vogue to consider it a constant, most likely because it being a constant is one of the two fundamental postulates of special relativity, which is a physical theory that has withstood a large variety of experimental tests. The wikipedia page also says that most of the measurement uncertainty of the speed of light was traced to the uncertainty in the length of the meter. Regardless, there is no a priori reason why it has to be a constant nor that it has been constant over time. Only experimental measurements can verify that. Similar considerations apply to the fundamental charge, which hasn't been an experimental constant historically as humorously related by Feynman.Invariants are things that don't change when one changes one's viewpoint. They're invariantly useful (pun intended). For example, in programming, a good programmer will know the invariants of an algorithm and use them to check that his code hasn't changed the invariant (and thus it serves as a useful check against bugs). In math, often a particular function (or functional) should be invariant regardless of what coordinate system you use to evaluate it. For example, a line integral over a path should yield the same value whether you use Cartesian or spherical coordinates. The modern parlance is that the invariant doesn't depend on what chart(s) you put on the manifold.
Invariants in physics are quite important and are often deeply related to the symmetries of a system. These can be discussed using the machinery of Noether's theorem or the work Lie did on the symmetries of differential equations in the 1800's. In a nutshell, the symmetries correspond to "constants of the motion". These, in turn, get related to the integration constants you learned about when (if) you studied differential equations. An example in mechanics is that if a certain function of coordinates and momenta (the "Hamiltonian") doesn't depend explicitly on time, then the total energy of the system is a constant. There's thus a deep connection between the constants and the points of view (i.e., the transformations).
I was recently cleaning out the garage and came across some work I did when I was a student in the early 70's on this symmetry stuff. The stuff is gibberish to me now, but at the time my teacher and I thought it was pretty neat. We were able to do things like show that the n dimensional free particle and n dimensional isotropic harmonic oscillator had symmetry groups that could be related through a number of different transformations -- and even integrate some of the local infinitesimal transformation generators to get the (rather complicated) finite transformations. They were all representations of the same SL(n+2, R) transformation group, so, in some sense, these physical systems were equivalent. I look at the complicated transformations now and wonder if they've ever been of any practical use to anyone (certainly not to me, as I pretty much ignored that stuff after I got out of school).
I think it's a great deal more than a matter of semantics.
Sticking for the moment with maths, let us consider the numbers
15, 7, 0
Now let us make the number base transformation from decimal to hex.
F, 7, 0
We see that the numbers 7 and 0 are the same before and after the transformation
So they are invariant under the transformation decimal \(\rightarrow\) hex
Whereas the number 15 is not.
If we enquire further we find that the invariants of number base transformations depend upon the bases. There are only two invariants common to all number base transformations viz 1 and 0.
So, Someone's Dad, how does this square with your comment about 3 and 7.2 being constants?
They are just numbers.
Studiot,
Based on your example above, in the math context, a constant is taken from the set of all integers or all real number. The term invariant is applied to a number in the context of some transformation process.
The transform of a constant is not always an invariant, while the transform of an invariant is alway a constant.
hgmjr
Based on your example above, in the math context, a constant is taken from the set of all integers or all real number. The term invariant is applied to a number in the context of some transformation process.
The transform of a constant is not always an invariant, while the transform of an invariant is alway a constant.
hgmjr
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someonesdad
- Joined Jul 7, 2009
- 1,583
It is precisely semantics.
As in all discussions, the terms need to be precisely defined (and agreed upon by the participants) before higher level discourse can take place. Otherwise, we all redouble our efforts after losing sight of our goals and I'm sure all of us have been in those situations.
So, studiot, I'd appreciate hearing your definitions of a "constant" and an "invariant". It will be interesting to see to what extent others use the same definitions.
Oh dear, I thought I had explained my definitions.
Obviously I didnt do it very well so I will have another go. I will try to take in Bills query as I go.
Both constants and invariants are basically mathematical ideas. When physicists apply mathematics to the real world they may add meaning ( and sometimes restrictions) not present in the original mathematical ideas.
A simple mathematical entity is an expression.
(3 tan 2x ) and (2 tan 3x ) are both expressions and different from each other.
Such expressions are formed by applying coefficients which we cant alter to variables, which we can. Hence we call the coefficients constants.
It is these coefficients, or constants, which define and distinguish expressions, not the variables.
We could substitute another variable (y, z, t etc) for x or allocate any value within the domain and the expression would be the same.
If, however, we changed any coefficient we would have a different expression.
Such are constants then - they are the coefficients which define mathematical expressions.
Mathematical expressions do not, of themselves, possess invariants.
Invariants arise when we consider mathematical processes. Such processes may apply to expressions with coefficients as above or to other mathematical entities such as shapes.
An operand of a process (the object being processed) is called an invariant if it is unchanged at the end of the process.
So, for instance, an equilateral triangle with its base on the x axis and apex on the y axis is unchanged or invariant under the transformation process of reflection in the y axis.
Note that in this example there is not a coefficient or constant to be seen.
When we consider transformations of mathematical expressions it is the variable that is the object being processed and the variable that is considered an invariant if it is unchanged by the transformation.
So for instance exp(x) is invariant under the process of integration.
This is as far as the mathematical definitions can take matters. As I previously observed physicists add meaning to the mathematical objects. I will deal with the additional layer(s) of structure imposed by applications to physics in a subsequent post.
Obviously I didnt do it very well so I will have another go. I will try to take in Bills query as I go.
Both constants and invariants are basically mathematical ideas. When physicists apply mathematics to the real world they may add meaning ( and sometimes restrictions) not present in the original mathematical ideas.
A simple mathematical entity is an expression.
(3 tan 2x ) and (2 tan 3x ) are both expressions and different from each other.
Such expressions are formed by applying coefficients which we cant alter to variables, which we can. Hence we call the coefficients constants.
It is these coefficients, or constants, which define and distinguish expressions, not the variables.
We could substitute another variable (y, z, t etc) for x or allocate any value within the domain and the expression would be the same.
If, however, we changed any coefficient we would have a different expression.
Such are constants then - they are the coefficients which define mathematical expressions.
Mathematical expressions do not, of themselves, possess invariants.
Invariants arise when we consider mathematical processes. Such processes may apply to expressions with coefficients as above or to other mathematical entities such as shapes.
An operand of a process (the object being processed) is called an invariant if it is unchanged at the end of the process.
So, for instance, an equilateral triangle with its base on the x axis and apex on the y axis is unchanged or invariant under the transformation process of reflection in the y axis.
Note that in this example there is not a coefficient or constant to be seen.
When we consider transformations of mathematical expressions it is the variable that is the object being processed and the variable that is considered an invariant if it is unchanged by the transformation.
So for instance exp(x) is invariant under the process of integration.
This is as far as the mathematical definitions can take matters. As I previously observed physicists add meaning to the mathematical objects. I will deal with the additional layer(s) of structure imposed by applications to physics in a subsequent post.
There is definitely semantics at play here. Invariant, like many other words, has a somewhat different meaning based on its context.
However, in the context of the subjects discussed in this forum, invariant means unaffected by a particular transformation. One could certainly find any number of physical models for which there are transformations wherein variables, or indeed entire functions, are invariant under those transformations, but they are not constants.
However, in the context of the subjects discussed in this forum, invariant means unaffected by a particular transformation. One could certainly find any number of physical models for which there are transformations wherein variables, or indeed entire functions, are invariant under those transformations, but they are not constants.
What does semantics have to do with it?
Let us consider the simplest binary combination of two objects, drawn from sets, which may be different sets.
This is a transformation of any object of the first set X by applying a single object from the second set A
Applying a different object would yield a different transformation.
x \(\rightarrow\) ax
Since we have introduced two distinct and different objects so we need two distinct and different names.
Since the first object can take on any value from the set X we call this a variable.
Since the second object is restricted to one particular object from the second set we call this a constant.
By way of example let the transformation be a doubling of value.
Then
x \(\rightarrow\) 2x
Where 2 is the constant, chosen from the set of integers ≠0 , and x is a real variable ≠0
We can easily see that there is no x which makes 2x = x
However if we choose a different constant such that
x \(\rightarrow\) 1x
We see that the result of the operation or transformation leaves any value of x unchanged.
We call this situation an invariant.
But realise that it does not occur for any old constant, just one particular one.
So constants and invariants really are different and represent or define two different ideas.
I think that some people use 'invariant' incorrectly in other contexts, when they really mean 'unvarying'.
For instance the teetotaller observes the unvarying level of whisky in his glass.
Note that unvarying is an adjective and invariant is a noun.
For instance the teetotaller observes the unvarying level of whisky in his glass.
Note that unvarying is an adjective and invariant is a noun.
When used as a noun, invariant has only one meaning as far as I know. However, despite the root word, variant, being only a noun, several dictionaries I checked, both on-line and printed, list invariant as both an adjective and a noun.
Here is one on-line entry I found:
invariant Definition
Here is another:
Noun1.invariant - a feature (quantity or property or function) that remains unchanged when a particular transformation is applied to it characteristic, feature - a prominent attribute or aspect of something; "the map showed roads and other features"; "generosity is one of his best characteristics"
math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
Adj.1.invariant - unaffected by a designated operation or transformation math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
invariable - not liable to or capable of change; "an invariable temperature"; "an invariable rule"; "his invariable courtesy"
2.invariant - unvarying in nature; "maintained a constant temperature"; "principles of unvarying validity" constant, unvarying, changeless
invariable - not liable to or capable of change; "an invariable temperature"; "an invariable rule"; "his invariable courtesy"
Here is one on-line entry I found:
invariant Definition
- (s) unaffected by a designated operation or transformation
- (s) persistent in occurrence and unvarying in nature; "maintained a constant temperature"; "a constant beat"; "principles of unvarying validity"; "a steady breeze"
Here is another:
Noun1.invariant - a feature (quantity or property or function) that remains unchanged when a particular transformation is applied to it characteristic, feature - a prominent attribute or aspect of something; "the map showed roads and other features"; "generosity is one of his best characteristics"
math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
Adj.1.invariant - unaffected by a designated operation or transformation math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
invariable - not liable to or capable of change; "an invariable temperature"; "an invariable rule"; "his invariable courtesy"
2.invariant - unvarying in nature; "maintained a constant temperature"; "principles of unvarying validity" constant, unvarying, changeless
invariable - not liable to or capable of change; "an invariable temperature"; "an invariable rule"; "his invariable courtesy"
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Not sure where either of those dictionaries resides, I found them on-line. I guess they mean you can say something like:
"Function F has at least one invariant term under transformation T.
or
"Only Bob possessed an invariant opinion after that otherwise convincing argument."
But to your point, language certainly shifts in semantics with changes in locale.
edit: BTW, I think the emboldened text is just there to explain some of the terms in the definition. Unfortunately, formattting may have been lost in the cut and paste operation.
Look up a work entitled "The Illusion of Invariant Quantities in Life Histories" penned by four British scentists. Apparently they have no problem using invariant as an adjective.
"Function F has at least one invariant term under transformation T.
or
"Only Bob possessed an invariant opinion after that otherwise convincing argument."
But to your point, language certainly shifts in semantics with changes in locale.
edit: BTW, I think the emboldened text is just there to explain some of the terms in the definition. Unfortunately, formattting may have been lost in the cut and paste operation.
Look up a work entitled "The Illusion of Invariant Quantities in Life Histories" penned by four British scentists. Apparently they have no problem using invariant as an adjective.
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