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Is the definite article required before a variable used to abbreviate a physical quantity?
The ft is the same for both cases at various values of the Lc. As the Lc increases, the fr also shifts.
ft is the same for both cases at various values of Lc. As Lc increases, fr also shifts.
If the second sentence is okay, I am still a bit weirded out by starting a sentence with a math notation.
(Lc = Length of the channel, ft = frequency of the transistor)
This is answered by recognising that Lc and ft (or fr) are names, referring to the quantities "channel length" and "transistor frequency" (and whatever the other one is).
You could call "channel length" Cuthbert. A sentence referring to Cuthbert wouldn't use an article:
Fred is the same for both cases at various values of Cuthbert. As Cuthbert increases, Fred also shifts.
Don't use an article for the names/symbols for the quantities.
There's no weirdness in starting a sentence with a name, however it's spelled. If it doesn't have a capital letter, don't use one. It's unusual, because names usually have a capital letter and sentences should start with one. But changing ft into Ft could substantially change what you are referring to. That is far more important than stylistic custom concerning capital letters. However, many don't like this flouting of convention, in which case, it's necessary to use "The value of ft ..."
See also the ELU question Is it okay to start a sentence with a Greek letter (variable)?
If ft, fr and Lc have been defined previously, the second sentence is clear, unambiguous, conventional and correct.
In mathematical and physical discourse, a function of a variable may be represented as a symbol. For example, we have your ft , or f(t) as it might often be represented. The value f of the function depends on the value of the determining variable t. The symbol ft represents the manner in which the value f of the function depends on the determining variable t. Once defined the symbol is understood as the representation of the functional relationship between the two. It is therefore redundant to use "The" with such defined symbols, whether or not they occur at the start of the sentence.
The alphanumeric expression of the function is irrelevant, as is the number of determining variables. This argument applies to any function. For example F(t), ft, pb, Φx, Gamma(x,y,z),Phiμ.
It is correct usage to refer to the function f(t) in examples such as the following. In each case the function is being defined on its first use.
"Consider the function f(t)=t*t, where f has a quadratic dependence on t."
"The function sin(t) has a regular periodicity in t"
As with many questions, the answer is consult your style guide.
Here is accepted answer to Is it okay to start a sentence with a Greek letter (variable)?. Note that it is asking the opposite of your question.
Another answer cites Chicago Manual of Style (but only addresses symbols):
But you are asking about about starting with a (non-Greek) variable name. Both my undergrad professor, Jon Louis Bentley, and my senior editor at IBM, were adamantly opposed to starting with a variable.
In the absence of a clarifying style guide, I would agree with the "weirded out" impression and go with the CMoS, 13th ed., advice for numbers, substituting variables:
At the beginning of a sentence any number that would ordinarily be set in figures is spelled out, regardless of any inconsistency this may create. (Section 8.9)
If this is impracticable or cumbersome, the sentence should be recast so that it does not begin with a number. (Section 8.10)
While it may be acceptable (or your style guide may be mute) to start with a variable, recast the sentence so that it does not start with a variable.
