On Mon, Feb 20, 2012 at 12:54 PM, Harold E. <[email protected]> wrote: > On 20 fév, 12:44, Sergiu Ivanov <[email protected]> wrote: >> Hello, >> >> On Mon, Feb 20, 2012 at 6:01 PM, Harold E. <[email protected]> wrote: >> >> > Concerning the use of constants as units, it's still not very clear in >> > my mind, but I think it would be possible to define an option to hide >> > it or not when printing the unit, and also to try to determine which >> > are the constants to add in an expression to recover the right one >> > (for example E = m to E = mc²). >> >> I'm far from being an expert in this field, so my questions arise >> rather from curiosity than from some serious background >> considerations: what uses do you envision for this feature of >> recovering the right constant? > > The use of constants as units is in the heart of theoretical physics, > since in some way it emphases the unification of several concepts. I > will use the example of special relativity: Einstein found the formula > E = mc², which means that the mass and the energy are exactly the same > thing and that we do not need to distinguish between these two > concepts, because if in some formula you use one concept, you can > immediately replace it with the other with the help of this formula. > The same happen for space and time (one can see this with the > Lorentz's transformation formula). > And, so instead of keeping all the factors c, saying that space and > time are different, it has been recognized that we don't have to keep > them and we can say that c = 1. This implies that velocities have no > units, for example v = 0.7. In fact, this means that this velocity is > 7/10 the speed of light, which cab be written v = 0.7 c, and then c > plays the role of an unit (but we never write it). In some way > velocity here is like the quantity of matter measured in mol: it's a > dimensionless unit, but meaningful. And then this implies that other > quantities have other units (m has energy unit, for example J or eV, > but it really means unit eV.c⁻²). > This phenomenon is very frequent: for example, at first heat and > energy was perceived as two different concepts, but then one found > that they was equivalent and one established and so no we use only > Joule for both concepts, and not anymore the calorie for the heat. We > also put k_b = 1 (temperature is energy), hbar = 1 (energy is > frequency...), G_N = 1, etc. > > Then, even if we do all the computations without writing explicitly > these constants, it's useful to write them at the end for several > reason: > 1. We are not accustomed to these units, because the value of > quantities are very different than the one of our daily experiences (v > = 0.3 c = 100000 km/s, v = 100 km/h ~ 10⁻¹⁰), and a formula will > generally "speak" us better if we can see these fundamental constants. > 2. Because of (1), our usual unit systems are very different, and so > it is easier for experiences and many other things to be able to > convert. > 3. Again because of (1) when you learn special relativity and so on, > you prefer to keep these factors.
If you're not going to use units until the very end, they're not very useful. Granted, there is a certain level of physical correctness that is lost without them, but I think the whole point of having a strong units system is that it can do dimensional analysis for you. This means that you use them in every point in the calculation, so that at the end, you know exactly the units of the end result. In other words, suppose you do some complicated computation with a quantity that contains c, but you assume that c = 1. Then, at the end, the only way to know where you should replace "1" with c would be to go back to the beginning and put it in explicitly as c, and redo the computation. Aaron Meurer > >> On the other hand, recovering the constant requires quite a bit of >> analysis of the environment. For example, in a certain environment, I >> may use E to denote something different from the total energy. > > Yes of course, here this formula was only an example. The idea to > recover the right factors are for example to say that in the left > side, the unit is J.c⁻², and in the right side it's kg. So we can > extract the good factors. If we don't have this possibility, we can > also say that "I want that this formula has the dimension X", which is > useful if people don't want to take in account the constants at the > beginning. > >> And further, how does using constants as units help? >> >> Again, I'm asking just because I know that when people ask me to >> explain something, I get to understand that something better. You >> said using constants as units isn't very clear yet, so, I'm here >> nagging you :-) >> >> Sergiu > > No problem, I posted this message with this goal! I hope that my > explanations help you. There is a very good article of Jean-Marc Lévy- > Leblond on this topic (www.springerlink.com/content/ > kh1n15r56q0682v8/), but more on the philosophical point of view. > > Harold > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
