Choppy said:
Short answer: their titles.
Not true.
First of all, it's going to depend where you are. For example, Applied mathematics in Britain was of course influenced by Newton (early on), then the Cambridge people in the 19-20th century, and then afterwards, Manchester in the 1950s. This had the effect of keeping it very much applied. In addition, applied maths is very much linked to industrial maths in Britain (with these Industry Study Groups, first initiated by Oxford in the 20th century).
In contrast, applied maths in America (and by extension, Canada) was more influenced by immigration of Germans and Russians. This had the effect of making it more pure. While some mathematics departments in the US have applied groups, they would perhaps be lumped into the Pure groups in the UK. So an applied mathematician coming over from UK to the US might even consider working with engineering or physics departments.
Then of course, you have variation between institutions. Doing applied maths at MIT or Brown is not the same as doing applied maths at say, Berkeley or Yale.
In my view, the simplest way to differentiate the physicist and engineer and the applied mathematician is by the following example.
Suppose you want to study the path of a ball thrown through the air. Both the physicist and the mathematician would know that it's modeled by a quadratic. Both would seek solutions -- numerical, analytical, or otherwise.
At this point, the mathematician may ask -- "Gee, instead of a quadratic, what happens if it's a cubic?"
To me, this fork in the road is the easiest way to distinguish between the two. Both fields study problems that are physically motivated. However, in mathematics, you tend to be more interested in the mathematical aspect of things: methodologies, structure, and so on. Your research will proceed according to what you consider mathematically interesting. Physical relevance -- while a bonus -- is not always demanded.
For example, I work on free-surface flows. Some of the regimes I'm interested in (low-speed, for example) are not priorities in Engineering applications. However, the mathematical structure is what I'm interested in.
Perhaps another way to think about it is through the following story: In the 1970s, the National Research Council of Canada realized they needed more mathematicians on their staff. So they decided to hire them. Today, there are almost no mathematicians working at the NRC.
Why? These people were required to justify -- on a scheduled basis -- the scope and expectations of their work (for funding purposes). But they could never do it! As an applied mathematician, you're often not confined to this strict adherence to physically related phenomenon...plus you're not always sure of where you're going and when you're going to be there.
will.c said:
My understanding was that applied mathematicians are more interested in creating existence and uniqueness theorems for the models, while physicists are more interested in specific solutions and their physical interpretation.
I'm an applied mathematician. I (and almost everybody I work with) have very little interest in existence/uniqueness theorems. I have never actually done a proof in the context of my research. In fact, of the literature that I read, I would say 98% (sort of random figure) have no theorem/proofs.
I can't actually name any theorem/proof paper I've read in the last 2-3 years. Okay, that's a lie. I can name one. Written by a Russian.