This assignment counts 5 per cent towards the final assessment, and is due on by 5 p.m. on Monday 30 April, to the box outside room 516, or to me directly. Mary Rees. This assignment concerns Euler's seminal text on Calculus of Variations problems ``Methodus Inveniendi..'', publication number 65 in the Dartmouth College archive
http://math.dartmouth.edu/~euler/tour/tour_17.html
You will also probably find useful pages 399-406 (photocopy provided) of
A Source Book in Mathematics 1200-1800 , edited by D.J. Struik QA21.D92
which describes some of the contents of Euler's book,and give annotated translations of key sections.
Most of questions relate to an example in the text which you are assigned. Here is your Individual assignment. We will restrict to examples in the Calculus of Variations without constraints. All examples come from Chapter 2 in the text. The examples fall into three groups:
GROUP 1: pages 30-41, which involve integrals of functions of x and y only
GROUP 2: pages 48-53 which involve integrals of functions of x, y and dy/dx=p
GROUP 3: pages 61-70 which involve integral of functions of x,y,dy/dx=p and d^2y/dx^2=q.
Note that Euler uses the notation p for dy/dx and q for d^2y/dx^2 throughout.
Give Euler's dates of birth and death and write down the full title of the archive publication number 65. Also write down the date and place of its first publication: this information is given in the Euler archive
Find your example in Euler's text. Write down the function Z such that the integral of Z over an interval is to be minimised or maximised in your example, in notation that you would normally use. You may use p instead of dy/dx and q instead of d^2y/dx^2, but if you do this, then say so. (Note that Euler always uses Z for the integrand.)
Special note for those who have been assigned Example I on pp 61-2: the Z given by Euler in this example is DIFFERENT from the Z in all the other examples. If we write Z_1 for the Z given by Euler, then the Z whose integral is to be minimised or maximised is Z_1q. Similarly if we write M_1, N_1,P_1 for the M, N,P given by Euler, then the quantities M, N and P which are the same as M, N and P in all the other sections are M_1q, N_1q and P_1q --- and Q is Z_1.
look at Proposition 2 in Chapter 2 of Euler's text and the notes on this in Struik's Source book. What are the quantities which Euler refers to as N and M in your example? Write down the necessary condition on N, given by proposition 2 for the integral being a maximum or minimum?
look at Proposition 3 in Chapter 2 of Euler's text, and the notes on this in Struik's Source book.
Write down the quantities which Euler refers to as M, N and P for your example.
Write down the necessary condition given on N and P in this Proposition 3 for the integral of Z to maximised or minimised, subject to fixed values of y at the ends of the interval of integration.
If M=0 in your example, EITHER explain why the condition of Proposition 3 on N and P gives Z=Pp+C for a constant C, OR identify the explanation in the text. (It is after the colloraries following proposition 3 but before the examples.) If N=0 in your example, explain why the condition of Proposition 3 on N and P gives P=constant. If M and N are both non-zero, you do not need to do anything
look at Proposition 4 and the following Corollary 1 in Euler's text, and it may also help to look at Struik's notes on Proposition 3 in the Source Book.
Write down the quantities which Euler refers to as M, N, P and Q in your example. Write down the necessary condition given on N, P and Q in Corollary 1 - or at the end of the proof of Proposition 4 - for the integral of Z to be maximised or minimised, subject to fixed values of y at the ends of the interval of integration.
Note for those who have been assigned example I of pp61-2: if we write Z_1 what Euler calls Z in this example, then the usual Z is Z_1q, the M, N and P given in this example are not the usual M, N, P. If these are written M_1, N_1, P_1, then the usual M, N and P are M_1q, N_1q and P_1q.
If M=N=0 in your example, EITHER explain why the condition of Proposition 4 on N, P and Q gives Z=Cp+D+Qq for constants C and D OR identify the explanation in the text. (It is in one of the later corollaries following proposition 4.) If M=0 in your example and N is not identically zero, EITHER explain why the condition of Proposition 4 implies that Z=pP+qQ+C-pdQ/dx OR find the explanation in the text (again in one of the later corollaries following proposition 4). If M and N are both non-zero in your example then you do not need to do anything.
For your example, and looking at the text, write down the equation given by Proposition 2 or 3 or 4 for your example, OR one of the ``first or second integrals'' Z=pP+C or Z=pP+D+qQ or Z=pP+qQ+C-pdQ/dx which needs to be solved to minimise/maximise the integral.
Also write down the solution(s) to the equation, in the form given in the text -- which may involve parameters and integral --- IF the solution is given in Euler's text. If your example is from group 1, also write down the (indefinite) integrals of Z for at least one of the solutions, using Euler's text. (For examples in group 1, the solution of the equation given by Proposition 2 is almost immediate.)
You are NOT expected to give any reasoning or verification. The solution is not given in Euler's text for a few examples (and I think in those cases there is no solution in terms of ``elementary'' functions) and in that case you are not expected to provide a solution, but the differential equations is complicated enough to compensate for that.
Notational point for some solutions: Euler uses A tang to denote the inverse tan function -- which is sometimes called arc tan.