Cálculo de Variaciones I Tarea # 3

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1 1 Universidad Autónoma Metropolitana - Iztapalapa Cálculo de Variaciones I Tarea # 3 Resuelva los siguientes problemas: De la sección 3.1: resolver los ejercicios 1, 2, 3 y 5. Nota: La ecuación (3.4) a la que hace referencia el ejercicio 2 es: H(y, y,y f d f f )=y y f =const. (3.4) y dx y y De la sección 3.2: resolver los ejercicios 1, 2 y 3. Nota: La ecuación (3.17) a la que hacen referencia los ejercicios 1 y 2 es: H = n k=1 q k L q k L =const. (3.17) De la sección 4.2: resolver los ejercicios 1 y 2. Prof. Antonio Hernández Garduño

2 3.1 Functionals Containing Higher-Order Derivatives 59 is x f(x, y, y,...,y (n) ) dx ( ) ( ) ( 1) n dn f n 1 dn 1 f dx n y n +( 1) dx n 1 y n f y =. (3.9) Exercises 3.1: 1. Find the general solution for the extremals to the functional J defined by x ( (y ) 2 y 2 +2yx 3) dx. 2. Conservation Law: Suppose the integrand f defining the functional J does not depend on x explicitly. Prove that equation (3.4) is satisfied along any extremal. 3. For the functional J defined by y 1+(y ) 2 dx, find an extremal satisfying the conditions y() =, y () =, y(1) = 1, and y (1) = Degenerate Case: Let J be a functional of the form x (A(x, y, y )y + B(x, y, y )) dx, where A and B are smooth functions of x, y, andy. Prove that the Euler- Lagrange equation for this functional is a differential equation of at most second order and that consequently any solutions can satisfy at most two arbitrary boundary conditions. 5. Let J and K be functionals defined by K(y) = x x1 x where, for some smooth function G, f(x, y, y,y ) dx F (x, y, y,y ) dx, F (x, y, y,y )=f(x, y, y,y )+ d dx G(x, y, y ). Prove that any extremals for J must also be extremals for K.

3 64 3 Some Generalizations which, in turn, lead to the relations d L = L, dt q k q k m q k = V q k, for k =1, 2, 3. Recall from Section 1.3 that the kth component of force, f k on the particle is given by f k = V q k. Hence, the Euler-Lagrange equations imply Newton s equation f = ma. where a = q is the acceleration and f =(f 1,f 2,f 3 )istheforceontheparticle. For this example note that if the potential energy V does not depend on time explicitly then neither does L. In this case, we have the conservation law (3.17), which gives H = 1 2 m( q2 1 + q q 2 3)+V (q) =const.; i.e., the total energy of the particle is conserved along an extremal. Exercises 3.2: 1. Let L(t, q, q) = 1 2 ( q q 2 2 ) gq2, where g is a constant. (a) Find the extremals for the functional J defined by J(q) = t1 t L(t, q, q) dt. (b) Verify that equation (3.17) is satisfied. 2. Prove equation (3.17). 3. Let L(t, q, q) = q q2 2 q2 2 kq 2, where k is a constant. Find the extremals for the functional J defined by J(q) = t1 t L(t, q, q) dt.

4 4.2 The Isoperimetric Problem 93 The Lagrange multiplier therefore corresponds to the rate of change of the extremum J(y) with respect to the isoperimetric parameter L. We note a certain duality that exists for the isoperimetric problem. Suppose that λ ; then any extremal y to the problem with F = f λg must also be an extremal to a problem with G = g ˆλf, whereˆλ =1/λ. More specifically, suppose that y minimizes J subject to the isoperimetric constraint I(y) = L. Let K denote the minimum value J(y). Then and thus K = J(y) λ(i(y) L), L = I(y) ˆλ(J(y) K). We have that J λi = λ(i ˆλJ), and this indicates that the minimum for the functional x 1 x Fdxcorresponds to the maximum for the functional x Gdx. A similar statement can be made if y produces a maximum for J subject to I(y) = L. Wethushavethefollowingresult. Theorem Suppose that y produces a minimum (maximum) value for J subject to the constraint I(y) =L and that λ.letk = J(y). Then y produces a maximum (minimum) for I subject to the constraint K, and I(y) =L. In view of the above result, suppose we revisit, for example, the catenary problem of Example We saw that the catenary is the curve along which the potential energy is an extremum subject to the condition that the cable is of length L. Infact,itcanbeshownthatthepotentialenergyisminimum along a catenary for the appropriate choice of ˆξ. Theorem4.2.3showsthat, for a fixed value of potential energy, the catenary is the curve along which the arclength is maximized. The duality relationship also helps to elucidate the condition that y not be an extremal for I in Theorem If y is an extremal for I, theninthe dual problem ˆλ =.ThismeansthatI(y) =L, independent of the constraint K, so that K can be prescribed without changing the extremum for I. Alternatively, if λ =, then K, independent of the constraint I(y) =L, so that the problem does not depend on the constraint. At any rate, if λ is not finite or if λ =theproblemisdegenerate. Exercises 4.2: 1. Let J and I be the functionals defined by y 2 dx, I(y) = ydx. Find the extremals for J subject to the conditions y() =, y(1) = 2, and I(y) =L.

5 94 4 Isoperimetric Problems 2. Dido s Problem in Polar Coördinates: LetJ and I be functionals defined by J(r) = 1 2 π r 2 dθ, I(r) = π r2 + r 2 dθ, where r = dr/dθ. FindanextremalforJ subject to the conditions r() =, r(π) =, and I(r) =L>. 3. Let J and I be the functionals defined by (yy ) 2 dx, I(y) =int 1 y 2 dx. Suppose that y is an extremal for J subject to the conditions y() = 1, y(1) = 2, and I(y) =L. (a) Find a first integral for the Euler-Lagrange equations for this problem and show that for L =3, y(x) = 4 3(x 1) 2. (b) For L =7/3 show that there exists a linear function that is an extremal for the problem. (c) For L =5/2 showthatthisproblemadmitsthesolutionλ =.Find the extremal corresponding to this value. 4. Let A(y) beasmoothfunctionandlet and I(y) = A(y)y dx 1+y 2 dx. Formulate the Euler-Lagrange equations for the isoperimetric problem with y() =, y(1) = 1, and I(y) =L> 2. Show that λ =,andthat there are an infinite number of solutions to the problem. Explain without using the Euler-Lagrange equations (or any conservation laws) why there must be an infinite number of solutions to this problem. 5. Let y be the extremal to the catenary problem of Example Show that for L sufficiently large there is an x (, 1) such that y(x) <. 4.3 Some Generalizations on the Isoperimetric Problem In this section we present some modest generalizations on the isoperimetric problem discussed in Section 4.2. Most of the details are left to the reader.