Gelfand, I. M. & Fomin, S. V. 1991 Calculus of Variations . New York: Dover.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(17) 9 must always hold; among all of its possible solutions, we should then pick the smoothest one. This reduces to a constrained minimization problem: u y ] dx dy; subject to E t E x u E y v = 0: 21) To solve this problem, we use the classical method of Lagrange multipliers [22] for a function of two variables. Let us consider the general problem min (u;v) J (u; v) subject to G(u; v) 0: 22) J (u; v) L(x; y; u; v; u x ; u y ; v x ; v y ) dx dx; 23) for L(x; y; u; v; u x ; u y ; v x ; v y ) u y ; 24) and where J (u; v) E t E x u E y v: 25) ....

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Clis, N. J., 1963.

....in both cases imply that the quadratic forms are transformed in a natural manner, and hence the algebraic characteristics (i.e. the rank, and, in the real case, the signature) are the complete invariants for the problem. More restrictively, classical convexity hypotheses on the Lagrangian, [20], or, what is essentially equivalent, ellipticity conditions on the Euler Lagrange equations, require that the Hessian matrix be positive definite. In the case of field theory in several variables, p 1, q 1) matters are far less clear cut. First of all, the Hessian tensor H derived by the ....

Gel'fand, I.M. and Fomin, S.V., Calculus of Variations, Prentice Hall, Englewood Cliffs, N.J., 1963. 34

....and give overflow (at s = T f ) f i (s) r i Gamma c)ds = b: 9) Constraint (9) can be introduced in the infimization by using a Lagrangian representation, with multiplier K, as explained in [20] in detail. Define x i (r i Gamma c) x 2 M; u 2 IR : 10) From the calculus of variations [10, 20] we find the first order necessary conditions for an optimal f , known as the Euler equations: x fi fi d u fi fi ; 11) for all s 2 [0; T f ] The optimal path f of problem (5) is a concatenation of the two subpaths found in these two steps. At time 0 the two subpaths meet in ....

I. Gelfand and S. Fomin. Calculus of Variations. Prentice-Hall, Englewood Cliffs, N.J., 1963.

....of all related paths. The di#culty is, the possible paths between two points spans an infinite dimensional space. In 18th century, Euler (1707 1783) and Lagrange (1736 1813) were concerned with similar problems and developed variational calculus to help solving them. The following is a key theorem [1] we will use in this paper: Theorem 2.1. Let J [y] be a function(al) of the form J [y] F (x, y, y # )dx defined on the set of functions y(x) which have continuous first derivatives in [a, b] and satisfy the boundary condition y(a) A, y(b) B. Then a necessary condition for J [y] to have ....

I. M. Gelfand and S. Fomin. Calculus of Variations. Prentice-Hall, Inc, Englewood Cli#s, New Jersey, 1963.

....or, more generally, the calculus of variations. Typically, Noether transformations are considered to be point transformations (they are considered to be functions of coordinates and time) but one can consider more general transformations depending also on velocities and higher derivatives [10] or within the broader context of dynamical symmetries [20] For an example of an integral of motion which comes from an invariance transformation depending on velocities, see [23] In most formulations of Noether s principle, the Noether transformations keep the integral functional invariant (cf. ....

I. M. Gelfand and S. V. Fomin. Calculus of variations. Dover Publications, Mineola, NY, 2000. Zbl 0964.49001

....in the calculus of variations was initiated in the early part of the XX century by Emmy Noether who, in uenced by the works of Klein and Lie on the transformation properties of di erential equations under continuous groups of transformations (see e.g. 2, Ch. 2] published in her seminal paper [13, 14] of 1918 two fundamental theorems, now classical results and known as the ( rst) Noether theorem and the second Noether theorem, showing that invariance with respect to a group of transformations of the variables t and x implies the existence of certain conserved quantities. These results, ....

.... of optimal control problems are available in [19, 21, 20] The second Noether theorem establishes the existence of k (m 1) rst integrals when the Lagrangian is invariant under an in nite continuous group of transformations which, rather than dependence on parameters, as in the rst theorem, depend 2000 Mathematics Subject Classi cation. 49K15, 49S05. Key words and phrases. optimal control, Pontryagin extremals, gauge symmetry, second Noether theorem, Noether currents. This research was supported in part by the Optimization and Control Theory Group of the R D Unit Mathematics and Applications, ....

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I. M. Gelfand and S. V. Fomin. Calculus of variations. Dover Publications, Mineola, NY, 2000. Zbl 0964.49001

....families of maps from finiteor infinite dimensional Lie group ( 11, 12] variational problems of (higher order) supermechanics (see [6] A generalization of the Noether s theorem which involves the maps depending on x has been considered in the textbook by I. M. Gelfand and S. V. Fomin [8]. Another interesting generalization can be found in the book of H. Rund [14] invariance of the Lagrangian up to addition of an exact differential d# (t, x) with # linear on the parameter s. H. Rund remarks however (cf. 14, Remark 2, p. 297] that this generalization is incompatible with the ....

I. M. Gelfand, S. V. Fomin. Calculus of variations, Prentice--Hall, (1963). Zbl 992.07815 MR 28:3353

....be called a conservation law . Definition 4. A quantity C(t, x, u, #) which is constant along every abnormal extremal (x( u( 0, #( of (P ) will be called an abnormal conservation law . 3 Main Results Both classical Noether s theorem for the calculus of variations (cf. e.g. 1] 2] [11], 15] 23] and respective versions for optimal control (cf. 16] 25] 26] have been formulated as implications: invariance implies a conservation law . Here we shall begin with a weak notion of invariance (5) and formulate a necessary and su#cient condition (3.1) Afterwards, in order to ....

....u, 0 , T (t, x, u) is a conservation law, where T (t, x, u) #T (t, x, u, s) # # X(t, x, u) #X (t, x, u, s) # # F (t, x, u) #F(t, x, u, s) # # . Remark 3. For the basic problem of the calculus of variations, and for 0, Theorem 3 coincides with the Noether s theorem in [11]. The triple (T (t, x, u) X(t, x, u) U(t, x, u) is called the tangent vector field of h and the constructed conservation law is the value of the Cartan di#erential 1 form w = # dx H dt on the tangent vector field. For a survey of these questions and the role of E. Noether s results in ....

Gelfand I. M., Fomin S. V. (2000) Calculus of Variations. Dover, New York

....that correspond to popular filter masks for smoothing and sharpening. A. Smoothing Our first example is a smoothing mask that is of practical use in signal processing. The smoothness measure #p ## (t)# dt is minimized if the corresponding Euler Lagrange equation p #### (t) 0 holds true [6]. The discrete version of the EulerLagrange equation, # p i = 0, is obtained by replacing differential operators with forward divided difference operators. It is well known that an iterative scheme using a local update rule p i (21) gradually adjusts the data points to approach the ....

I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, 1963.

....and averaging in the radial direction (see the upper part of figure 1) d. 4 The modulation Lagrangian can be written as a constrained one using only first derivatives: t(L) L( t4 , with M being the Lagrange multiplier. 3.3. 1 Conservation Laws Based on Noether s theorem [11], we can use the transformation groups that leave the action integral I: t)2 4( 3 j dz dt (23) invariant to derive conservation laws for the modulation system. Invariance to phase (i.e. the identity) time and space translations leads to mass, energy and momentum conservation: dt = ....

I. Gelfand and S. Fomin. Calculus of Variations. Prentice-Hall, New-Jersey, 1963.

....fil ters that correspond to popular filter masks for smoothing and sharpening. A. Smoothing Our first example is a smoothing mask that is of practical use in signal processing. The smoothness measure dt is minimized if the corresponding Euler Lagrange equation #t# # # holds true [5]. The discrete version of the EulerLagrange equation, # # # ##, is obtained by replacing differential operators with forward divided difference operators. It is well known that an iterative scheme using a local update rule # # # # # # # # (21) gradually adjusts the data points to approach ....

I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, 1963.

....correspond to popular filter masks for smoothing and sharpening. A. Smoothing Our first example is a smoothing mask that is of practical use in signal processing. The smoothness measure # ## ## #t## # dt is minimized if the corresponding Euler Lagrange equation # #### #t# # # holds true [6]. The discrete version of the EulerLagrange equation, # # # # ##, is obtained by replacing differential operators with forward divided difference operators. It is well known that an iterative scheme using a local update rule # # # # # # # # # (21) gradually adjusts the data points to ....

I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice-Hall, 1963.

....accelerations into the framework, consider what happens when the Lagrangian is a function of q, q and q, i.e the Lagrangian is some function L (2) q, q, q) 16 Hamilton s principle (2. 1) and simple variational calculus (see for example Chapter 2, page 42, equation 22 of Gelfand and Fomin [4]) ensures that the solutions q(t) satisfy the fourth order Euler Lagrange equations d 2 dt 2 #L (2) #q i d dt #L (2) # q i #L (2) #q i = 0. 4.2) This is fourth order because in general the term d 2 dt 2 #L (2) #q i (q, q, q) 4.3) involves a fourth derivative ....

Gelfand, I.M and Fomin, S.V. Calculus of Variations. Prentice-Hall, 1963.

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Gelfand, I. M. & Fomin, S. V. 1991 Calculus of Variations . New York: Dover.

No context found.

Gelfran. I.M. and S.V. Fomin. Calculus of Variations. Prentice Hall, 1963.

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I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, Mineola, NY, 2000.

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Gelfand, M. Fomin, S. "Calculus of Variations." Prentice-Hall Inc. 1963.

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Gelfand, I.M., and Fomin, S.V. Calculus of Variations. Prentice-Hall, Inc., Englewood Clis, New Jersey (1963).

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Gelfand, I.M., and Fomin, S.V. Calculus of Variations, Prentice-Hall, Inc., Englewood Cli#s, New Jersey (1963).

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I. M. GEL'FAND AND S. V. FOMIN, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.

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I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, N. J., 1963.

No context found.

I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice Hall, 1963.

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I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, N. J., 1963.

No context found.

I. M. Gelland and S. V. Fomin, Calculus of variations. Dover Publications, Mine- ola, NY, 2000.

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I. M. Gelfand and S. V. Fomin. Calculus of variations. Dover Publications, Mineola, NY, 2000. Zbl 0964.49001

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