The Calculus of Variations (Universitext)
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Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether’s theorem. The text includes numerous examples along with problems to help students consolidate the material.
functional deﬁned by π J(y) = 0 (y 2 − ky 2 ) dx, with endpoint conditions y(0) = 0 and y(π) = 0. If y is an extremal for J then 2.2 The Euler-Lagrange Equation 35 d (2y ) + 2ky = 0; dx i.e., y + ky = 0. The general solution to the Euler-Lagrange equation is √ √ y(x) = c1 cos( kx) + c2 sin( kx). √ Now √ y(0) = 0 implies that c1 = 0, and y(π) = 0 implies that c2 sin( kπ)√= 0. If k is not an integer, √ then c2 = 0, and the only extremal is y = 0. If k is an integer, then sin( kπ) = 0 and c2
luxury of knowing the general solution before we investigate these questions. Even if we cannot solve the Euler-Lagrange equation analytically, qualitative properties such as existence and uniqueness of solutions to the boundary-value problem are nonetheless important. These properties test the veracity of the model especially when experiment shows that a solution must exist. Moreover, the investigation of solution existence and uniqueness highlights any special parameter regions where no
(4.24) lead to the expression x1 η2 x0 d ∂F ∂F − dx ∂y ∂y dx = 0, which is always satisﬁed for any η2 provided equation (4.24) is satisﬁed. 86 4 Isoperimetric Problems The above analysis is valid provided condition (4.21) is satisﬁed. Suppose that ∇Ξ = 0 at = 0. The above calculations show that in this case x1 η1 ∂g d ∂g − ∂y dx ∂y dx = 0 η2 ∂g d ∂g − ∂y dx ∂y dx = 0. x0 and x1 x0 The former equation must be valid for arbitrary η1 ; hence, by Lemma 2.2.2 we have that d ∂g ∂g − =
time explicitly are called rheonomic. Condition (6.10) can thus be called a scleronomic holonomic constraint. Need we say more? 6.2 Nonholonomic Constraints 125 Let Σ denote the surface described by equation (6.11) and let P0 and P1 be two distinct points on Σ. A geodesic on Σ from P0 to P1 is a curve on Σ with endpoints P0 and P1 such that the arclength is stationary. Assuming that such a curve can be represented by a (single) parametric function of the form (6.11) with r(t0 ) = P0 and r(t1
from Geometry 1.4.1 Dido’s Problem Dido was a Carthaginian queen (ca. 850 B.C.?) who came from a dysfunctional family. Her brother, Pygmalion, murdered her husband (who was also her uncle) and Dido, with the help of various gods, ﬂed to the shores of North Africa with Pygmalion in pursuit. Upon landing in North Africa, legend has it that she struck a deal with a local chief to procure as much land as an oxhide could contain. She then selected an ox and cut its hide into very narrow strips, which