Mathematical Tools for Physics - Department of Physics - University

16—Calculus of Variations 394 γis a free parameter in this calculation, replacing the originalβ. To minimize this energy, set th ...

16—Calculus of Variations 395 Not quite yet. What abouty′(a)andy′(b)? The endpointsy(a)andy(b)aren’t changing, but that doesn’t ...

16—Calculus of Variations 396 Translate this into the notation that I’ve been using and you have Eq. (16.10). Why did I divide b ...

16—Calculus of Variations 397 The potential energy is a functionUofrandφ. With the of the Lagrangian defined asT−U, the variatio ...

16—Calculus of Variations 398 This produces an expression for∆S ∆S=L ( (tb,y(tb),y ̇(tb) ) ∆tb−L ( (ta,y(ta),y ̇(ta) ) ∆ta + ∂L ...

16—Calculus of Variations 399 If the physical phenomenon described by this equation is invariant under spacial translation, then ...

16—Calculus of Variations 400 These represent the contributions to the variation just abovetmand just below it. This has to vani ...

16—Calculus of Variations 401 If the slope is not continuous, the second factor must vanish. (y′+)^2 + (y′+)(y′−) + (y′−)^2 −β/ ...

16—Calculus of Variations 402 16.10 Second Order Except for a couple of problems in optics in chapter two,2.35and2.39, I’ve most ...

16—Calculus of Variations 403 Is it really that simple? No. First theδyterms can be important too, and secondycan itself have se ...

16—Calculus of Variations 404 In the integral forT, where the starting point and the ending point are the source and image point ...

16—Calculus of Variations 405 Exercises 1 For the functionalF[x] =x(0) + ∫π 0 dt ( x(t)^2 +x ̇(t)^2 ) and the functionx(t) = 1 + ...

16—Calculus of Variations 406 Problems 16.1 You are near the edge of a lake and see someone in the water needing help. What path ...

16—Calculus of Variations 407 16.13 On a right circular cylinder, find the path that represents the shortest distance between tw ...

16—Calculus of Variations 408 16.22 The equation (16.25) is an approximate solution to the path for light above a hot road. Is t ...

Densities and Distributions . Back in section12.1I presented a careful and full definition of the word “function.” This is usefu ...

17—Densities and Distributions 410 You can even think of this as a new kind of functionm(V): input a specification for a volume ...

17—Densities and Distributions 411 17.2 Functionals F[φ]= ∫∞ −∞ dxf(x)φ(x) defines a scalar-valued function of a function variab ...

17—Densities and Distributions 412 (17.7) would have a simpler appearance, such as ̄g=F 1 [g]. For the present case of molecular ...

17—Densities and Distributions 413 I’ve arranged it so that the integral ofφnover allxis one. If I want the value offatx 0 I can ...