Wheeler, Mechanics

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so vanishing δs requires 0 = εh(λ 0 ) ⎡ ⎣ d dλ ⎛ ⎞⎤ ⎝ y 0 ′ √ ⎠⎦ 1 + (y 0 ′ )2 Since εh (λ 0 ) > 0, we may divide it out, leaving ⎡ ⎛ ⎞⎤ ⎣ d ⎝ y 0 ′ √ ⎠ = 0 dλ 1 + (y 0 ⎦λ=λ ′ )2 0 But the point λ 0 was arbitrary, so we can drop the λ = λ 0 condition. The argument holds at every point of the interval. Therefore, the function y 0 (λ) that makes the length extremal must satisfy ⎛ ⎞ d ⎝ y 0 ′ √ ⎠ = 0 dλ 1 + (y 0 ′ )2 This is just what we need, if it really gives the condition we want! Integrating once gives y ′ 0 √1 + (y ′ 0 )2 = c = const. λ=λ 0 so that y 0 ′ = √ c 1 − c ≡ b = const. y 0 (λ) = a + bλ Finally, with initial conditions y 0 (0) = y (A) and y 0 (1) = B, we find y A = a y B = a + b y 0 (λ) = y A + (y B − y A ) λ This is the unique straight line connection (0, y A ) and (1, y B ). Find the form of the function x (λ) that makes the variation, δf, of each of the following functionals vanish: 1. f[x] = ∫ λ 0 ẋ2 dλ where ẋ = dx dλ . 2. f[x] = ∫ λ (ẋ2 + x 2) dλ 0 3. f[x] = ∫ λ 0 (ẋn − x m ) dλ 4. f[x] = ∫ λ (ẋ2 (λ) − V (x(λ)) ) dλ where V (x) is an arbitrary function of x(λ). 0 5. f[x] = ∫ λ 0 ẋ2 dλ where ẋ = (ẋ, ẏ, ż) and ẋ 2 = ẋ 2 + ẏ 2 + ż 2 . Notice that there are three functions to be determined here: x(λ), y(λ) and z(λ). 24
2.3.2 Formal definition of functional differentiation There are several reasons to go beyond the simple variation of the previous section. First, we have only extracted a meaningful result in the case of vanishing variation. We would like to define a functional derivative that exists even when it does not vanish! Second, we would like a procedure that can be applied iteratively to define higher functional derivatives. Third, the procedure can become difficult or impossible to apply for functionals more general than curve length. It is especially important to have a definition that will generalize to field theory. Finally, the added rigor of a formal definition allows careful proofs of the properties of functionals, including generalizations to a complete functional calculus. In this section we look closely at the calculation of the previous section to formally develop the functional derivative, the generalization of differentiation to functionals. We will give a general definition sufficient to for any functional of the form find δf δx f [x] = ∫ 1 0 ( L x, dx ) dλ , d2 x dλ 2 , . . . dn x dλ n dλ One advantage of treating variations in this more formal way is that we can equally well apply the technique to relativistic systems and classical field theory. There are two principal points to be clarified: 1. We would like to allow fully arbitrary variations, h (λ) . 2. We require a rigorous way to remove the integral, even when the variation, δf, does not vanish. We treat these points in turn. Arbitrary variations: one parameter families of functions We would like the functional derivative to formalize finding the extremum of a functional. Suppose we have a functional, f [x (λ)] . We argued that we can look at the difference between f [x (λ) + h (λ)] and f [x (λ)] , demanding that for an extremum, δf ≡ f[x + h] − f[x] = 0 (3) to first order in h (λ) . We want to generalize this to a definition of a functional derivative, δf[x (λ)] δx (λ) (4) such that at extrema, δf[x(λ)] δx(λ) = 0 is equivalent to vanishing variation of f [x] . Let f be given by f[x (λ)] = ∫ 1 0 L(x (λ) , x (1) (λ) , . . . , x (n) (λ))dλ (5) where L is a function of x(λ) and its derivatives up to some maximum order, n. Because L is a function, we found δf by expanding f[x (λ) + h (λ)] to first order in h (λ) . We can get the same result without making any assumptions about h(x) by introducing a parameter α along with h. Thus, we write f [x (λ, α)] ≡ f [x (λ) + αh (λ)] (6) and think of x(λ, α) as a function of two variables for any given h (λ). Equivalently, we may think of x(λ, α) as a 1-parameter family of functions, one for each value of α. Then, although f[x (λ) + αh (λ)] is a functional of x (λ) and h (λ) , it is a simple function of α: f [x (λ, α)] = = ∫ ∫ ( ) L x (λ, α) , x (1) (λ, α) , . . . , x (n) (λ, α) dλ ( ) L x (λ, α) + αh (λ, α) , x (1) + αh (1) , . . . dλ (7) 25
Page 1 and 2: Mechanics James T. Wheeler Septembe
Page 3 and 4: 12 Special Relativity 122 12.1 Spac
Page 5 and 6: Part I Preliminaries Geometry is ph
Page 7 and 8: The first case is immediate because
Page 9 and 10: in contradiction with x and y being
Page 11 and 12: 1. For all A, B in τ, their inters
Page 13 and 14: 1.3 The Euclidean group We now retu
Page 15 and 16: Now, we see that the double sum of
Page 17 and 18: To recover 3-dim space, we first id
Page 19 and 20: Sometimes it is useful to think of
Page 21 and 22: functional. Whereas a function retu
Page 23: = ∣ ∣ ∣∣∣∣∣ ∣∣∣
Page 27 and 28: The development of the theory of di
Page 29 and 30: where h(λ) is arbitrary. In partic
Page 31 and 32: Suppose g (x (β)) = ∫ h (x, x
Page 33 and 34: = 1 ∫ ∫ dλ 1 2 + 1 ∫ ∫ dλ
Page 35 and 36: ( ) If we want to know the position
Page 37 and 38: 4.1.2 Vector transformations To hel
Page 39 and 40: 4.1.4 Some second rank tensors Mome
Page 41 and 42: The squared separation is therefore
Page 43 and 44: 1. V is closed under addition. If u
Page 45 and 46: 2. Functions on M. A function on a
Page 47 and 48: together with the coordinate invari
Page 49 and 50: 4.3 The metric We have already intr
Page 51 and 52: Formally, we are treating the map g
Page 53 and 54: When we do this, the only distincti
Page 55 and 56: ( 2. Express the orthonormal basis
Page 57 and 58: By constrast, suppose we know that
Page 59 and 60: and so on so that e i ⊗ e j has a
Page 61 and 62: y a Lie group. In the next sections
Page 63 and 64: = = ∫ ∫ ( ∂L ∂x k ∂x k
Page 65 and 66: where ε i (x) is a fixed function
Page 67 and 68: is ∫ ( ) ∂L ∂L δS = δq +
Page 69 and 70: Now consider the (vanishing) variat
Page 71 and 72: so x i and v i are always parallel
Page 73 and 74: Using scalings of the variables, we
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We map this point into a 1-dim cont
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for all δx. This p i is the genera
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is constant in time for all antisym
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7 The physical Lagrangian We have s
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where M i j and ai are constant. Th
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in the last step. We can give this
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7.3 Gauging Newton’s law Newton
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There are distinct advantages to th
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8 Motion in central forces While th
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Alternatively, after solving for
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Notice that now any transform of r
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1. At any given value of the energy
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so the condition requires −Ak 2 s
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Since j must be constant while x =
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Using the force law and the angular
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8.5.1 Conic sections We can charact
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Now, for the integral ∫ I = lim a
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do this. Since the force that maint
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Notice that the magnitude is mg cos
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Finally, substituting the expressio
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so The kinetic energy is therefore
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Newtonian mechanics. this is the on
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Now, dividing by ẋ and integratin
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where v = √ v 2 then the momenta
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Our most important tool is the inva
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and τ is invariant, we can find u
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12.3 Acceleration Next we consider
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Notice that P α β = δ β α + 1
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4. Show, using the constraint freel
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Add the first two and subtract the
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Once again cycling the indices, add
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The argument is the same as above,
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and the metric is ⎛ 1 η AB = ⎜
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and demanding that W m transform to
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or in a basis (dx α , dy α )as W
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acket we find 0 = δ { t, ξ A} = {
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This means that each p n 0 is suffi
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[ V i ] A dξA = − ∂H ∂p i dt
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is just 0 = dθ = dF i ∧ dx i =
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Note that Hamilton’s equations ar
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Consider what happens to Hamilton
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= = N∑ j=1 N∑ j=1 = ∂H ∂p i
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and let π j be the momentum variab
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This establishes that replacing the
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Write the quantum wave function as
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arbitrary functions, but we need to
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Hamilton’s principal function is
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Substituting into the expression fo
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16.2 Statistics and pseudomechanics
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we see that B b ε a bc is an eleme
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17 Gauge theory Recall our progress
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The identity is present because = A
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in the infinitesimal transformation
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y inserting identities to write [ A
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These properties - a vector space w
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matrix that depend on, say, N conti
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−J a (J c J b ) + (J a J c ) J b
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then For any odd-order form, ω, we
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17.4 The exterior derivative We may
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and further require the star to be
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where ( ) ∂x m J = det ∂y n is
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so ∇f = ∂f ∂ρ ˆρ + 1 ∂f
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= d ( e ijk ω i dx j dx k) = d (
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From Maxwell’s equations, ∗ d
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[33] Gorringe, V. M. and Leach, P.
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[79] Morgan, J. A., Spin and statis
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Wheeler, Mechanics

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