Wheeler, Mechanics

More documents

Recommendations

Info

17.5 The Hodge dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 17.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 17.7 The Levi-Civita tensor in arbitrary coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 194 17.8 Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 17.8.1 Grad, Div, Curl and Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4
Part I Preliminaries Geometry is physics; physics is geometry. It is human nature to unify our experience, and one of the important specializations of humans to develop language to describe our world. Thus, we find the unifying power of geometric description a powerful tool for spotting patterns in our experience, while at the same time, the patterns we find lead us to create new terms, concepts, and most importantly, pictures. Applying geometric abstractions to model things in the world, we discover new physics. Creating new pictures, we invent new mathematics. Our ability to make predictions based on perceived patterns is what makes physics such an important subject. The more mathematical tools we have at hand, the more patterns we can model, and therefore the more predictions we can make. We therefore start this adventure with some new tools, Lie groups and the calculus of variations. We will also cast some old familiar tools in a new form. [DESCRIBE EACH] With this extension of the familiar calculus, along with some new ways to look at curves and spaces, we will be able to demonstrate the naturalness and advantages of the most elegant formulation of Newton’s laws of mechanics: the phase space formulation due to Euler, Lagrange, Hamilton and many others, in which the position and the momentum of each particle are treated as independent variables coupled by a system of first-order differential equations. We will take a gradual approach to this formulation. Our starting point centers on the idea of space, an abstraction which started in ancient Greece. Basing our assumptions in our direct experience of the world, we provisionally work in a 3-dimensional space, together with time. Because we want to associate physical properties with objects which move in this space rather than with the space itself, we demand that the space be homogeneous and isotropic. This leads us to the construction of (among other possibilities) Euclidean 3-space. In order to make measurements in this space, we introduce a metric, which allows us to characterize what is meant by uniform motion. We also treat the description of matter from a fundamental approach, carefully defining the mathematical structures that might be used to model the stuff of the world. Finally, we develop techniques to describe the motion of matter. Each of these developments involves the introduction of new mathematics. The description of uniform motion leads to the calculus of variations, the description of matter leads to a discussion of vectors and tensors, and our analysis of motion requires techniques of differential forms, connections on manifolds, and gauge theory. Once these tools are in place, we derive various well-known and not so well-known techniques of classical (and quantum) mechanics. Numerous examples and exercises are scattered throughout. [MORE HERE] Enjoy the ride! 1 The physical arena We have presented general arguments that we can reconcile our visual and tactile experiences of the world by choosing a 3-dim model, together with time. We still need to specify what we mean by a space. Returning to Aristotle’s question, we observe that we can maintain the idea of the “space” where an object was as independent of the body if we insist that “space” contain no absolute information. Thus, the orientation of a body is to be a property of the body, not of the space. Moreover, it should not matter whether a body is at this or that location in space. This notion of space lets us specify, for example, the relative nearness or farness of two bodies without any dependence on the absolute positions of the bodies. These properties are simply expressed by saying that space should have no preferred position, direction or scale. We therefore demand a 3-dim space which is homogeneous, isotropic and scale invariant. 5
Page 1 and 2: Mechanics James T. Wheeler Septembe
Page 3: 12 Special Relativity 122 12.1 Spac
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 and 24: = ∣ ∣ ∣∣∣∣∣ ∣∣∣
Page 25 and 26: 2.3.2 Formal definition of function
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
Page 75 and 76:
We map this point into a 1-dim cont
Page 77 and 78:
for all δx. This p i is the genera
Page 79 and 80:
is constant in time for all antisym
Page 81 and 82:
7 The physical Lagrangian We have s
Page 83 and 84:
where M i j and ai are constant. Th
Page 85 and 86:
in the last step. We can give this
Page 87 and 88:
7.3 Gauging Newton’s law Newton
Page 89 and 90:
There are distinct advantages to th
Page 91 and 92:
8 Motion in central forces While th
Page 93 and 94:
Alternatively, after solving for
Page 95 and 96:
Notice that now any transform of r
Page 97 and 98:
1. At any given value of the energy
Page 99 and 100:
so the condition requires −Ak 2 s
Page 101 and 102:
Since j must be constant while x =
Page 103 and 104:
Using the force law and the angular
Page 105 and 106:
8.5.1 Conic sections We can charact
Page 107 and 108:
Now, for the integral ∫ I = lim a
Page 109 and 110:
do this. Since the force that maint
Page 111 and 112:
Notice that the magnitude is mg cos
Page 113 and 114:
Finally, substituting the expressio
Page 115 and 116:
so The kinetic energy is therefore
Page 117 and 118:
Newtonian mechanics. this is the on
Page 119 and 120:
Now, dividing by ẋ and integratin
Page 121 and 122:
where v = √ v 2 then the momenta
Page 123 and 124:
Our most important tool is the inva
Page 125 and 126:
and τ is invariant, we can find u
Page 127 and 128:
12.3 Acceleration Next we consider
Page 129 and 130:
Notice that P α β = δ β α + 1
Page 131 and 132:
4. Show, using the constraint freel
Page 133 and 134:
Add the first two and subtract the
Page 135 and 136:
Once again cycling the indices, add
Page 137 and 138:
The argument is the same as above,
Page 139 and 140:
and the metric is ⎛ 1 η AB = ⎜
Page 141 and 142:
and demanding that W m transform to
Page 143 and 144:
or in a basis (dx α , dy α )as W
Page 145 and 146:
acket we find 0 = δ { t, ξ A} = {
Page 147 and 148:
This means that each p n 0 is suffi
Page 149 and 150:
[ V i ] A dξA = − ∂H ∂p i dt
Page 151 and 152:
is just 0 = dθ = dF i ∧ dx i =
Page 153 and 154:
Note that Hamilton’s equations ar
Page 155 and 156:
Consider what happens to Hamilton
Page 157 and 158:
= = N∑ j=1 N∑ j=1 = ∂H ∂p i
Page 159 and 160:
and let π j be the momentum variab
Page 161 and 162:
This establishes that replacing the
Page 163 and 164:
Write the quantum wave function as
Page 165 and 166:
arbitrary functions, but we need to
Page 167 and 168:
Hamilton’s principal function is
Page 169 and 170:
Substituting into the expression fo
Page 171 and 172:
16.2 Statistics and pseudomechanics
Page 173 and 174:
we see that B b ε a bc is an eleme
Page 175 and 176:
17 Gauge theory Recall our progress
Page 177 and 178:
The identity is present because = A
Page 179 and 180:
in the infinitesimal transformation
Page 181 and 182:
y inserting identities to write [ A
Page 183 and 184:
These properties - a vector space w
Page 185 and 186:
matrix that depend on, say, N conti
Page 187 and 188:
−J a (J c J b ) + (J a J c ) J b
Page 189 and 190:
then For any odd-order form, ω, we
Page 191 and 192:
17.4 The exterior derivative We may
Page 193 and 194:
and further require the star to be
Page 195 and 196:
where ( ) ∂x m J = det ∂y n is
Page 197 and 198:
so ∇f = ∂f ∂ρ ˆρ + 1 ∂f
Page 199 and 200:
= d ( e ijk ω i dx j dx k) = d (
Page 201 and 202:
From Maxwell’s equations, ∗ d
Page 203 and 204:
[33] Gorringe, V. M. and Leach, P.
Page 205 and 206:
[79] Morgan, J. A., Spin and statis
show all

Wheeler, Mechanics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?