Classical Mechanics

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 wellknown
techniques of classical (and quantum) mechanics.
Numerous examples and exercises are scattered throughout.
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Enjoy the ride!
1. Physical theories
Within any theory of matter and motion we may distinguish two conceptually
different features: dynamical laws and measurement theory. We discuss each in
turn.
By dynamical laws, we mean the description of various motions of objects,
both singly and in combination. The central feature of our description is generally
some set of dynamical equations. In classical mechanics, the dynamical
equation is Newtons second law,
v
F = m d dt
or its relativistic generalization, while in classical electrodynamics two of the
Maxwell equations serve the same function:
1 dE
∇ B = 0
c dt
1 dB
+ ∇ E = 4
J
c dt
c
The remaining two Maxwell equations may be regarded as constraints on the
initial eld conguration. In general relativity the Einstein equation gives the
time evolution of the metric. Finally, in quantum mechanics the dynamical law
is the Schrödinger equation
H ˆ = i
∂
∂t
which governs the time evolution of the wave function, .
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