Single-Particle Electrodynamics - Assassination Science

such systems. On the other hand, for systems of arbitrary mass, the four
components of the mechanical momentum p α are clearly independent, since
the identity
p 2 (τ) ≡ m 2 (τ)
is now a definition of the function m 2 (τ), rather than a constraint. But any
use of the proper-time τ as the “time parameter” in the variational framework
will always lead to the constraint (2.75) for the generalised velocities, and
hence cannot yield a formalism embodying four truly independent degrees of
freedom.
The way out of this dilemma is to define a new time parameter, θ, related
differentially to the proper-time by
dθ(τ) ≡
dτ
m(τ) , (2.76)
and use this new time parameter as the “time” parameter in the Euler–Lagrange
formalism. The generalised velocities of the translational motion of
the particle, in this new formalism, can be obtained by means of the chain
rule:
d θ z ≡ (d θ τ)(d τ z)
≡ m(τ)U(τ)
≡ p(τ);
hence, in terms of the theta-time, the generalised velocities are simply the
components of the mechanical momentum:
d θ z(θ) ≡ p(θ).
An additional benefit of using the theta-time θ is that massless particles
may (in principle) be described by the formalism, since both θ and p are
well-defined for such particles, but τ and U are not.
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