We can write the free particle Lagrangian using only scalar reductions of
suitable 4-vectors:
L_free = - mc&gamma#gamma; U_&alpha#alpha;U^&alpha#alpha;
(which is still
). The action is thus
A = -mc &int#int;_&tau#tau;_0^&tau#tau;_1 U_&alpha#alpha;U^&alpha#alpha; d&tau#tau;.
The variations on this action must be carried out subject to the constraint
U_&alpha#alpha;U^&alpha#alpha;= c^2
which severely limits the allowed solutions. We write this as
d(U_&alpha#alpha;U^&alpha#alpha;)d&tau#tau; & = & 0

dU_&alpha#alpha;d&tau#tau; U^&alpha#alpha;+ U_&alpha#alpha;dU^&alpha#alpha;d&tau#tau; & = & 0

dU_&alpha#alpha;d&tau#tau; g^&alpha#alpha;&beta#beta;g_&beta#beta;&alpha#alpha;U^&alpha#alpha;+ U_&alpha#alpha;dU^&alpha#alpha;d&tau#tau; & = & 0

U_&beta#beta;dU^&beta#beta;d&tau#tau; + U_&alpha#alpha;dU^&alpha#alpha;d&tau#tau; & = & 0

2 U_&alpha#alpha;dU^&alpha#alpha;d&tau#tau; & = & 0

U_&alpha#alpha;dU^&alpha#alpha;d &tau#tau; & = & 0

Now,
U_&alpha#alpha;U^&alpha#alpha; d&tau#tau;= dx_&alpha#alpha;d&tau#tau;
dx^&alpha#alpha;d&tau#tau; d&tau#tau;= g^&alpha#alpha;&beta#beta; dx_&alpha#alpha;dx_&beta#beta;
which is an infinitesimal length in four-space. The latter expression does
not explicitly contain
. We can thus parameterize the action in terms
of a path-parameter
that increases monotonically with
but is
otherwise arbitrary. Then
A = -mc &int#int;_s_0^s_1 g^&alpha#alpha;&beta#beta; dx_&alpha#alpha;ds
dx_&beta#beta;ds ds .
We are clearly making progress. We have taken a perfectly good expression and
made in unrecognizable. To make you a little happier, note that this has now
got the form of
A = &int#int;L ds
where
is a scalar ``Lagrangian'' written in terms of an *independent* free parameter. This might be progress after all, since we have
quashed the annoying
.

If we now do the calculus of variations thing and get the Euler-Lagrange
equations in *four* dimensions:
d ds ( dL &part#partial;(
dx_&alpha#alpha;ds ) ) - &part#partial;^&alpha#alpha;L = 0
(for
). Applying them to the Langrangian in this action,
they turn out to be:
mc d ds
&part#partial;
{g^&delta#delta;&beta#beta; dx_&beta#beta;ds dx_&delta#delta;ds }^12
&part#partial;( dx_&alpha#alpha;ds ) & = & 0

mc2d ds {
dx^&alpha#alpha;ds + dx^&alpha#alpha;ds
dx_&beta#beta;ds dx^&beta#beta;ds } & = & 0

mc d ds { dx^&alpha#alpha;ds
dx_&beta#beta;ds dx^&beta#beta;ds } & = & 0 .

This still does not have the constraint above imposed on it. We impose the
constraint by identifying
with
in such a way that the constraint
is simultaneously satisfied:
dx_&alpha#alpha;ds dx^&alpha#alpha;ds ds & = & c d&tau#tau;

d d&tau#tau; & = & cdx_&alpha#alpha;dsdx^&alpha#alpha;ds
d ds
(which requires both
and
). If you like,
this constraint picks out of all possible path parameterizations the one that
follows the proper time while keeping the four vector velocity scalar product
Lorentz invariant. For free particles this is a lot of work, but it is paid
back when we include an interaction.

If we multiply the Euler-Lagrange equation (in terms of ) from the left by: cdx_&alpha#alpha;dsdx^&alpha#alpha;ds and use the constraint to convert to , the result (for the equation of motion) is: mcdx_&beta#beta;dsdx^&beta#beta;ds d ds { cdx_&beta#beta;ds dx^&beta#beta;ds d dsx^&alpha#alpha;} = 0 or m d^2x^&alpha#alpha;d&tau#tau;^2 = 0 which certainly looks it has the right form.

We can include an interaction. Just as before, must be a Lorentz scalar. When we make a parameterized version of the Lagrangian, the part under the integral must be a 4-scalar. The covariant form of the result is (hopefully obviously) A = - &int#int;_s_0^s_1 { mc g^&delta#delta;&beta#beta; d_&delta#delta;ds dx_&beta#beta;ds + qc dx_&beta#beta;dsA^&beta#beta;} ds . The ``four Lagrangian'' in this equation is L = - { mc g^&delta#delta;&beta#beta; d_&delta#delta;ds dx_&beta#beta;ds + qc dx_&beta#beta;dsA^&beta#beta;}.

As before we construct the Euler-Lagrange equation.
mc d ds { dx^&alpha#alpha;ds
dx_&beta#beta;ds dx^&beta#beta;ds +
qcA^&alpha#alpha;} -
qcx_&beta#beta;ds&part#partial;^&alpha#alpha;A^&beta#beta;& = & 0

Again we multiply through from the left by cdx_&alpha#alpha;dsdx^&alpha#alpha;ds and convert to to get: m d^2x^&alpha#alpha;d&tau#tau;^2 + qc dA^&alpha#alpha;d&tau#tau; - qcdx_&beta#beta;d&tau#tau; &part#partial;^&alpha#alpha;A^&beta#beta;= 0

The derivative is a bit jarring. However, if we express this total derivative in terms of partials we observe that: dA^&alpha#alpha;d&tau#tau; = dx_&beta#beta;d&tau#tau; x_&beta#beta; A^&alpha#alpha;= dx_&beta#beta;d&tau#tau; &part#partial;^&beta#beta;A^&alpha#alpha; Substituting, the equation of motion becomes: d (mU^&alpha#alpha;)d&tau#tau; = m d^2x^&alpha#alpha;d&tau#tau;^2 = qc (&part#partial;^&alpha#alpha;A^&beta#beta;- &part#partial;^&beta#beta;A^&alpha#alpha;) dx_&beta#beta;d &tau#tau; = qcF^&alpha#alpha; &beta#beta; U_&beta#beta;. which is, as expected, the Lorentz force law in covariant form! How lovely!

To make a Hamiltonian in this notation, we must first make the canonical
momentum:
P^&alpha#alpha;= - &part#partial;L &part#partial;( x_&alpha#alpha;ds
) = m U^&alpha#alpha;+ qc A^&alpha#alpha;
which is a covariant version of the *complete* set of interaction
equations from the previous section (it does both energy and
3-momentum).

There are several ways to make a Hamiltonian (recall that in general
there is what amounts to gauge freedom, minimally the ability to add an
arbitrary constant which naturally does not affect the resulting
differential equations). One is^{19.1}:
H = U_&alpha#alpha;P^&alpha#alpha;+ L
Again, we must eliminate:
U^&alpha#alpha;= 1m(P^&alpha#alpha;- qcA^&alpha#alpha;)
in favor of
. Thus:
L = -mc1m^2(P_&alpha#alpha;-
qcA_&alpha#alpha;)(P^&alpha#alpha;- qcA^&alpha#alpha;) -
qmc(P_&alpha#alpha;- qcA_&alpha#alpha;)A^&alpha#alpha;
and
H & = & 1m(P_&alpha#alpha;-
qcA_&alpha#alpha;)P^&alpha#alpha;-mc1m^2(P_&alpha#alpha;-
qcA_&alpha#alpha;)(P^&alpha#alpha;- qcA^&alpha#alpha;)

& & - qmc(P_&alpha#alpha;- qcA_&alpha#alpha;)A^&alpha#alpha;

& = & 1m(P_&alpha#alpha;- qcA_&alpha#alpha;)
(P^&alpha#alpha;- qcA^&alpha#alpha;) - c(P_&alpha#alpha;-
qcA_&alpha#alpha;)(P^&alpha#alpha;- qcA^&alpha#alpha;)

This Hamiltonian in four dimensions is no longer an energy since it is
obviously a 4-scalar and energy transforms like the time-component of
a four vector. However, it works. Hamilton's equations (in four
dimensions) lead again directly to the relativistic Lorentz force law:
dx^&alpha#alpha;d&tau#tau; = HP_&alpha#alpha; & = &
1m(P^&alpha#alpha;- qcA^&alpha#alpha;)

dP^&alpha#alpha;d&tau#tau; = - Hx_&alpha#alpha; = -
&part#partial;^&alpha#alpha;H & = &
qmc(P_&alpha#alpha;- qcA_&alpha#alpha;) &part#partial;^&alpha#alpha;
A^&beta#beta;

There is a bit of algebra involved in deriving this result. For
example, one has to recognize that:
p^&alpha#alpha;= mU^&alpha#alpha;= P^&alpha#alpha;- qcA^&alpha#alpha;
and
and apply this to eliminate unwanted
terms *judiciously*, that is *after* differentiation. If you
apply it too early (for example at the beginning) you observe the
puzzling result that:
H & = & 1mp_&alpha#alpha;p^&alpha#alpha;- c p_&alpha#alpha;
p^&alpha#alpha;

& = & 1m m^2c^2 - c m^2c^2

& = & m c^2 - mc^2

& = & 0
which leads one to the very Zen conclusion that the cause of all things
is Nothing (in four dimensions, yet)!

We are left with a rather mystified feeling that the algebraic hand is
quicker than the eye. Somehow an equation whose *four-scalar value*
is zero has a functional form, a *structure*, that leads to
non-zero, covariant equations of motion. Also (as already remarked)
this Hamiltonian is not unique. Obviously one can add an arbitrary
four-scalar constant to the equation and get no contribution from the
derivatives (just as one can in nonrelativistic classical physics).
There are other gauge freedoms - ultimately there several other ways of
writing the Hamiltonian associated with the given Lagrangian; all of
them yield a constant value that is not the energy when evaluated and
yield the correct equations of motion when processed.

Finally there exist what are called *singular Lagrangians* -
Lagrangians for which the generalized coordinates do not always map into
generalized conjugate variables! Dirac was (unsurprisingly) the first
to formally identify this in the context of constrained systems (systems
described by a Lagrangian and constraints with Lagrange multipliers for
which the Hesse determinant vanishes); Bergmann (at Syracuse) also made
major contributions to the formal development of the concept. However
the roots of the problem date much further back to e.g. Noether's
theorem. I have a couple of papers on this that I've collected from the
web, although the idea is also discussed in various monographs and
textbooks on mechanics.

It is worth pointing out that there was at one point considerable work being done here at Duke on the idea - N. Mukunda, Max Lohe, (both friends of mine) worked on the idea with Larry Biedenharn (my advisor); Biedenharn also published work with Louck on the subject, and of course Mukunda and Sudarshan's book on classical mechanics remains a ``classic''. Since Dirac's time the notion that the ``right'' unified field theory will have certain interesting properties related to this has been batted around.

This points out an ongoing problem in relativistic quantum theories. These theories are generally based on a Hamiltonian, but manifestly covariant Hamiltonians for a given system cannot in general be uniquely derived from first principles as the mapping between velocities and momenta is not always one-to-one. Thus even when a covariant Lagrangian density can be constructed, the associated Hamiltonian is not obvious or necessarily unique. This is just one (although it is one of the most fundamental) obstacles to overcome when developing a relativistic quantum field theory.