The Relativistic Lagrangian

In "The Relativity of Phase Space" we saw that we can express the action carried out by a particle moving between two points between two given instants as a product of two four-vectors,

(Eq’n 1)

Hamilton’s principle, the principle of least action, requires that

(Eq’n 2)

which tells us that the particle follows a trajectory that makes the action a
minimum relative to the actions of variant trajectories that run near it. Those
variant trajectories differ from the true trajectory by the minuscule amount
δ**x**,
which equals zero at the endpoints of the trajectory.

Combining Equations 1 and 2 gives us Hamilton’s principle as

(Eq’n 3)

That relation leads us, for convenience, to define the Lagrangian function as

(Eq’n 4)

In Newtonian physics, with T representing the particle’s kinetic energy and U representing the particle’s potential energy, we have

(Eq’n 5)

which gives us the conventional description of the Lagrangian.

Hamilton’s principle now takes the form

(Eq’n 6)

in which the variation operation (represented by the curly delta) commutes
with the integration because it has the nature of a differentiation. Because it
depends on a kinetic energy (which depends on the velocity of the particle) and
a potential energy (which depends on the location of the particle), the
Lagrangian has the dependency L=L(**x**,**v**), so when we write out the
variation of the Lagrangian in its total differential form we get

(Eq’n 7)

Note that in that equation I have made use of the fact that δt=0. So now we get Equation 6 as

(Eq’n 8)

In the first line of that equation the second term on the right side of the
equality sign drops out because the integration simply produces the expression
in parentheses evaluated between the endpoints of the path, where
δx_{i}=0.
The other terms zeroes out in the integration, as required by Hamilton’s
principle, if and only if

(Eq’n 9)

which gives us the Euler-Lagrange equations for all the values of the index i.

For the fully relativistic version we use Einstein’s
theorem (E=mc^{2}) to rewrite Equation 4 as

(Eq’n 10)

Recalling the relation between proper time (τ) elapsed for the particle and the measured time (t) that the observers read off their clocks,

(Eq’n 11)

and the fact that the action remains invariant under a Lorentz Transformation, we write the action accrued by a particle in potential-free space as

(Eq’n 12)

That equation tells us that Hamilton’s principle of least action corresponds to the principle of least proper time (Fermat’s principle). It also tells us that we can use measured time, rather than proper time, as our independent variable, which means that the derivation in Equations 7 and 8 remains valid in fully relativistic situations, so that we continue to use the Euler-Lagrange equations in the form of Equation 9 to derive the equations of motion from the Lagrangian function describing the system.

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