27
2.1. Relativistic classical mechanics
third identity (2.21), the canonical momentum components can be
written as
P i = γmv i + qA i ,
i = 1, . . . , 3.
(2.24)
Equivalently, from (2.18), this is
P i = p i + q A i .
(2.25)
The canonical momentum is thus the sum of the kinetic momentum plus the charge times the magnetic vector potential. Obviously, the canonical momentum and kinetic momentum are identical in the case where the magnetic vector potential is zero.
Next we consider the special case where the potentials A and φ
have no explicit time dependence; i.e., the fields are static. From
(2.9) it follows that the right side of (2.23) vanishes, and
dH = 0.
(2.26)
dt
This means that H is a conserved quantity in this case. From (2.9,
2.19, 2.24, 2.26) it follows that
H = γmc
2 + qφ = const,
(2.27)
and H is a constant of the motion. We will see in the following
that H can be identified with the total energy.
The energy H does not depend on the magnetic vector potential
A, because the magnetic Lorentz force in (2.15) acts in a direction
perpendicular to the particle velocity v. As a result, the magnetic
force alters the direction of the velocity v, but not the magnitude.
Consequently, the magnetic force cannot cause a change in energy.
We now proceed to define two quantities which will prove very
useful later. We define a quantity E by
E = γmc
2 ,
(2.28)
where mc
2 is the rest energy. We further define the kinetic energy
T by
E = T + mc
2 .
(2.29)
2.1. Relativistic classical mechanics
third identity (2.21), the canonical momentum components can be
written as
P i = γmv i + qA i ,
i = 1, . . . , 3.
(2.24)
Equivalently, from (2.18), this is
P i = p i + q A i .
(2.25)
The canonical momentum is thus the sum of the kinetic momentum plus the charge times the magnetic vector potential. Obviously, the canonical momentum and kinetic momentum are identical in the case where the magnetic vector potential is zero.
Next we consider the special case where the potentials A and φ
have no explicit time dependence; i.e., the fields are static. From
(2.9) it follows that the right side of (2.23) vanishes, and
dH = 0.
(2.26)
dt
This means that H is a conserved quantity in this case. From (2.9,
2.19, 2.24, 2.26) it follows that
H = γmc
2 + qφ = const,
(2.27)
and H is a constant of the motion. We will see in the following
that H can be identified with the total energy.
The energy H does not depend on the magnetic vector potential
A, because the magnetic Lorentz force in (2.15) acts in a direction
perpendicular to the particle velocity v. As a result, the magnetic
force alters the direction of the velocity v, but not the magnitude.
Consequently, the magnetic force cannot cause a change in energy.
We now proceed to define two quantities which will prove very
useful later. We define a quantity E by
E = γmc
2 ,
(2.28)
where mc
2 is the rest energy. We further define the kinetic energy
T by
E = T + mc
2 .
(2.29)