Dynamics of Relativistic Particles and EM Fields

The canonical momentum ⃗ P conjugate to the position coordinate ⃗x is
obtained by the definition
Thus the conjugate momentum is
P i ≡ ∂L
∂u i
= γmu i + e c A i (13)
⃗P = ⃗p + e c ⃗ A (14)
where ⃗p = γm⃗u is the ordinary kinetic momentum.
The Hamiltonian H is a function of the coordinate ⃗x and its conjugate
momentum ⃗ P and is a constant of motion if the Lagrangian is not an
explicit function of time, in terms of the Lagrangian is :
H = ⃗ P · ⃗u − L (15)
by eliminating ⃗u in favor of ⃗ P and ⃗x we find (HOW?) that
cP ⃗u =
⃗ − eA
√ ⃗ (⃗P )
(16)
−
e ⃗ 2
A
c + m2 c 2
Dynamics of Relativistic Particles and EM Fields