Single-Particle Electrodynamics - Assassination Science

2.4 Lagrangian mechanics
An alternative, and most fruitful, formulation of classical mechanics is that
due to Lagrange and Hamilton. One postulates a Lagrangian function, L,
that depends on some number n of generalised coördinates of the system,
labelled q a (where a runs from 1 to n); their first time-derivatives, denoted
˙q a ; and the time t. One then further postulates that the motion of the system
from time t 1 to time t 2 is such that the line integral
I ≡
∫ t2
t 1
L dt, (2.21)
referred to as the action, has a stationary value for the actual motion of the
system, where the two end-points are fixed. From this principle, it a merely
mathematical problem to show [96] that the system evolves according to the
Euler–Lagrange equations:
(
d t ∂ ˙qa L ) − ∂ qa L = 0. (2.22)
Because the end-points of the variation are fixed, the Euler–Lagrange equations
are unchanged if a total time derivative is added to the Lagrangian:
L −→ L + d t Γ .
We thus seen that, in Lagrangian mechanics, the fundamental mathematical
quantity is the Lagrangian function; equally important is a knowledge of
the appropriate degrees of freedom, q a , for the physical system in question.
We shall now describe various definitions and specifications, for the use
of the Lagrangian formulation of mechanics in this thesis, as was done in
Section 2.3 for the Newtonian formulation.
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