Lagrange's equations have the same form regardless of
- coordinate system (Cartesian, Cylindrical or spherical) [Sym, p.366, l.20-l.22; Coh, p.1485, l.-15]
- the type of system (a particle or a rigid body [Coh, pp.1482-1485] and
- the type of external field (a particle in a central potential [Coh, p.1488, §4.a] or a charged particle placed in an electromagnetic field [Coh, p.191, §4.b]).
Hamilton's equations are more advantageous than Lagrange's equations because
- Hamilton's equations are simpler and more symmetric [Lan1, p.132, l.6-l.10]
- Hamilton's equations retain the same form under a wider range of transformations [Lan1, p.143, l.20-p.144, l.11 & p.144, l.-4-l.-1]. Therefore, we adopt Hamilton's equations as the equations of motion in quantum mechanics.
Remark.
The transformation L→ L' leaves
By searching for a transformation that leaves the form of Maxwell’s equations unaltered [Rob, p.2, l.13-l.28], we establish the relationship between time and space [Lan2, p.10, (4.3)].
Remark.
Choices of vector and scalar potential:
The transformation (A,f )→ (A',f ') leaves E and H invariant.