The gyromagnetic relation—that is, the proportionality between the angular momentum L and the magnetization M—is evidence of the intimate connections between the magnetic properties and the inertial properties of ferromagnetic bodies. However, inertia is absent from the dynamics of a magnetic dipole: The Landau–Lifshitz equation, the Gilbert equation, and the Bloch equation contain only the first derivative of the magnetization with respect to time. In order to investigate this paradoxical situation, the Lagrangian approach, proposed originally by Gilbert, is revisited keeping an arbitrary nonzero inertia tensor. The corresponding physical picture is a generalization to three dimensions of Ampère’s hypothesis of molecular currents. A dynamic equation generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation and the well-known Landau–Lifshitz–Gilbert equation are recovered in the kinetic limit, that is, for time scales longer than the relaxation time of the angular momentum.

1.
R.
Feynman
,
R.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics
(
CIT
,
1963
), Vol.
I
, 20.6–7 for the description of the gyroscope and Vol. II 34–2 to 34–5 for the magnetic precession.
2.
J.
Stöhr
and
H. C.
Siegmann
,
Magnetism: From Fundamental to Nanoscale Dynamics
(
Springer
,
Berlin
,
2006
).
3.
This is not the case for non-uniform magnetization (domain walls or antiferromagnets), for which a magnetic mass is defined. See the pioneering work of
W.
Döring
: “
Uber die trägheit der Wände zwischen Weisschen Bezirken
” (On the inertia of walls between Weiss domains),
Z. Naturforsch.
3a
,
373
379
(
1948
). Döring introduced the magnetic Lagrangian in this paper.
4.
M.-C.
Ciornei
,
J. M.
Rubí
, and
J.-E.
Wegrowe
, “
Magnetization dynamics in the inertial regime: Nutation predicted at short time scales
,”
Phys. Rev. B
83
,
020410
R
1
4
(
2011
).
5.
Fähnle
,
D.
Steiauf
, and
Ch.
Illg
, “
Generalized Gilbert equation including inertial damping: Derivation from an extended breathing Fermi surface model
,”
Phys. Rev. B
84
,
172403
(
2011
).
6.
L. P.
Williams
, “
Why Ampère did not discover electromagnetic induction
,”
Am. J. Phys.
54
,
306
311
(
1986
).
7.
The gyromagnetic relation M=γL has been established through static magnetomechanical measurements. See
S. J.
Barnett
, “
Gyromagnetic and electron-inertia effects
,”
Rev. Mod. Phys.
7
,
129
167
(
1935
);
and
A.
Einstein
and
W. J.
de Haas
, “
Experimenteller Nachweis der Ampereschen Molekularströme
,”
Verhandl. Dtsch. Phys. Ges.
17
,
152
170
(
1915
).
8.
V. Ya.
Frenkel’
On the history of the Einstein-de Haas effect
,”
Sov. Phys. Usp.
22
,
580
587
(
1979
).
9.
P.
Galison
,
How Experiments End
(
The University of Chicago Press
,
Chicago
,
1987
), Chapter 2.
10.
H. C.
Ohanian
, “
What is spin?
,”
Am. J. Phys.
54
,
500
505
(
1986
).
11.
L.
Landau
and
E.
Lifshitz
, “
On the theory of dispersion of magnetic permeability in ferromagnetic bodies
,”
Phys. Z. Sowjetunion
8
,
153
169
(
1935
).
12.
N.
Bloembergen
, “
On the ferromagnetic resonance in Nickel and supermalloy
,”
Phys. Rev.
78
,
572
580
(
1950
).
13.
T. L.
Gilbert
, “
Formulation, foundations and applications of the phenomenological theory of ferromagnetism
,” Ph.D. dissertation,
Illinois Institute of Technology
,
1956
, Appendix B.
14.
T. L.
Gilbert
, “
A phenomenological theory of damping in ferromagnetic materials
,”
IEEE Trans. Mag.
40
,
3443
3449
(
2004
). The discussion related to the assumption I1=I2=0 is confined in footnotes 7 and 8. Note that the original reference in Physical Review is only an abstract:
T. L.
Gilbert
, “
A Lagrangian formulation of the gyromagnetic equation of the magnetization fields
,”
Phys. Rev.
100
,
1243
(
1955
).
15.
J.
Miltat
,
G.
Alburquerque
, and
A.
Thiaville
, “
An Introduction to Micromagnetics in the Dynamics Regime
,” in
Spin dynamics in confined magnetic structures I
, edited by
B.
Hillebrands
and
K.
Ounadjela
(
Springer
,
Berlin
,
2002
). The kinetic energy is introduced through the Lagrangian on page 19, Eq. (37). The Lagrangian is such that (with our notation) I1=I2=0 and Ω3=φ·cosθ.
16.
W. F.
Brown
 Jr.
, “
Single-domain particles: New uses of old theorems
,”
Am. J. Phys.
28
,
542
551
(
1960
). See page 549.
17.
A. H.
Morrish
,
The Physical Principles of Magnetism
(
John Wiley & Sons
,
New York
,
1965
; reprinted by IEEE Press, New York, 2001). See end of page 551.
18.
T. F.
Ricci
and
C.
Scherer
, “
A stochastic model for the dynamics of classical spin
,”
J. Stat. Phys.
67
,
1201
1208
(
1992
), page 1204: “In order to simulate the behavior of a classical spin, we take, for these equations, the limit I10, I30, and ψ·, but maintaining I3ψ·=S(t)= finite.” In our notation SMs.
19.
J. M.
Rubí
and
A.
Pérez-Madrid
, “
Inertial effects in non-equilibrium thermodynamics
,”
Physica A
264
,
492
502
(
1999
).
20.
H.
Goldstein
,
Classical Mechanics
(
Addison-Wesley
,
Reading, MA
,
1980
).
21.
D. W.
Condiff
and
J. S.
Dahler
, “
Brownian motion of polyatomic molecules: The coupling of rotational and translational motions
,”
J. Chem. Phys.
44
,
3988
4005
(
1966
).
22.
J.-E.
Wegrowe
, “
Spin transfer from the point of view of the ferromagnetic degrees of freedom
,”
Solid State Commun.
150
,
519
523
(
2010
).
23.
The full calculation with ψ·0 gives the same result. It is consistent with that found with a different approach in
M.-C.
Ciornei
 et al., “
Magnetization dynamics in the inertial regime: Nutation predicted at short time scales
,”
Phys. Rev. B
83
,
020410
R
(
2011
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.