Jacobi elliptic functions allow to solve many physical problems. Today I will review briefly some features. Let me first highlight that the simple pendulum, Euler asymmetric top, the heavy top, the Duffing oscillator, the Seiffert spiral motion, and the Ginzburg-Landau theory of superconductivity are places where you can find Jacobi functions to arise.
Firstly, you can know there are three special Jacobi functions, named 
, 
and 
. The addition formulae for these 3 functions resembles those from euclidean or hyperbolic geometry:
(1) 
(2) 
(3) 
and where 
is the modulus fo the Jacobi elliptic function. To prove these addition theorems, we can take some hard paths. Let me define the derivatives:
(4) 
(5) 
(6) 
and where
(7) 
and where the initial conditions 
, 
, 
are often assumed. A more symmetric form of these equations can be deduced (exercise!):
(8) 
(9) 
(10) 
Using that
(11) 
you can derive the third form of the addition theorem for Jacobi elliptic functions:
(12) 
(13) 
(14) 
Finally the fourth form of the addition theorem for these functions can be found from algebra, to yield:
(15) 
(16) 
(17) 
Du Val showed long ago that these 4 forms can be derived from a language of five 4d vectors that are parallel to each other. The vectors are
(18) 
(19) 
(20) 
(21) 
(22) 
Du Val also grouped the vectors 
to 
in a compact matrix 
invented by Glaisher in 1881:
![\[ \begin{pmatrix} s_1^2-s_2^2 & s_1c_1d_2+s_2c_2d_1 & s_1c_2d_1+s_2c_1d_2 & s_1c_2d_2+s_2c_1d_1\\ s_1c_1d_2-s_2c_2d_1 & c_2^2d_1^2-(1-k^2)s_1^2 & c_1c_2d_1d_2-(1-k^2)s_1s_2 & c_1c_2-s_1s_2d_1d_2\\ s_1c_2d_1-s_2c_1d_2 & c_1c_2d_1d_2+(1-k^2)s_1s_2 & (1-k^2)s_2^2+c_2^2d_1^2 & d_1d_2-k^2s_1s_2c_1c_2\\ s_1c_2d_2-s_2c_1d_1 & c_1c_2+s_1s_2d_1d_2 & d_1d_2+k^2s_1s_2c_1c_2 & 1-k^2s_1^2s_2^2 \end{pmatrix} \]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-2863329168562420c995e34c8cf7c61d_l3.png "Rendered by QuickLaTeX.com")
This matrix has a very interesting symmetry 
. You can also define the antisymmetric tensor 
from any vector pair 
, 
. In fact, you can prove that the tensor
(23) 
where 
equals to 
, and the 
tensor is the Levi-Civita tensor, holds as identity between the matrix , and the division in two couples the quartets of vectors. It rocks!
How, a refresher of classical mechanics. The first order hamiltonian Mechanics reads
(24) 
From these equations, you get the celebrated Hamilton equations
(25) 
(26) 
Strikingly similar to 
, or 
. First order lagrangian theory provides
(27) 
Also, it mimics classical newtonian mechanics if you allow
(28) 
There is a relation between the lagrangian 
and the hamiltonian function 
via Legendre transforamations:
(29) 
where the generalized momentum is
(30) 
There is also routhian mechanics, by Routh, where you have 
degrees of freedom chosen to be 
and 
, such as
(31) 
and where there are 
routhian-ham-equations, and routhian-lag-equations. The routhian energy reads off easily
(32) 
and
(33) 
Finally, the mysterious Nambu mechanics. Yoichiru Nambu, trying to generalize quantum mechanics and Poisson brackets, introduced the triplet mechanics (and by generalization the N-tuplet) with two hamiltonians 
as follows. For a single N=3 (triplets):
(34) 
and for several triplets
(35) 
and where 
. Sometimes it is written as 
. In the case of N-n-plets, you have
(36) 
and also you get an invariant form for the triplet Nambu mechanics
(37) 
This 3-form is the 3-plet analogue of the symplectic 2-form
(38) 
The analogue for N-n-plets can be easily derived:
(39) 
The quantization of Nambu mechanics is a mystery, not to say what is its meaning or main applications. However, Nambu dynamics provides useful ways to solve some hard problems, turning them into superintegrable systems.
See you in other blog post!
