Schemannian

As a scheme/Racket based package for symbolic mathematics for physicist, "Schemannian" currently supports a realization of Euler-Lagrangian Equation is classical physics, Riemannian geometry and General Relativity calculations, and simple Grassmannian Calculus.

"Schemannian" won the first Jane Street prize of Lisp In Summer Projects 2013.

Test Environment

"Schemannian" is written and debugged using a Racket v5.3.1 in a 64-bit Ubuntu 13.04 (Raring Ringtail) computer, and a v5.3.5 in a 32-bit Ubuntu 12.04 (Precise Pangolin) computer. Racket is installed by the default setting of sudo apt-get install racket.

Highlights

"Schemannian" can calculate the Lagrangian and the equation of motion (by Euler-Lagrangian equation) of a classical mechanical system. For example, this piece of code will give you the equation of motion of the double pendulum.

(define pendulum1
  (make-pendulum 'm1 'l1 'pivotX1 'pivotY1 (make-function 'theta1 't)))
(define pendulum2
  (make-pendulum 'm2 'l2 (pendulum1 'X) (pendulum1 'Y) (make-function 'theta2 't)))

(define L (lagrangian (list pendulum1 pendulum2)))
(define euler-lagrangian-L
  (euler-lagrangian-equation L
                             (list (make-function 'theta1 't) (make-function 'theta2 't))
                             (list (deriv (make-function 'theta1 't) 't)
                                   (deriv (make-function 'theta2 't) 't))
                             't))

That is interesting, because Lagrangian formulation and Euler-Lagrangian equation are extremely important for loop calculations in quantum field theory. Those calculations are really tedious, and currently there is NO general propose package to do them.

"Schemannian" gives an interface to virtualize the motion of mechanical objects by Euler-Lagrangian equation.

"Schemannian" is capable to calculate typical General Relativity expressions such as Christoffel symbols, Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar from the metric. For example, the following code calculate the curvature on the surface of a sphere.

(define g (make-tensor '((_ a) (_ b))
                       '(((** r 2) 0)
                         (0 (* (** r 2) (** (sin theta) 2))))))

(define Gamma^a_bc (christoffel '((^ a) (_ b) (_ c)) g '(theta phi)))
(define R^a_bcd (riemann-tensor '((^ a) (_ b) (_ c) (_ d)) Gamma^a_bc '(theta phi)))
(define R_ab (ricci-curvature-tensor '((_ a) (_ b)) R^a_bcd))
(ricci-scalar g R_ab)