pendulum_nonlinear_exact

pendulum_nonlinear_exact, a MATLAB code which evaluates an exact formula y(t) for the solution of the the ordinary differential equations (ODE) that represent the behavior of a nonlinear pendulum of length L under a gravitational force of strength G, y''=-G/L sin(y).

The formula relies on the evaluation of Jacobi elliptic functions cn(x,k), dn(x,k), sn(x,k), and their inverses.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

pendulum_nonlinear_exact is available in a MATLAB version and an Octave version.

Related Data and codes:

pendulum_nonlinear_exact_test

elfun, a MATLAB code which evaluates elliptic integrals and Jacobi elliptic functions cn(), dn(), sn(), by Milan Batista.

elliptic_integral, a MATLAB code which evaluates complete elliptic integrals of first, second and third kind, using Carlson's elliptic integral functions.

matlab_exact, a MATLAB code which evaluates exact solutions to a few selected examples of ordinary differential equations (ODE) and partial differential equations (PDE).

Reference:

• Milton Abramowitz, Irene Stegun,
  Handbook of Mathematical Functions,
  National Bureau of Standards, 1964,
  ISBN: 0-486-61272-4,
  LC: QA47.A34.
• Augusto Belendez, Carolina Pascual, David Mendez, Tarsicio Belendez Vazquez, Cristian Neipp,
  Exact solution of the nonlinear pendulum problem,
  Revista Brasileira de Ensino de Fisica,
  Volume 29, Number 4, pages, 645-648, 2007.
• Roland Bulirsch,
  Numerical Calculation of Elliptic Integrals and Elliptic Functions,
  Numerische Mathematik,
  Volume 7, pages 78-90, 1965.
• Bille Carlson,
  Numerical computation of real or complex elliptic integrals,
  Numerical Algorithms,
  Volume 10, 1995, pages 13-26.
• Wolfgang Ehrhardt,
  The AMath and DAMath Special functions: Reference manual and implementation notes,
  https://www.swmath.org/software/29423.
• Karlheinz Ochs,
  A comprehensive analytical solution of the nonlinear pendulum,
  European Journal of Physics,
  Volume 32, pages 479-490, 2011.
• William Reinhardt, Peter Walker,
  Jacobian Elliptic Functions,
  Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
  NIST Handbook of Mathematical Functions,
  Cambridge University Press, 2010,
  ISBN: 978-0521192255,
  LC: QA331.N57.
• Paul D. Williams,
  Achieving seventh-order amplitude accuracy in leapfrog integrations,
  Monthly Weather Review,
  Volume 141, Number 9, pages 3037-3051, September 2013.

Source Code:

• ijsn.m, inverse of Jacobi elliptic function SN.
• jacobi_cnk.m, Jacobi's elliptic function cn(x,k).
• jacobi_dnk.m, Jacobi's elliptic function dn(x,k).
• jacobi_snk.m, Jacobi's elliptic function sn(x,k).
• jacobi_snk_inverse.m, inverse of Jacobi's elliptic function sn(x,k).
• jcn.m, evaluates the Jacobi elliptic function cn(x,k).
• jdn.m, evaluates the Jacobi elliptic function dn(x,k).
• jsn.m, evaluates the Jacobi elliptic function sn(x,k).
• melf.m, evaluates incomplete elliptic integrals of the 1st kind.
• melk.m, evaluates complete elliptic integrals of the 1st kind.
• mijsn.m, computes the inverse of Jacobi elliptic function sn(x,m).
• pendulum_nonlinear_conserved.m, evaluates a quantity that should be conserved by the nonlinear pendulum ODE.
• pendulum_nonlinear_deriv.m, evaluates the right hand side of the nonlinear pendulum ODE.
• pendulum_nonlinear_exact.m, returns the exact solution of the nonlinear pendulum ODE;
• pendulum_nonlinear_parameters.m, returns parameters of the nonlinear pendulum ODE.
• r8_sign.m, returns the sign (+1 or -1 only) of a real value.
• rf.m, evaluates Carlson's elliptic integral of the first kind.
• sncndn.m, evaluates the Jacobi elliptic functions sn, cn, dn.
• ufun2.m, mimics the elemental behavior of a function FUN(X,Y).