pendulum_nonlinear_exact
pendulum_nonlinear_exact,
an Octave code which
evaluates an exact formula 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.
The formula relies on the evaluation of Jacobi elliptic functions
cn(x,k), dn(x,k), sn(x,k), and their inverses.
Licensing:
The computer code and data files described and made available on this web page
are 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,
an Octave code which
evaluates elliptic integrals and Jacobi elliptic functions
cn(), dn(), sn(),
by Milan Batista.
elliptic_integral,
an Octave code which
evaluates complete elliptic integrals of first, second and third kind,
using Carlson's elliptic integral functions.
pendulum_comparison,
an Octave code which
compares the linear and nonlinear ordinary differential equations (ODE)
that represent the behavior of a pendulum of length L under a
gravitational force of strength G.
pendulum_nonlinear_ode,
an Octave code which
sets up the ordinary differential equations (ODE) that represent
the behavior of a nonlinear pendulum of length L under a
gravitational force of strength G.
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).
Last revised on 04 July 2023.