\hl nSrSSKJr SSKJrJrJr SSKJr SSK J r SSK J r SSK Jr SSKJr S S jrg ) zX This module implements a method to find Euler-Lagrange Equations for given Lagrangian. )combinations_with_replacement) DerivativeFunctiondiff)Eq)S)Symbolsympify)iterablec [U5(a [U5OU4nU(d[UR[55nO,UH&n[ U[5(aM[ SU-5e [U5(a [U5OU4nU(dUSR nO[SU55n[SU55(d[ SU-5eUH%nX#R :XaM[SU<SU<35e [UR[5Vs/sH*nURU;dM[UR5PM, snS/-5n/nUHn[X5n[SUS-5HCn[!X(5H1n U["R$U-[U[U/U Q76/U Q76--nM3 ME ['US5n [ U [&5(dMUR)U 5 M U$s snf) a Find the Euler-Lagrange equations [1]_ for a given Lagrangian. Parameters ========== L : Expr The Lagrangian that should be a function of the functions listed in the second argument and their derivatives. For example, in the case of two functions $f(x,y)$, $g(x,y)$ and two independent variables $x$, $y$ the Lagrangian has the form: .. math:: L\left(f(x,y),g(x,y),\frac{\partial f(x,y)}{\partial x}, \frac{\partial f(x,y)}{\partial y}, \frac{\partial g(x,y)}{\partial x}, \frac{\partial g(x,y)}{\partial y},x,y\right) In many cases it is not necessary to provide anything, except the Lagrangian, it will be auto-detected (and an error raised if this cannot be done). funcs : Function or an iterable of Functions The functions that the Lagrangian depends on. The Euler equations are differential equations for each of these functions. vars : Symbol or an iterable of Symbols The Symbols that are the independent variables of the functions. Returns ======= eqns : list of Eq The list of differential equations, one for each function. Examples ======== >>> from sympy import euler_equations, Symbol, Function >>> x = Function('x') >>> t = Symbol('t') >>> L = (x(t).diff(t))**2/2 - x(t)**2/2 >>> euler_equations(L, x(t), t) [Eq(-x(t) - Derivative(x(t), (t, 2)), 0)] >>> u = Function('u') >>> x = Symbol('x') >>> L = (u(t, x).diff(t))**2/2 - (u(t, x).diff(x))**2/2 >>> euler_equations(L, u(t, x), [t, x]) [Eq(-Derivative(u(t, x), (t, 2)) + Derivative(u(t, x), (x, 2)), 0)] References ========== .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation zFunction expected, got: %src38# UHn[U5v M g7fNr ).0vars L/var/www/auris/envauris/lib/python3.13/site-packages/sympy/calculus/euler.py "euler_equations..Vs2TcWS\\Tsc3B# UHn[U[5v M g7fr) isinstancer )rvs rrrXs3dz!V$$dsz!Variables are not symbols, got %sz Variables z do not match args: )r tupleatomsrr TypeErrorargsall ValueErrormaxrexprlen variablesrrangerr NegativeOnerappend) Lfuncsvarsfdordereqnseqipnew_eqs reuler_equationsr1st%UOOE%L%E aggh'(Aa** ,+>a66U?"Q[[!+>,/0c2 3E D  !Zq%!)$A24;!--*44;A;+C+CCC<%B fb ! ! KK  K,s $H<HN)r2)__doc__ itertoolsrsympy.core.functionrrrsympy.core.relationalrsympy.core.singletonrsympy.core.symbolr sympy.core.sympifyr sympy.utilities.iterablesr r1r2rr<s*4<<$"$&.^r;