+ il pRt^RIHt^RIHtHtHt^RIHt^RI H t ^RI H t ^RI Ht^RIHtR RltR #) zX This module implements a method to find Euler-Lagrange Equations for given Lagrangian. )combinations_with_replacement) DerivativeFunctiondiff)Eq)S)Symbolsympify)iterablec \V4'd \V4MV3pV'g \VP\44pM1VF+p\ V\4'dK\ RV,4h \V4'd \V4MV3pV'gV^,P pM,\;QJd.RV4FNK 5M !RV44p\;QJdRV4F 'dK RM RM !RV44'g\ RV,4hVF&pW#P 8XdK\RV: RV: 24h \VP\4Uu.uF+pVPV9gK\VP4NK- up^.,4p.pVFp\W4p\^V^,4FQp\!W(4F?p V\"P$V,\V\V.V O5!.V O5!,,pKA KS \'V^4p \ V \&4'gKVP)V 4K V#uupi) 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: %sc38"TFp\V4xK R#5iNr ).0vars& t/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/calculus/euler.py "euler_equations..Vs2TcWS\\Tsc3B"TFp\V\4xK R#5ir) isinstancer)rvs& rrrXs3dz!V$$dsFTz!Variables are not symbols, got %sz Variables z do not match args: )r tupleatomsrr TypeErrorargsall ValueErrormaxrexprlen variablesrrangerr NegativeOnerappend) Lfuncsvarsfdordereqnseqipnew_eqs &&& reuler_equationsr/st%UOOE%L%E aggh'(Aa** ,+>a66U?"Q[[!+>,/0c2 3E D  !Zq%!)$A24;!--*44;A;+C+CCC<%B fb ! ! KK  K,s 3I( I(N)r0)__doc__ itertoolsrsympy.core.functionrrrsympy.core.relationalrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr sympy.utilities.iterablesr r/r0rr:s*4<<$"$&.^r9