Otjl ndZddlmZddlmZmZmZddlmZddl m Z ddl m Z ddl mZddlmZd d Zd S) zX This module implements a method to find Euler-Lagrange Equations for given Lagrangian. )combinations_with_replacement) DerivativeFunctiondiff)Eq)S)Symbolsympify)iterablec trtnfs(t|tn,D])}t |tst d|z*t|rt|n|f}|sdj}ntd|D}td|Dst d|zD]"}||jkstd|d|#tfd|tDdgz}g}D]}t||}td |d zD]A}t||D].}|tj|zt|t|g|Rg|Rzz}/Bt!|d} t | t r|| |S) a5 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: %src34K|]}t|VdSNr ).0vars R/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/calculus/euler.py z"euler_equations..Vs(22cWS\\222222c3@K|]}t|tVdSr) isinstancer )rvs rrz"euler_equations..Xs,33z!V$$333333rz!Variables are not symbols, got %sz Variables z do not match args: cJg|]}|jv t|j Sr )exprlen variables)rdfuncss r z#euler_equations.._s5,,,a6U??Q[!!*??r)r tupleatomsrr TypeErrorargsall ValueErrormaxrrrangerr NegativeOnerappend) Lrvarsfordereqnseqipnew_eqs ` reuler_equationsr4s_t%UOO 9E%LLL%E Baggh''(( B BAa** B +>,,,/0c2 3 3E D    !QZZq%!)$$ D DA24;; D D!-*44;A;;;+C+C+C+CCC DB fb ! ! KK    KrN)r r )__doc__ itertoolsrsympy.core.functionrrrsympy.core.relationalrsympy.core.singletonrsympy.core.symbolr sympy.core.sympifyr sympy.utilities.iterablesr r4r rrr=s433333<<<<<<<<<<$$$$$$""""""$$$$$$&&&&&&......^^^^^^r