o 8Va @sNdZddlmZddlmZmZmZmZmZm Z m Z ddl m Z d ddZ dS) zX This module implements a method to find Euler-Lagrange Equations for given Lagrangian. )combinations_with_replacement)FunctionsympifydiffEqSSymbol Derivative)iterablec s\trtnfst|tnD] }t|ts$td|qt|r-t|n|f}|s8dj}n tdd|D}tdd|DsPtd|D]}||jksatd||fqRt fdd|t D}g}D]7}t ||}t d |d D] }t ||D]}|tj|t |t |g|Rg|R}qq|t|dqt|S) af Find the Euler-Lagrange equations [1]_ for a given Lagrangian. Parameters ========== L : Expr The Lagrangian that should be a function of the functions listed in the second argument and their derivatives. For example, in the case of two functions `f(x,y)`, `g(x,y)` and two independent variables `x`, `y` the Lagrangian would have the form: .. math:: L\left(f(x,y),g(x,y),\frac{\partial f(x,y)}{\partial x}, \frac{\partial f(x,y)}{\partial y}, \frac{\partial g(x,y)}{\partial x}, \frac{\partial g(x,y)}{\partial y},x,y\right) In many cases it is not necessary to provide anything, except the Lagrangian, it will be auto-detected (and an error raised if this couldn't be done). funcs : Function or an iterable of Functions The functions that the Lagrangian depends on. The Euler equations are differential equations for each of these functions. vars : Symbol or an iterable of Symbols The Symbols that are the independent variables of the functions. Returns ======= eqns : list of Eq The list of differential equations, one for each function. Examples ======== >>> from sympy import Symbol, Function >>> from sympy.calculus.euler import euler_equations >>> x = Function('x') >>> t = Symbol('t') >>> L = (x(t).diff(t))**2/2 - x(t)**2/2 >>> euler_equations(L, x(t), t) [Eq(-x(t) - Derivative(x(t), (t, 2)), 0)] >>> u = Function('u') >>> x = Symbol('x') >>> L = (u(t, x).diff(t))**2/2 - (u(t, x).diff(x))**2/2 >>> euler_equations(L, u(t, x), [t, x]) [Eq(-Derivative(u(t, x), (t, 2)) + Derivative(u(t, x), (x, 2)), 0)] References ========== .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation zFunction expected, got: %srcss|]}t|VqdSN)r).0varr r 6/usr/lib/python3/dist-packages/sympy/calculus/euler.py Ssz"euler_equations..css|]}t|tVqdSr ) isinstancer)r vr r rrUsz!Variables are not symbols, got %sz!Variables %s don't match args: %sc3s$|] }|jvrt|jVqdSr )exprlenZ variables)r dfuncsr rr\s)r tupleZatomsrr TypeErrorargsall ValueErrormaxr rrangerrZ NegativeOneappendr) LrvarsfZorderZeqnseqipr rreuler_equations s6;      .r'N)r r )__doc__ itertoolsrZsympyrrrrrrr Zsympy.core.compatibilityr r'r r r rs  $