ó ~9­\c@ssdZddlmZddlmZmZmZmZmZm Z m Z ddl m Z m Z ddd„ZdS( sX This module implements a method to find Euler-Lagrange Equations for given Lagrangian. iÿÿÿÿ(tcombinations_with_replacement(tFunctiontsympifytdifftEqtStSymbolt Derivative(titerabletrangec såtˆƒrtˆƒnˆf‰ˆs?t|jtƒƒ‰n3x0ˆD](}t|tƒsFtd|ƒ‚qFqFWt|ƒrŠt|ƒn|f}|s©ˆdj}ntd„|Dƒƒ}td„|Dƒƒsètd|ƒ‚nx6ˆD].}||jksïtd||fƒ‚qïqïWt ‡fd†|jt ƒDƒƒ}g}x’ˆD]Š}t ||ƒ}x_t d|dƒD]J}xAt ||ƒD]0}|tj|t |t ||Œ|Œ}q’Wq|W|jt|ƒƒqSW|S( sf 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 sFunction expected, got: %sicss|]}t|ƒVqdS(N(R(t.0tvar((s3lib/python2.7/site-packages/sympy/calculus/euler.pys Sscss|]}t|tƒVqdS(N(t isinstanceR(R tv((s3lib/python2.7/site-packages/sympy/calculus/euler.pys Uss!Variables are not symbols, got %ss!Variables %s don't match args: %sc3s-|]#}|jˆkrt|jƒVqdS(N(texprtlent variables(R td(tfuncs(s3lib/python2.7/site-packages/sympy/calculus/euler.pys \si(RttupletatomsRR t TypeErrortargstallt ValueErrortmaxRRR RRt NegativeOnetappendR( tLRtvarstftorderteqnsteqtitp((Rs3lib/python2.7/site-packages/sympy/calculus/euler.pyteuler_equations s0;! ! % 2N(((t__doc__t itertoolsRtsympyRRRRRRRtsympy.core.compatibilityRR R$(((s3lib/python2.7/site-packages/sympy/calculus/euler.pyts4