o o hl @sndZddlmZddlmZmZmZddlmZddl m Z ddl m Z ddl mZddlmZd d 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 sptrtnfst|tnD] }t|ts$td|qt|r-t|n|f}|s8dj}n tdd|D}tdd|DsPtd|D]}||jksatd||fqRt fdd |t Ddg}g}D]>}t ||}t d |d D] }t ||D]}|tj|t |t |g|Rg|R}qqt|d} t| tr|| qw|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: %srcss|]}t|VqdSNr ).0varr r h/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/calculus/euler.py Vsz"euler_equations..css|]}t|tVqdSr ) isinstancer)rvr r rrXsz!Variables are not symbols, got %sz"Variables %s do not match args: %scs g|] }|jvrt|jqSr )exprlen variables)rdfuncsr r _s z#euler_equations..)r tupleatomsrr TypeErrorargsall ValueErrormaxrrrangerr NegativeOnerappend) Lrvarsfordereqnseqipnew_eqr rreuler_equationss@:      .   r/N)r r )__doc__ itertoolsrsympy.core.functionrrrsympy.core.relationalrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr sympy.utilities.iterablesr r/r r r rs