from itertools import combinations_with_replacement
from sympy import Function, sympify, diff, Eq, S, Symbol, Derivative
from sympy.core.compatibility import (iterable, range)
[docs]def euler_equations(L, funcs=(), vars=()):
r"""
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, t), 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, t) + Derivative(u(t, x), x, x), 0)]
References
==========
.. [1] http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation
"""
funcs = tuple(funcs) if iterable(funcs) else (funcs,)
if not funcs:
funcs = tuple(L.atoms(Function))
else:
for f in funcs:
if not isinstance(f, Function):
raise TypeError('Function expected, got: %s' % f)
vars = tuple(vars) if iterable(vars) else (vars,)
if not vars:
vars = funcs[0].args
else:
vars = tuple(sympify(var) for var in vars)
if not all(isinstance(v, Symbol) for v in vars):
raise TypeError('Variables are not symbols, got %s' % vars)
for f in funcs:
if not vars == f.args:
raise ValueError("Variables %s don't match args: %s" % (vars, f))
order = max(len(d.variables) for d in L.atoms(Derivative)
if d.expr in funcs)
eqns = []
for f in funcs:
eq = diff(L, f)
for i in range(1, order + 1):
for p in combinations_with_replacement(vars, i):
eq = eq + S.NegativeOne**i*diff(L, diff(f, *p), *p)
eqns.append(Eq(eq))
return eqns
