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detools::euler -- Euler operator of variational calculus

Introduction

detools::euler(L,t,z) applies the Euler operator to the Lagrangian L and returns the left hand side of the corresponding Euler-Lagrange equations. The Lagrangian can be of any order and there can be any number of independent variables t and of dependent variables z.

Call(s)

detools::euler(L, t, z)
detools::euler(L, DV)

Parameters

L - the Lagrangian: an expression or an element of a domain of Cat::DifferentialFunction(DV).
t - the independent variable(s): either a single (indexed) identifier or a list of (indexed) identifiers.
z - the dependent variable(s): either a single identifier or a list of identifiers.
DV - the domain of the differential variables: DV must belong to Cat::DifferentialVariable.

Returns

either a single expression or a list ofexpressions; if for L an element of a domain of Cat::DifferentialFunction was given, the output will also consist of elements of this domain.

Details

Example 1

This is a finite dimensional example computing the equations of motion of a particle moving in the plane under the influence of a potential V.

>> L := 1/2*(diff(x(t), t)^2 + diff(y(t), t)^2) - V(x(t), y(t)):
   detools::euler(L, t, [x, y])
        [x([t, t]) + D([1], V)(x, y), y([t, t]) + D([2], V)(x, y)]

Example 2

This is a simple example for the generation of field equations. The field u depends here on two variables t,x.

>> L := 1/2*(diff(u(t, x), t)^2 + diff(u(t, x), x)^2) - u(t, x)^2:
   detools::euler(L, [x, t], u)
                        u([t, t]) + u([x, x]) + 2 u




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