Euler equation

A linear ordinary differential equation of order $ n $ of the form

$$ \tag{1 } \sum _ { i= 0} ^ { n } a _ {i} x ^ {i} \frac{d ^ {i} y }{d x ^ {i} } = f ( x ) , $$

where $ a _ {i} $, $ i = 0, \dots, n $, are constants and $ a _ {n} \neq 0 $. This equation was studied in detail by L. Euler, starting from 1740.

The change of the independent variable $ x = e ^ {t} $ transforms (1) for $ x > 0 $ to the linear equation of order $ n $ with constant coefficients

$$ \sum _ { i= 0} ^ { n } a _ {i} D ( D - 1 ) \dots ( D- i+ 1) y = f ( e ^ {t} ) ,\ \ D = \frac{d}{dt} . $$

The characteristic equation of the latter is called the indicial equation of the Euler equation (1). The point $ x = 0 $ is a regular singular point of the homogeneous Euler equation. A fundamental system of (real) solutions of the real homogeneous equation (1) on the semi-axis $ x > 0 $ consists of functions of the form

$$ \tag{2 } x ^ \alpha \cos ( \beta \mathrm{ln} x ) \mathrm{ln} ^ {m} x ,\ \ x ^ \alpha \sin ( \beta \mathrm{ln} x ) \mathrm{ln} ^ {m} x . $$

If $ x < 0 $, then (1) requires the substitution $ x = - e ^ {t} $, and in (2) $ x $ is replaced by $ | x | $.

A more general equation than (1) is the Lagrange equation

$$ \sum _ { j= 0} ^ { n } a _ {j} ( \alpha x + \beta ) ^ {j} y ^ {(j)} = f ( x) , $$

where $ \alpha $, $ \beta $ and $ a _ {j} $ are constants and $ \alpha \neq 0 $, $ a _ {n} \neq 0 $, which can also be reduced to a linear equation with constant coefficients by means of the substitution

$$ \alpha x + \beta = e ^ {t} \ \ \textrm{ or } \ \alpha x + \beta = - e ^ {t} . $$

References

[1]	E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07

The Euler equation is a necessary condition for an extremum in problems of variational calculus; it was obtained by L. Euler (1744). Later J.L. Lagrange (1759) derived it by a different method. For this reason it is sometimes called the Euler–Lagrange equation. The Euler equation is a necessary condition for the vanishing of the first variation of a functional.

One of the problems of variational calculus consists in finding an extremum of the functional

$$ \tag{1 } J ( x) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F ( t , x , \dot{x} ) dt $$

for prescribed conditions at the end points:

$$ \tag{2 } x ( t _ {1} ) = x _ {1} ,\ \ x ( t _ {2} ) = x _ {2} . $$

If a continuously-differentiable function $ x ( t) $, $ t _ {1} \leq t \leq t _ {2} $, is a solution of (1) and (2), then $ x ( t) $ satisfies the Euler equation

$$ \tag{3 } F _ {x} - \frac{d}{dt} F _ {\dot{x} } = 0 , $$

or, in expanded form,

$$ \tag{4 } F _ {x} - F _ {t \dot{x} } - F _ {x \dot{x} } \dot{x} - F _ {\dot{x} \dot{x} } \ddot{x} = 0 . $$

A smooth solution of (3) or (4) is called an extremal. If $ F _ {\dot{x} \dot{x} } \neq 0 $ at a point $ ( t , x ) $ on an extremal, then at this point the extremal has a continuous second derivative $ \ddot{x} $. An extremal such that $ F _ {\dot{x} \dot{x} } \neq 0 $ at all its points is called non-singular. For a non-singular extremal the Euler equation can be written in a form that is solvable with respect to the second derivative $ \ddot{x} $.

The solution of the variational problem (1), (2) need not be continuously differentiable. In general, the optimal solution $ x ( t) $ may be a piecewise-differentiable function. Then at the corner points of $ x ( t) $ the Weierstrass–Erdmann corner conditions must be satisfied, which ensure the continuity of $ F _ {\dot{x} } $ and $ F - \dot{x} F _ {\dot{x} } $ at the passage through a corner point, while on the segments between consecutive corner points the function $ x ( t) $ must satisfy the Euler equation. The piecewise-smooth curves consisting of pieces of extremals and satisfying the Weierstrass–Erdmann corner conditions are called polygonal (broken) extremals. In general, the differential Euler equation is an equation of the second order. Hence, its general solution depends on two arbitrary constants $ c _ {1} $ and $ c _ {2} $:

$$ x = f ( t , c _ {1} , c _ {2} ) . $$

These arbitrary constants can be determined from the boundary conditions (2):

$$ \tag{5 } f ( t _ {1} , c _ {1} , c _ {2} ) = x _ {1} ,\ \ f ( t _ {2} , c _ {1} , c _ {2} ) = x _ {2} . $$

If the functional depends on several functions, that is,

$$ \tag{6 } J ( x ^ {1}, \dots, x ^ {n} ) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } F ( t , x ^ {1}, \dots, x ^ {n} ,\ \dot{x} ^ {1} \dots \dot{x} ^ {n} ) dt , $$

then one obtains a system of $ n $ Euler equations instead of one:

$$ \tag{7 } F _ {x ^ {i} } - \frac{d}{dt} F _ {\dot{x} ^ {i} } = 0 ,\ \ i = 1, \dots, n . $$

The general solution of (7) depends on 2n arbitrary constants, which are determined from given $ 2n $ boundary conditions (in a problem with fixed end points).

In variational problems with variable end points, where the left-hand and right-hand end points of the extremal can move on given hypersurfaces, the missing boundary conditions, which make it possible to obtain a closed system of relations of the type (5), are determined by means of the necessary transversality condition.

For functionals containing higher-order derivatives (not just the first one, as in (1) and (6)), a necessary condition analogous to the Euler equation can be written in the form of the Euler–Poisson differential equation (see [1]).

For variational problems concerning the extremum of functionals that depend on functions of several variables, a necessary condition analogous to the Euler equation is written in the form of the Euler–Ostrogradski equation, which is a partial differential equation (see [2]).

In the case of variational problems for a conditional extremum the system of Euler equations is obtained by means of Lagrange multipliers. For example, for the Bolza problem, which requires one to find the extremum of a functional depending on $ n $ functions $ x = ( x ^ {1}, \dots, x ^ {n} ) $,

$$ \tag{8 } J ( x) = \int\limits _ { t _ {1} } ^ { {t _ 2 } } f ( t , x , \dot{x} ) dt + g ( t _ {1} , x ( t _ {1} ) ,\ t _ {2} , x ( t _ {2} ) ) $$

under differential constraints of the form

$$ \tag{9 } \phi _ {i} ( t , x , \dot{x} ) = 0 ,\ \ i = 1, \dots, m ,\ m < n , $$

and boundary conditions

$$ \tag{10 } \psi _ \mu ( t _ {1} , x ( t _ {1} ) , t _ {2} , x ( t _ {2} ) ) = 0 ,\ \ \mu = 1, \dots p ,\ p \leq 2n + 2 , $$

after using Lagrange multipliers $ \lambda _ {0} $ and $ \lambda _ {i} ( t) $, $ i = 1, \dots, m $, to construct from $ f $ and $ \phi _ {i} $ the function

$$ F ( t , x , \dot{x} , \lambda ) = \lambda _ {0} f ( t , x , \dot{x} ) + \sum _ { i= } 1 ^ { m } \lambda _ {i} \phi _ {i} ( t , x , \dot{x} ) , $$

the Euler equations can be written in the from

$$ \tag{11 } \left . \begin{array}{c} F _ {\lambda _ {i} } - \frac{d}{dt} F _ {\dot \lambda _ {i} } \equiv \phi _ {i} ( t , x , \dot{x} ) = 0 ,\ i = 1, \dots, m , \\ F _ {x ^ {i} } - \frac{d}{dt} F _ {\dot{x} ^ {i} } = 0 ,\ \ i = 1, \dots, n. \end{array} \right \} $$

In this way, an optimal solution of the variational problem (8)–(10) must satisfy the system of $ m + n $ Euler differential equations (11) the first $ m $ of which coincide with the given constraints (9). The additional use of the necessary transversality condition leads to a closed boundary value problem for determining the solution of (8)–(10).

Besides the Euler equation and the transversality condition, the solution of a variational problem must also satisfy the remaining necessary conditions, that is, those of Clebsch (Legendre), Weierstrass and Jacobi.

References

[1]	N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian) MR0142019 Zbl 0718.49001
[2]	M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

I.B. Vapnyarskii

References

[a1]	W.H. Fleming, R.W. Rishel, "Deterministic and stochastic optimal control" , Springer (1975) MR0454768 Zbl 0323.49001
[a2]	I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian) Zbl 0534.58024

The Euler equation is a differential equation of the form

$$ \frac{dx}{\sqrt X } + \frac{dy}{\sqrt Y } = 0 , $$

where

$$ X ( x) = a _ {0} x ^ {4} + a _ {1} x ^ {3} + a _ {2} x ^ {2} + a _ {3} x + a _ {4} , $$

$$ Y ( x) = a _ {0} y ^ {4} + a _ {1} y ^ {3} + a _ {2} y ^ {2} + a _ {3} y + a _ {4} . $$

L. Euler considered this equation in a number of papers, starting from 1753. He showed that its general solution has the form $ F ( x , y ) = 0 $, where $ F ( x , y ) $ is a symmetric polynomial of degree 4 in $ x $ and $ y $.

BSE-3

Comments

In fluid mechanics, the system of equations of motion are also called the Euler equations.

For instance, for an inviscuous fluid the Euler equations of motion are

$$ \rho \left ( \frac{\partial u _ {i} }{\partial t } + u _ \alpha \frac{\partial u _ {i} }{\partial x _ \alpha } \right ) = \ - \frac{\partial p }{\partial x _ {i} } + \rho X _ {i} , $$

where $ t $ is time, $ \rho $ is the density of the fluid, $ p $ is the pressure, $ X _ {i} $ is the $ i $- th component of the body force per unit mass, and $ u _ {i} $ is the velocity component in the direction of $ x _ {i} $, $ i = 1 , 2 , 3 $. Here $ x _ {1} , x _ {2} , x _ {3} $ are Cartesian coordinates, and repeated indices in the equation above indicate summation.

Finally, partial differential equations of the type

$$ \tag{a1 } \frac{\partial ^ {2} F }{\partial t _ {1} \partial t _ {2} } + \frac{1}{t _ {1} - t _ {2} } \left ( a \frac \partial {\partial t _ {1} } - b \frac \partial {\partial t _ {2} } \right ) F = 0 , $$

where $ a $, $ b $ are constants, are called Euler partial differential equations. Certain solutions can be expressed as integrals

$$ \int\limits \frac{dx}{( Q ( x) ) ^ {1/n} } $$

on Riemann surfaces given by $ Y ^ {n} = Q ( X , t _ {1} , t _ {2} ) $ depending on two parameters $ t _ {1} $, $ t _ {2} $. (Here $ Q ( X , t _ {1} , t _ {2} ) $ is a polynomial in $ X $.)

There are connections between the monodromy of these solutions and automorphic functions on the $ 2 $- dimensional unit ball. Cf. [a3] for a great deal of material on this topic. The Euler partial differential equation

is also called Euler–Darboux–Poisson equation (cf. Mixed and boundary value problems for hyperbolic equations and systems) and Euler–Poisson–Darboux equation (cf. Differential equation, partial, with singular coefficients).

References

[a1]	A.J. Chorin, J.E. Marsden, "A mathematical introduction to fluid dynamics" , Springer (1979)
[a2]	C.-S. Yih, "Stratified flows" , Acad. Press (1980) MR0569474 Zbl 0458.76095
[a3]	R.-P. Holzapfel, "Geometry and arithmetic around Euler partial differential equations" , Reidel (1986) MR0867406 MR0849778 Zbl 0595.14017 Zbl 0595.14016