Euler–Lagrange equation

History

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.

Statement

The Euler–Lagrange equation is an equation satisfied by a function q of a real argument t, which is a stationary point of the functional

S ( q ) = ∫ a b L ( t , q ( t ) , q ˙ ( t ) ) d t

where:

q is the function to be found:

q : [ a , b ] ⊂ R → X t ↦ x = q ( t ) such that q is differentiable, q ( a ) = x a , and q ( b ) = x b ;

q ˙ ; is the derivative of q : q ˙ : [ a , b ] → T q ( t ) X t ↦ v = q ˙ ( t )

T q ( t ) X denotes the tangent space to X at the point q ( t ) .

L is a real-valued function with continuous first partial derivatives: L : [ a , b ] × T X → R ( t , x , v ) ↦ L ( t , x , v ) .

TX being the tangent bundle of X defined by T X = ⋃ x ∈ X { x } × T x X  ;

The Euler–Lagrange equation, then, is given by

where L_x and L_v denote the partial derivatives of L with respect to the second and third arguments, respectively.

If the dimension of the space X is greater than 1, this is a system of differential equations, one for each component:

∂ L ∂ q i ( t , q ( t ) , q ˙ ( t ) ) − d d t ∂ L ∂ q ˙ i ( t , q ( t ) , q ˙ ( t ) ) = 0 for  i = 1 , … , n .

Examples

A standard example is finding the real-valued function f on the interval [a, b], such that f(a) = c and f(b) = d, for which the path length along the curve traced by f is as short as possible.

ℓ ( f ) = ∫ a b 1 + ( f ′ ( x ) ) 2 d x ,

the integrand function being L(x, y, y′) = √1 + y′ ² evaluated at (x, y, y′) = (x, f(x), f′(x)).

The partial derivatives of L are:

∂ L ( x , y , y ′ ) ∂ y ′ = y ′ 1 + y ′ 2 and ∂ L ( x , y , y ′ ) ∂ y = 0.

By substituting these into the Euler–Lagrange equation, we obtain

d d x f ′ ( x ) 1 + ( f ′ ( x ) ) 2 = 0 f ′ ( x ) 1 + ( f ′ ( x ) ) 2 = C = constant ⇒ f ′ ( x ) = C 1 − C 2 := A ⇒ f ( x ) = A x + B

that is, the function must have constant first derivative, and thus its graph is a straight line.

Single function of single variable with higher derivatives

The stationary values of the functional

I [ f ] = ∫ x 0 x 1 L ( x , f , f ′ , f ″ , … , f ( k ) )   d x   ;     f ′ := d f d x ,   f ″ := d 2 f d x 2 ,   f ( k ) := d k f d x k

can be obtained from the Euler–Lagrange equation

∂ L ∂ f − d d x ( ∂ L ∂ f ′ ) + d 2 d x 2 ( ∂ L ∂ f ″ ) − ⋯ + ( − 1 ) k d k d x k ( ∂ L ∂ f ( k ) ) = 0

under fixed boundary conditions for the function itself as well as for the first k − 1 derivatives (i.e. for all f ( i ) , i ∈ { 0 , . . . , k − 1 } ). The endpoint values of the highest derivative f ( k ) remain flexible.

Several functions of single variable with single derivative

If the problem involves finding several functions ( f 1 , f 2 , … , f m ) of a single independent variable ( x ) that define an extremum of the functional

I [ f 1 , f 2 , … , f m ] = ∫ x 0 x 1 L ( x , f 1 , f 2 , … , f m , f 1 ′ , f 2 ′ , … , f m ′ )   d x   ;     f i ′ := d f i d x

then the corresponding Euler–Lagrange equations are

∂ L ∂ f i − d d x ( ∂ L ∂ f i ′ ) = 0 i

Single function of several variables with single derivative

A multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, then

I [ f ] = ∫ Ω L ( x 1 , … , x n , f , f , 1 , … , f , n ) d x   ;     f , j := ∂ f ∂ x j

is extremized only if f satisfies the partial differential equation

∂ L ∂ f − ∑ j = 1 n ∂ ∂ x j ( ∂ L ∂ f , j ) = 0.

When n = 2 and L is the energy functional, this leads to the soap-film minimal surface problem.

Several functions of several variables with single derivative

If there are several unknown functions to be determined and several variables such that

I [ f 1 , f 2 , … , f m ] = ∫ Ω L ( x 1 , … , x n , f 1 , … , f m , f 1 , 1 , … , f 1 , n , … , f m , 1 , … , f m , n ) d x   ;     f i , j := ∂ f i ∂ x j

the system of Euler–Lagrange equations is

∂ L ∂ f 1 − ∑ j = 1 n ∂ ∂ x j ( ∂ L ∂ f 1 , j ) = 0 1 ∂ L ∂ f 2 − ∑ j = 1 n ∂ ∂ x j ( ∂ L ∂ f 2 , j ) = 0 2 ⋮ ⋮ ⋮ ∂ L ∂ f m − ∑ j = 1 n ∂ ∂ x j ( ∂ L ∂ f m , j ) = 0 m .

Single function of two variables with higher derivatives

If there is a single unknown function f to be determined that is dependent on two variables x₁ and x₂ and if the functional depends on higher derivatives of f up to n-th order such that

I [ f ] = ∫ Ω L ( x 1 , x 2 , f , f , 1 , f , 2 , f , 11 , f , 12 , f , 22 , … , f , 22 … 2 ) d x f , i := ∂ f ∂ x i , f , i j := ∂ 2 f ∂ x i ∂ x j , …

then the Euler–Lagrange equation is

∂ L ∂ f − ∂ ∂ x 1 ( ∂ L ∂ f , 1 ) − ∂ ∂ x 2 ( ∂ L ∂ f , 2 ) + ∂ 2 ∂ x 1 2 ( ∂ L ∂ f , 11 ) + ∂ 2 ∂ x 1 ∂ x 2 ( ∂ L ∂ f , 12 ) + ∂ 2 ∂ x 2 2 ( ∂ L ∂ f , 22 ) − ⋯ + ( − 1 ) k ∂ k ∂ x 2 k ( ∂ L ∂ f , 22 … 2 ) = 0

which can be represented shortly as:

∂ L ∂ f + ∑ j = 1 n ∑ μ 1 ≤ … ≤ μ j ( − 1 ) j ∂ j ∂ x μ 1 … ∂ x μ j ( ∂ L ∂ f , μ 1 … μ j ) = 0

wherein μ 1 … μ j are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the μ 1 … μ j indices is only over μ 1 ≤ μ 2 ≤ … ≤ μ j in order to avoid counting the same partial derivative multiple times, for example f , 12 = f , 21 appears only once in the previous equation.

Several functions of several variables with higher derivatives

If there are p unknown functions f_i to be determined that are dependent on m variables x₁ ... x_m and if the functional depends on higher derivatives of the f_i up to n-th order such that

I [ f 1 , … , f p ] = ∫ Ω L ( x 1 , … , x m ; f 1 , … , f p ; f 1 , 1 , … , f p , m ; f 1 , 11 , … , f p , m m ; … ; f p , m … m ) d x f i , μ := ∂ f i ∂ x μ , f i , μ 1 μ 2 := ∂ 2 f i ∂ x μ 1 ∂ x μ 2 , …

where μ 1 … μ j are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is

∂ L ∂ f i + ∑ j = 1 n ∑ μ 1 ≤ … ≤ μ j ( − 1 ) j ∂ j ∂ x μ 1 … ∂ x μ j ( ∂ L ∂ f i , μ 1 … μ j ) = 0

where the summation over the μ 1 … μ j is avoiding counting the same derivative f i , μ 1 μ 2 = f i , μ 2 μ 1 several times, just as in the previous subsection. This can be expressed more compactly as

∑ j = 0 n ∑ μ 1 ≤ … ≤ μ j ( − 1 ) j ∂ μ 1 … μ j j ( ∂ L ∂ f i , μ 1 … μ j ) = 0

Generalization to Manifolds

Let M be a smooth manifold, and let C ∞ ( [ a , b ] ) denote the space of smooth functions f : [ a , b ] → M . Then, for functionals S : C ∞ ( [ a , b ] ) → R of the form

S [ f ] = ∫ a b ( L ∘ f ˙ ) ( t ) d t

where L : T M → R is the Lagrangian, the statement d S f = 0 is equivalent to the statement that, for all t ∈ [ a , b ] , each coordinate frame trivialization ( x i , X i ) of a neighborhood of f ˙ ( t ) yields the following dim ⁡ M equations:

∀ i : d d t ∂ F ∂ X i | f ˙ ( t ) = ∂ F ∂ x i | f ˙ ( t )