Euler – Lagrange equation

The Euler – Lagrange equations (in physics also the Lagrange – Euler equations or the Lagrange equations ) are the basic formulas of the calculus of variations , by which stationary points and extremes of functionals are sought. In particular, these equations are widely used in optimization problems, and, together with the principle of stationarity of action , they are used to calculate trajectories in mechanics. In theoretical physics in general, these are (classical) equations of motion in the context of obtaining them from an explicitly written expression for an action ( Lagrangian ).

Using the Euler – Lagrange equations to find the extremum of a functional in some sense is similar to using the differential calculus theorem, which states that only at the point where the first derivative of the function vanishes does a smooth function have an extremum (in the case of a vector argument, the gradient of the function is equal to zero, i.e. derivative with vector argument). More precisely, this is a direct generalization of the corresponding formula to the case of functionals — functions of an infinite-dimensional argument.

The equations were obtained by Leonard Euler and Joseph-Louis Lagrange in the 1750s .

Content

Statement

Let the functional be given

J=\int \limits _{a}^{b}F(x,f(x),f'(x))\,dx.

{\ displaystyle J = \ int \ limits _ {a} ^ {b} F (x, f (x), f '(x)) \, dx.}

with integrand function $F(x,f(x),f'(x))$ ${\ displaystyle F (x, f (x), f '(x))}$ possessing continuous first partial derivatives and called the Lagrange function or Lagrangian , where through $f^{'}$ ${\ displaystyle f '}$ marked first derivative $f$ ${\ displaystyle f}$ by $x$ ${\ displaystyle x}$ . If this functional reaches an extremum on some function $f$ ${\ displaystyle f}$ then an ordinary differential equation must be satisfied for it

{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}=0,

{\ displaystyle {\ frac {\ partial F} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial F} {\ partial f '}} = 0,}

which is called the Euler – Lagrange equation .

Examples

Consider the standard example: find the shortest path between two points of a plane. The answer, obviously, is the segment connecting these points. Let's try to get it using the Euler-Lagrange equation. Let the points to be connected have coordinates $(a,c)$ ${\ displaystyle (a, c)}$ and $(b,d)$ ${\ displaystyle (b, d)}$ . Then the length of the path $y(x)$ ${\ displaystyle y (x)}$ connecting these points can be written as follows:

L=\int \limits _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx.

{\ displaystyle L = \ int \ limits _ {a} ^ {b} {\ sqrt {1+ \ left ({\ frac {dy} {dx}} \ right) ^ {2}} dx.}

The Euler – Lagrange equation for this functional takes the form:

{\frac {d}{dx}}{\frac {\partial }{\partial y'}}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}=0,

{\ displaystyle {\ frac {d} {dx}} {\ frac {\ partial} {\ partial y '}} {\ sqrt {1+ \ left ({\ frac {dy} {dx}} \ right) ^ {2}}} = 0,}

where do we get that

{\frac {dy}{dx}}=C\Rightarrow y=Cx+D.

{\ displaystyle {\ frac {dy} {dx}} = C \ Rightarrow y = Cx + D.}

Thus, we get a straight line. Considering that $y(a)=c$ ${\ displaystyle y (a) = c}$ , $y(b)=d$ ${\ displaystyle y (b) = d}$ , that is, that it passes through the original points, we get the correct answer: the segment connecting the points.

Multidimensional variations

There are also many multidimensional versions of the Euler – Lagrange equations.

If a $q(t)$ ${\ displaystyle q (t)}$ - the path to $n$ ${\ displaystyle n}$ dimensional space, it delivers an extremum to the functional

J=\int \limits _{t1}^{t2}L(t,q(t),q'(t))\,dt

{\ displaystyle J = \ int \ limits _ {t1} ^ {t2} L (t, q (t), q '(t)) \, dt}

only if it satisfies the condition

{\frac {d}{dt}}{\frac {\partial L}{\partial q'_{k}}}-{\frac {\partial L}{\partial q_{k}}}=0

{\ displaystyle {\ frac {d} {dt}} {\ frac {\ partial L} {\ partial q '_ {k}}} - {\ frac {\ partial L} {\ partial q_ {k}}} = 0}

\forall k=1,2,\dots n

{\ displaystyle \ forall k = 1,2, \ dots n}

In physical applications when $L$ ${\ displaystyle L}$ is a Lagrangian (meaning the Lagrangian of some physical system; that is, if J is an action for this system), these equations are the (classical) equations of motion of such a system. This statement can be directly generalized to the case of infinite-dimensional q .

Another multidimensional generalization is obtained by looking at the function. $n$ ${\ displaystyle n}$ variables. If a $\Omega$ ${\ displaystyle \ Omega}$ - any, in this case, n- dimensional, surface,

J=\int \limits _{\Omega }L(f,x_{1},\dots ,x_{n},f_{x_{1}},\dots ,f_{x_{n}})\,d\Omega ,

{\ displaystyle J = \ int \ limits _ {\ Omega} L (f, x_ {1}, \ dots, x_ {n}, f_ {x_ {1}}, \ dots, f_ {x_ {n}}) \, d \ Omega,}

Where $x_{i}=x_{1},x_{2},x_{3},\dots ,x_{n}$ ${\ displaystyle x_ {i} = x_ {1}, x_ {2}, x_ {3}, \ dots, x_ {n}}$ - independent coordinates, $f=f(x_{1},x_{2},x_{3},\dots ,x_{n})$ ${\ displaystyle f = f (x_ {1}, x_ {2}, x_ {3}, \ dots, x_ {n})}$ , $f_{x_{i}}\equiv {\frac {\partial f}{\partial x_{i}}}$ ${\ displaystyle f_ {x_ {i}} \ equiv {\ frac {\ partial f} {\ partial x_ {i}}}}$ ,

delivers an extremum if only $f$ ${\ displaystyle f}$ satisfies the partial differential equation

{\frac {\partial L}{\partial f}}-\sum _{i=1}^{n}{\frac {\partial }{\partial x_{i}}}{\frac {\partial L}{\partial f_{x_{i}}}}=0.

{\ displaystyle {\ frac {\ partial L} {\ partial f}} - \ sum _ {i = 1} ^ {n} {\ frac {\ partial} {\ partial x_ {i}}} {\ frac { \ partial L} {\ partial f_ {x_ {i}}}} = 0.}

If a $n=2$ ${\ displaystyle n = 2}$ and $L$ ${\ displaystyle L}$ - energy functional, this task is called “minimization of the surface of a soap film”.

The obvious combination of the two cases described above is used to derive the equations of motion for distributed systems, such as physical fields, oscillating strings or membranes, etc.

In particular, instead of the static equilibrium equation of the soap film, given as an example in the previous paragraph, we have in this case the dynamic equation of motion of such a film (if, of course, we were able to initially write down the action for it, that is, kinetic and potential energy).

History

The Euler – Lagrange equation was obtained in the 1750s by Euler and Lagrange when solving the isochron problem. This is the problem of determining the curve by which a heavy particle hits a fixed point in a fixed time, regardless of the starting point.

Lagrange solved this problem in 1755 and sent the solution to Euler. The subsequently developed Lagrange method and its application in mechanics led to the formulation of Lagrangian mechanics . The correspondence of scientists led to the creation of the calculus of variations (the term suggested by Euler in 1766 ).

Proof

The derivation of the one-dimensional Euler – Lagrange equation is one of the classical proofs in mathematics. It is based on the main lemma of the calculus of variations .

We want to find such a function. $f$ ${\ displaystyle f}$ that satisfies the boundary conditions $f(a)=c$ ${\ displaystyle f (a) = c}$ , $f(b)=d$ ${\ displaystyle f (b) = d}$ and delivers an extremum to the functional

J=\int \limits _{a}^{b}F(x,f(x),f'(x))\,dx.

{\ displaystyle J = \ int \ limits _ {a} ^ {b} F (x, f (x), f '(x)) \, dx.}

Let's pretend that $F$ ${\ displaystyle F}$ has continuous first derivatives. Weaker conditions are also sufficient, but the proof for the general case is more complicated.

If a $f$ ${\ displaystyle f}$ gives an extremum to the functional and satisfies the boundary conditions, then any weak perturbation $f$ ${\ displaystyle f}$ which preserves boundary conditions should increase the value $J$ ${\ displaystyle J}$ (if a $f$ ${\ displaystyle f}$ minimizes it) or reduce $J$ ${\ displaystyle J}$ (if a $f$ ${\ displaystyle f}$ maximizes).

Let be $\eta (x)$ ${\ displaystyle \ eta (x)}$ - any differentiable function satisfying the condition $\eta (a)=\eta (b)=0$ ${\ displaystyle \ eta (a) = \ eta (b) = 0}$ . Define

J(\varepsilon )=\int \limits _{a}^{b}F(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,dx.

{\ displaystyle J (\ varepsilon) = \ int \ limits _ {a} ^ {b} F (x, f (x) + \ varepsilon \ eta (x), f '(x) + \ varepsilon \ eta' ( x)) \, dx.}

Where $\varepsilon$ ${\ displaystyle \ varepsilon}$ - arbitrary parameter.

Insofar as $f$ ${\ displaystyle f}$ gives extremum for $J(0)$ ${\ displaystyle J (0)}$ then $J'(0)=0$ ${\ displaystyle J '(0) = 0}$ , i.e

J'(0)=\int \limits _{a}^{b}\left[\eta (x){\frac {\partial F}{\partial f}}+\eta '(x){\frac {\partial F}{\partial f'}}\right]\,dx=0.

{\ displaystyle J '(0) = \ int \ limits _ {a} ^ {b} \ left [\ eta (x) {\ frac {\ partial F} {\ partial f}} + \ eta' (x) {\ frac {\ partial F} {\ partial f '}} \ right] \, dx = 0.}

Integrating in parts the second term, we find that

0=\int \limits _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,dx+\left[\eta (x){\frac {\partial F}{\partial f'}}\right]_{a}^{b}.

{\ displaystyle 0 = \ int \ limits _ {a} ^ {b} \ left [{\ frac {\ partial F} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial F} {\ partial f '}} \ right] \ eta (x) \, dx + \ left [\ eta (x) {\ frac {\ partial F} {\ partial f'}} \ right] _ {a } ^ {b}.}

Using boundary conditions on $\eta$ ${\ displaystyle \ eta}$ get

0=\int \limits _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,dx.

{\ displaystyle 0 = \ int \ limits _ {a} ^ {b} \ left [{\ frac {\ partial F} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial F} {\ partial f '}} \ right] \ eta (x) \, dx.}

From here, since $\eta (x)$ ${\ displaystyle \ eta (x)}$ - any, the Euler – Lagrange equation follows:

{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}=0.

{\ displaystyle {\ frac {\ partial F} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial F} {\ partial f '}} = 0.}

If you do not enter the boundary conditions on $f(x)$ ${\ displaystyle f (x)}$ , then transversality conditions are also required:

{\frac {\partial F}{\partial f'}}(a,f(a),f'(a))=0

{\ displaystyle {\ frac {\ partial F} {\ partial f '}} (a, f (a), f' (a)) = 0}

{\frac {\partial F}{\partial f'}}(b,f(b),f'(b))=0

{\ displaystyle {\ frac {\ partial F} {\ partial f '}} (b, f (b), f' (b)) = 0}

Generalization to the case of higher derivatives

Lagrangian can also depend on derivatives $f$ ${\ displaystyle f}$ order higher than the first.

Let the functional whose extremum needs to be found is given in the form:

J=\int \limits _{a}^{b}F(x,f(x),f'(x),f''(x),...,f^{(n)}(x))\,dx.

{\ displaystyle J = \ int \ limits _ {a} ^ {b} F (x, f (x), f '(x), f' '(x), ..., f ^ {(n)} (x)) \, dx.}

If we impose boundary conditions on $f$ ${\ displaystyle f}$ and its derivatives to order $n-1$ ${\ displaystyle n-1}$ inclusive and also assume that $F$ ${\ displaystyle F}$ has continuous first derivatives, it is possible, applying integration by parts several times, to derive an analogue of the Euler-Lagrange equation for this case:

{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}+{\frac {d^{2}}{dx^{2}}}{\frac {\partial F}{\partial f''}}-\cdots +(-1)^{n}{\frac {d^{n}}{dx^{n}}}{\frac {\partial F}{\partial f^{(n)}}}=0.

{\ displaystyle {\ frac {\ partial F} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial F} {\ partial f '}} + {\ frac {d ^ {2}} {dx ^ {2}}} {\ frac {\ partial F} {\ partial f ''}} - \ cdots + (- 1) ^ {n} {\ frac {d ^ {n}} {dx ^ {n}}} {\ frac {\ partial F} {\ partial f ^ {(n)}}} = 0.}

This equation is often called the Euler-Poisson equation .

Two Lagrangians differing by the full derivative will give the same differential equations, however the maximum order of the derivatives in these Lagrangians may be different. For example, $L_{1}=(f^{\prime }(x))^{2}~,~L_{2}=-f(x)f^{\prime \prime }(x)~,~L_{1}-L_{2}={\frac {d}{dx}}(f(x)f^{\prime }(x))$ ${\ displaystyle L_ {1} = (f ^ {\ prime} (x)) ^ {2} ~, ~ L_ {2} = - f (x) f ^ {\ prime \ prime} (x) ~, ~ L_ {1} -L_ {2} = {\ frac {d} {dx}} (f (x) f ^ {\ prime} (x))}$ . To get the differential equation on the extremum, to $L_{1}$ ${\ displaystyle L_ {1}}$ it is enough to apply the “ordinary” Euler – Lagrange equation, and for $L_{2}$ ${\ displaystyle L_ {2}}$ since it depends on the second derivative, it is necessary to use the Euler – Poisson equation with the corresponding term:

{\frac {\partial L_{1}}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L_{1}}{\partial f'}}=-2f^{\prime \prime }(x),

{\ displaystyle {\ frac {\ partial L_ {1}} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial L_ {1}} {\ partial f '}} = -2f ^ {\ prime \ prime} (x),}

{\frac {\partial L_{2}}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L_{2}}{\partial f'}}+{\frac {d^{2}}{dx^{2}}}{\frac {\partial L_{2}}{\partial f^{\prime \prime }}}=-2f^{\prime \prime }(x),

{\ displaystyle {\ frac {\ partial L_ {2}} {\ partial f}} - {\ frac {d} {dx}} {\ frac {\ partial L_ {2}} {\ partial f '}} + {\ frac {d ^ {2}} {dx ^ {2}}} {\ frac {\ partial L_ {2}} {\ partial f ^ {\ prime \ prime}}} = - 2f ^ {\ prime \ prime} (x),}

and in both cases you get the same differential equation $-2f^{\prime \prime }(x)=0$ ${\ displaystyle -2f ^ {\ prime \ prime} (x) = 0}$ .

Necessary and sufficient condition for the existence and uniqueness of the Euler-Lagrange equation

A necessary and sufficient condition for the existence and uniqueness of the Euler-Lagrange equation is $det\left\|{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{k}}}\right\|_{j,k=1}^{n}\neq 0$ ${\ displaystyle det \ left \ | {\ frac {\ partial ^ {2} L} {\ partial {\ dot {q}} _ {j} \ partial {\ dot {q}} _ {k}}} \ right \ | _ {j, k = 1} ^ {n} \ neq 0}$ . Here $L$ ${\ displaystyle L}$ - Lagrangian , ${\dot {q}}_{j}$ ${\ displaystyle {\ dot {q}} _ {j}}$ - derivative $j$ ${\ displaystyle j}$ - from the generalized time coordinate ^[1] .

Notes

↑ Aizerman M. A. Classical mechanics. - M., Science, 1980. - p. 165

Literature

Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal Control. - M .: Science, 1979
Dubrovin B. A., Novikov S. P., Fomenko A. T. Modern geometry: Methods and applications. - M .: Science, 1979
Elsgolts L.. Differential equations and calculus of variations. - M .: Science, 1969.
Zelikin M.I. Homogeneous spaces and the Riccati equation in calculus of variations, - Factorial, Moscow, 1998.
Zelikin M.I. Optimal control and calculus of variations, - URSS, Moscow, 2004.

Links

Weisstein, Eric W. Euler-Lagrange (Eng.) On the Wolfram MathWorld website.
Calculus of Variations (eng.) On PlanetMath .
Summary with some historical information
Examples are tasks from variational calculus.

[1] Aizerman M. A. Classical mechanics. - M., Science, 1980. - p. 165