Calculus of variations and optimal control; optimization Definition

Euler–Poisson equation for higher derivatives

Builds on Integral functional with a Lagrangian density Euler–Lagrange equation Partial derivative

Definition

Let $k \ge 1$ and $a < b$ . Let $L \colon [a, b] \times \R^{k+1} \to \R$ , written $L(x, z_0, z_1, \dots, z_k)$ , be a function of class $C^{k+1}$ , and write $L_{z_j}$ for its partial derivative in $z_j$ . Consider the functional $J(y) = \int_a^b L\big(x, y(x), y'(x), \dots, y^{(k)}(x)\big) \, dx.$ The Euler–Poisson equation of $J$ is the equation $\sum_{j=0}^{k} (-1)^j \frac{d^j}{dx^j} \Big( L_{z_j}\big(x, y(x), y'(x), \dots, y^{(k)}(x)\big) \Big) = 0, \qquad a \le x \le b,$ for an unknown $y \in C^{2k}([a, b])$ .

Remarks

For $k = 1$ it is the Euler–Lagrange equation $L_{z_0} - \frac{d}{dx} L_{z_1} = 0$ of an integral functional. A minimiser $y \in C^{2k}([a, b])$ of $J$ among functions with prescribed $y^{(j)}(a)$ and $y^{(j)}(b)$ for $0 \le j \le k - 1$ solves it. To see this, take variations $\eta$ with $\eta^{(j)}(a) = \eta^{(j)}(b) = 0$ for $j < k$ , integrate the term with $\eta^{(j)}$ by parts $j$ times, and apply the fundamental lemma of the calculus of variations.

Example: for $k = 2$ and $L = \frac{1}{2} z_2^2$ , one has $L_{z_0} = L_{z_1} = 0$ and $L_{z_2} = z_2$ , so the equation is $y'''' = 0$ . Every cubic polynomial solves it. Non-example: $y(x) = x^4$ does not solve this equation, since $y'''' = 24$ .

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Euclidean space object
Euler–Lagrange equation concept
Function concept
Integral functional with a Lagrangian density concept
Function of several variables concept
Partial derivative concept
Euler–Poisson equation for higher derivatives definition

Tag	`74DV24XGKZ`
URI	`m2://74DV24XGKZ`
Kind	definition
Writing	W0 stub
MSC	49K05, 34B15
Defines	Euler–Poisson equation for higher derivatives
Uses	Integral functional with a Lagrangian density, Euler–Lagrange equation, Partial derivative

Euler–Poisson equation for higher derivatives

Definition

Remarks

Reading path

Further reading

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