functionalDerivative
Functional derivative (variational derivative)
Description
returns the functional derivative
of the functional with respect to the function y = y(x), where x represents one or more independent variables. The
functional derivative relates the change in the functional S[y] with respect to a small variation in y(x).The functional derivative is also known as the variational derivative.G
= functionalDerivative(f
,y
)
If y
is a vector of symbolic functions,
functionalDerivative
returns a vector of functional derivatives with
respect to the functions in y
, where all functions in
y
must depend on the same independent variables.
Examples
Functional Derivative with Respect to Single Function
Find the functional derivative of the functional with respect to the function , where the integrand is .
Declare y(x)
as a symbolic function and define f
as the integrand of . Use f
and y
as the parameters of functionalDerivative
.
syms y(x)
f = y*sin(y);
G = functionalDerivative(f,y)
G(x) =
Functional Derivative with Respect to Two Functions
Find the functional derivative of the functional with respect to the functions and , where the integrand is .
Declare u(x)
and v(x)
as symbolic functions, and define f
as the integrand of .
syms u(x) v(x) f = u^2*diff(v,x) + v*diff(u,x,x);
Specify a vector of symbolic functions [u v]
as the second input argument in functionalDerivative
.
G = functionalDerivative(f,[u v])
G(x) =
functionalDerivative
returns a vector of symbolic functions containing the functional derivatives of the integrand f
with respect to u
and v
, respectively.
Euler–Lagrange Equation of Simple Mass-Spring System
Find the Euler–Lagrange equation of a mass m
that is connected to a spring with spring constant k
.
Define the kinetic energy T
, potential energy V
, and Lagrangian L
of the system. The Lagrangian is the difference between the kinetic and potential energy.
syms m k x(t) T = 1/2*m*diff(x,t)^2; V = 1/2*k*x^2; L = T - V
L(t) =
In Lagrangian mechanics, the action functional of the system is equal to the integral of the Lagrangian over time, or . The Euler–Lagrange equation describes the motion of the system for which is stationary.
Find the Euler–Lagrange equation by taking the functional derivative of the integrand L
and setting it equal to 0
.
eqn = functionalDerivative(L,x) == 0
eqn(t) =
eqn
is the differential equation that describes mass-spring oscillation.
Solve eqn
using dsolve
. Assume the mass m
and spring constant k
are positive. Set the initial conditions for the oscillation amplitude as and the initial velocity of the mass as .
assume(m,'positive') assume(k,'positive') Dx(t) = diff(x(t),t); xSol = dsolve(eqn,[x(0) == 10, Dx(0) == 0])
xSol =
Clear assumptions for further calculations.
assume([k m],'clear')
Differential Equation of Brachistochrone Problem
The Brachistochrone problem is to find the quickest path of descent of a particle under gravity without friction. The motion is confined to a vertical plane. The time for a body to move along a curve from point to under gravity is given by
Find the quickest path by minimizing the change in with respect to small variations in the path . The condition for a minimum is .
Compute the functional derivative to obtain the differential equation that describes the Brachistochrone problem. Use simplify
to simplify the equation to its expected form.
syms g y(x) assume(g,'positive') f = sqrt((1 + diff(y)^2)/(2*g*y)); eqn = functionalDerivative(f,y) == 0; eqn = simplify(eqn)
eqn(x) =
This equation is the standard differential equation of the Brachistochrone problem. To find the solutions of the differential equation, use dsolve
. Specify the 'Implicit'
option to true
to return implicit solutions, which have the form .
sols = dsolve(eqn,'Implicit',true)
sols =
The symbolic solver dsolve
returns general solutions in the complex space. Symbolic Math Toolbox™ does not accept the assumption that the symbolic function is real.
Depending on the boundary conditions, there are two real-space solutions to the Brachistochrone problem. One of the two solutions below describes a cycloid curve in real space.
solCycloid1 = sols(3)
solCycloid1 =
solCycloid2 = sols(5)
solCycloid2 =
Another solution in real space is a horizontal straight line, where is a constant.
solStraight = simplify(sols(4))
solStraight =
To illustrate the cycloid solution, consider an example with boundary conditions and . In this case, the equation that can satisfy the given boundary conditions is solCycloid1
. Substitute the two boundary conditions into solCycloid1
.
eq1 = subs(solCycloid1,[x y(x)],[0 5]); eq2 = subs(solCycloid1,[x y(x)],[4 1]);
The two equations, eq1
and eq2
, have two unknown coefficients, and . Use vpasolve
to find numerical solutions for the coefficients. Substitute these solutions into solCycloid1
.
coeffs = vpasolve([eq1 eq2]); eqCycloid = subs(solCycloid1,{'C1','C3'},{coeffs.C1,coeffs.C3})
eqCycloid =
The implicit equation eqCycloid
describes the cycloid solution of the Brachistochrone problem in terms of and .
You can then use fimplicit
to plot eqCycloid
. Since fimplicit
only accepts implicit symbolic equations that contain symbolic variables and , convert the symbolic function to a symbolic variable . Use mapSymType
to convert to . Plot the cycloid solution within the boundary conditions and .
funToVar = @(obj) sym('y'); eqPlot = mapSymType(eqCycloid,'symfun',funToVar); fimplicit(eqPlot,[0 4 1 5])
Minimal Surface Equation in 3-D Space
For a function that describes a surface in 3-D space, the surface area can be determined by the functional
where and are the partial derivatives of with respect to and .
Find the functional derivative of the integrand f
with respect to u
.
syms u(x,y)
f = sqrt(1 + diff(u,x)^2 + diff(u,y)^2);
G = functionalDerivative(f,u)
G(x, y) =
The result is the equation G
that describes the minimal surface of a 3-D surface defined by u(x,y)
. The solutions to this equation describe minimal surfaces in 3-D space, such as soap bubbles.
Input Arguments
f
— Integrand of functional
symbolic variable | symbolic function | symbolic expression
Integrand of a functional, specified as a symbolic variable, function, or expression. The
argument f
represents the density of the functional.
y
— Differentiation function
unassigned symbolic function | vector of unassigned symbolic functions | matrix of unassigned symbolic functions | multidimensional array of unassigned symbolic functions
Differentiation function, specified as an unassigned symbolic function or a vector,
matrix, or multidimensional array of unassigned symbolic functions. The argument
y
can be a function of one or more independent variables. If
y
is a vector of symbolic functions,
functionalDerivative
returns a vector of functional derivatives with
respect to the functions in y
, where all functions in
y
must depend on the same independent variables.
Example: syms y(x); G = functionalDerivative(y*sin(y),y)
Output Arguments
G
— Functional derivative
symbolic function | vector of symbolic functions
Functional derivative, returned as a symbolic function or a vector of symbolic functions.
If input y
is a vector, matrix, or an array, then G
is a
vector.
More About
Functional Derivative
Consider a functional
which can take any path from a to b in the x-space.
For a small variation in the path y(x), define the change as in which ϕ(x) is an arbitrary test function. The change in the functional S is
The expression is the functional derivative of S with respect to y. The linear functional DS[y] is also known as the first variation or the Gateaux differential of the functional S.
One method to calculate the functional derivative is to apply Taylor expansion to the expression S[y + εϕ] with respect to ε. By keeping the first order terms in ε, performing integration by parts, and choosing the boundary conditions ϕ(a) = ϕ(b) = ϕ'(a) = ϕ'(b) = ... = 0, the functional derivative becomes
Version History
Introduced in R2015a
MATLAB 命令
您点击的链接对应于以下 MATLAB 命令:
请在 MATLAB 命令行窗口中直接输入以执行命令。Web 浏览器不支持 MATLAB 命令。
Select a Web Site
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
You can also select a web site from the following list:
How to Get Best Site Performance
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.
Americas
- América Latina (Español)
- Canada (English)
- United States (English)
Europe
- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)
- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)