Derive the Euler-Lagrange Equations
The Euler–Lagrange equations are fundamental in the calculus of variations and are used to find the functions that make a functional stationary. A common context in which they arise is in classical mechanics, where they are used to derive the equations of motion from a Lagrangian.To derive the Euler–Lagrange equations for a functional of the form: where is the Lagrangian, a function of , its derivative (x), and the independent variable , follow these steps:1. Consider a Variation: Assume a small variation in the function such that , where is a small parameter and is an arbitrary smooth function satisfying .2. Expand the Functional: The change in the functional due to this variation is: Expanding this to the first order in , we get: 3. Integration by Parts: To eliminate the derivative of , use integration by parts: x–ηx The boundary term vanishes because .4. The Euler–Lagrange Equation: Substituting back, the requirement for arbitrary gives: –0This equation is the Euler–Lagrange equation, which the function must satisfy to make the functional stationary.If you want to see how to implement this in the Wolfram Language using the function, let me know!
J[y]L(y(x),(x),x)x
x
2
∫
x
1
′
y
L
y(x)
′
y
x
y(x)
y(x)y(x)+ϵη(x)
ϵ
η(x)
η()η()0
x
1
x
2
J[y]
δJJ[y+ϵη]
d
dϵ
|
ϵ0
ϵ
δJη+dx
x
2
∫
x
1
∂L
∂y
∂L
∂
′
y
dη
dx
η
x
2
∫
x
1
∂L
∂
′
y
dη
dx
x
2
η
∂L
∂
′
y
x
1
x
2
∫
x
1
d
dx
∂L
∂
′
y
η()η()0
x
1
x
2
δJ0
η(x)
∂L
∂y
d
dx
∂L
∂
′
y
y(x)
J[y]
EulerEquations
Yes, please
To derive the Euler–Lagrange equations using the Wolfram Language, you can use the . This function helps you find the Euler–Lagrange differential equations from a given functional.Here's a basic example of how to use it:Suppose you have a functional with a Lagrangian , where is the mass and is a potential energy function. You want to derive the Euler–Lagrange equation for the function .Here's how you can do it:
This code will return the Euler–Lagrange equation for the given Lagrangian. You can replace the lagrangian with any other Lagrangian of interest, and the function will provide the corresponding equations. Let me know if you need further assistance or have a specific example in mind!
ResourceFunction["EulerEquations"]
Lm–V(y(x))
1
2
2
((x))
′
y
m
V(y(x))
y(x)
(*DefinetheLagrangian*)lagrangian=(1/2)m(y'[x])^2–V[y[x]];(*UseEulerEquationstoderivetheequationofmotion*)eulerLagrangeEquation=
| |||||||||
|
In[]:=
(*DefinetheLagrangian*)lagrangian=(1/2)m(y'[x])^2–V[y[x]];(*UseEulerEquationstoderivetheequationofmotion*)eulerLagrangeEquation=
[lagrangian,y[x],x]
|
Out[]=
–[y[x]]–m[x]0
′
V
′′
y
Derive the Euler-Lagrange Equations for a pendulum
|
For a simple pendulum, the Lagrangian is typically defined by the kinetic energy minus the potential energy. The kinetic energy and potential energy are given by: where: The Lagrangian for the pendulum is: To derive the Euler–Lagrange equation for this Lagrangian using Wolfram Language, you can use the as follows:
This will provide the equation of motion for the simple pendulum. Let me know if you have any questions or need further assistance!
• Kinetic Energy:
Tm
1
2
2
r
2
()
θ
• Potential Energy:
Umgrcos(θ)
• is the mass of the pendulum,
m
• is the length of the pendulum,
r
• is the angle from the vertical,
θ
• is the acceleration due to gravity.
g
L
LT–Um–mgrcos(θ)
1
2
2
r
2
()
θ
ResourceFunction["EulerEquations"]
(*DefinetheLagrangianforasimplependulum*)lagrangian=(1/2)mr^2θ'[t]^2–mgrCos[θ[t]];(*UseEulerEquationstofindtheequationsofmotion*)eulerLagrangeEquation=
| |||||||||
|
In[]:=
(*DefinetheLagrangianforthependulum*)lagrangian=(1/2)mr^2θ'[t]^2+mgrCos[θ[t]];(*DerivetheEuler–Lagrangeequation*)eulerLagrangeEquation=
[lagrangian,θ[t],t]
|
Out[]=
–Sin[θ[t]]–[t]0
981
100
′′
θ
Plot the solution assuming θ[0] = 1 and θ'[0]=0
|
To plot the solution of the pendulum's equation of motion, you need to solve the differential equation obtained from the Euler–Lagrange equation. The equation is: Given the initial conditions and , we can use the function to find the numerical solution and then plot it.Here's how you can do this in the Wolfram Language:
This code snippet will plot the angle of the pendulum over time from to seconds. Let me know if you need further assistance or adjustments!
NDSolve
(*Solvethedifferentialequationnumerically*) solution=NDSolve[{θ''[t]+(981/100)Sin[θ[t]]==0,θ[0]==1,θ'[0]==0},θ,{t,0,10}]; (*Plotthesolution*) Plot[Evaluate[θ[t]/.solution],{t,0,10},PlotLabels–>{"θ(t)"},PlotLabel–>"Pendulum Angle θ(t)",AxesLabel–>{"Time t","Angle θ"},PlotRange–>All] | |||||||||
|
In[]:=
(*Solvethedifferentialequationnumerically*) solution=NDSolve[{θ''[t]+(981/100)Sin[θ[t]]==0,θ[0]==1,θ'[0]==0},θ,{t,0,10}]; (*Plotthesolution*) Plot[Evaluate[θ[t]/.solution],{t,0,10},PlotLabels–>{"θ(t)"},PlotLabel–>"Pendulum Angle θ(t)",AxesLabel–>{"Time t","Angle θ"},PlotRange–>All]
Out[]=
