Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Use first variations to obtain an Euler–Lagrange equation.
- Justify the conclusion "u(x)=x minimizes J and J=1/2" using the stated assumptions.
Before you start
Integration by parts and variations with fixed endpoints.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Minimize J[u]=∫₀¹(u′)²/2 dx subject to u(0)=0,u(1)=1.
Why this math matters
Use first variations to obtain an Euler–Lagrange equation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Admissible functions have square-integrable weak derivatives.
- Endpoint values are fixed.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Derive an equation from stationary energy
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Minimize J[u]=∫₀¹(u′)²/2 dx subject to u(0)=0,u(1)=1.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
For u+εη with η(0)=η(1)=0, dJ/dε at zero is ∫u′η′ dx
Variations preserve the endpoint constraints.
Work through the mathematics
Integration by parts gives −∫u″η dx=0 for all such tests
Stationarity forces u″=0 in the appropriate weak or classical sense.
Check the conclusion
u(x)=x minimizes J and J=1/2
Writing u=x+v with v zero at both endpoints gives J=1/2+∫(v′)²/2, proving global minimality.
The result
u(x)=x minimizes J and J=1/2
Writing u=x+v with v zero at both endpoints gives J=1/2+∫(v′)²/2, proving global minimality.
Common mistakes to catch
- Boundary terms vanish only for the permitted variations.
- Stationarity and minimization are different conclusions.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What changes for endpoints u(0)=2,u(1)=5?
Show a hint
Solve the affine boundary-value problem.
Reveal answer and explanation
u=2+3x
The minimizing slope is constant three.
Practice 2
Does every stationary variational solution minimize its functional?
Show a hint
Recall maxima and saddles.
Reveal answer and explanation
No
Here convexity and the explicit nonnegative remainder establish minimality.
Take the idea with you
Interpret a straight interpolation as the least-gradient-energy connection between two endpoint values.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use an energy bound to support existence and uniqueness
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