Skills

Physics 7: Classical Mechanics II

14 weeks · 3 milestones

Lagrangian and Hamiltonian mechanics, phase space and symmetries. Course 7 of the Varsity Curriculum: Physics. Free sources only.

Part of: Physics · Proves for: UG

Milestone map

3 milestones

The Lagrangian

5 weeks (about 12 hrs/week)

Work through the 8.223 lectures and the 8.09 notes on the calculus of variations and the Euler–Lagrange equations. Then solve 8.223 problem sets 1–2. Self-check: MIT 8.223 problem sets with solutions.

Proof required

Link the public repo with 8.223 problem sets 1–2 written up (a typeset PDF, LaTeX source or notebook), and show the self-check: which answers matched and how you fixed the ones that did not.

What gets checked

Generalised coordinates are chosen and justified.
Constraints are handled correctly.
Each set is marked against its solutions.

Common mistakes

Answers copied from the key with no working.
Wrong units or missing signs.

Powstik Guide

A public write-up of 8.223 problem sets 1–2, with the self-check shown.

Steps

Study the linked course and chapters.
Solve the problems on your own first.
Check them against the self-check and rework the misses.
Typeset the write-up (LaTeX or a notebook) in a public repo.
Ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II, unit 1: The Lagrangian
Problems:
Self-check result (matched / reworked):
What I got wrong first and why:

What gets sent back

No working shown.
The self-check result is missing.

Resources

Foundationstart here

Depthgo deeper

What a verifier looks for

Generalised coordinates are chosen and justified.
Constraints are handled correctly.
Each set is marked against its solutions.
Is the physics right, not just the final number?

You'll sign in first, then come straight back here.

Hamiltonians, phase space and symmetry

5 weeks (about 12 hrs/week)

Work through the 8.09 notes on Hamilton's equations, Poisson brackets and Noether's theorem. Then solve 8.223 problem sets 3–4. Self-check: MIT 8.223 problem sets with solutions.

Proof required

Link the public repo with 8.223 problem sets 3–4 written up (a typeset PDF, LaTeX source or notebook), and show the self-check: which answers matched and how you fixed the ones that did not.

What gets checked

Each Hamiltonian comes from its Lagrangian by a Legendre transform.
Every conserved quantity is tied to a symmetry.
Each set is marked against its solutions.

Common mistakes

Answers copied from the key with no working.
Wrong units or missing signs.

Powstik Guide

A public write-up of 8.223 problem sets 3–4, with the self-check shown.

Steps

Study the linked course and chapters.
Solve the problems on your own first.
Check them against the self-check and rework the misses.
Typeset the write-up (LaTeX or a notebook) in a public repo.
Ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II, unit 2: Hamiltonians, phase space and symmetry
Problems:
Self-check result (matched / reworked):
What I got wrong first and why:

What gets sent back

No working shown.
The self-check result is missing.

Resources

Foundationstart here

Depthgo deeper

What a verifier looks for

Each Hamiltonian comes from its Lagrangian by a Legendre transform.
Every conserved quantity is tied to a symmetry.
Each set is marked against its solutions.
Is the physics right, not just the final number?

You'll sign in first, then come straight back here.

Simulate a double pendulum from its Lagrangian

5 weeks (about 12 hrs/week)

Make a notebook that derives the double pendulum's equations of motion from its Lagrangian, integrates them, and shows chaos. Integrate two runs from large angles (both arms at 120°) that start 10⁻⁸ rad apart, plot the log of their separation over time, and fit its slope.

Proof required

Link the public repo with the notebook, the result with its uncertainty, and a README that says what you did and what limits the result.

What gets checked

Energy stays constant to a stated accuracy.
The separation grows exponentially, and its rate has an uncertainty from several starting pairs.
Small swings match the normal modes (equal arms and masses: ω² = (2 ± √2) g/l).

Common mistakes

A result with no uncertainty.
Code or data that cannot be rerun from the repo.

Powstik Guide

A notebook that derives the double pendulum's equations of motion from its Lagrangian, integrates them, and shows chaos, in a public repo.

Steps

Write down what you expect before you start.
Integrate two runs from large angles (both arms at 120°) that start 10⁻⁸ rad apart, plot the log of their separation over time, and fit its slope.
Work out the uncertainty from your own data.
Compare with the accepted value and explain any gap.
Write the README and ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II — Simulate a double pendulum from its Lagrangian
What I expected:
Method:
Result (value ± uncertainty):
Accepted value and comparison:
What limits the result:

What gets sent back

No uncertainty, or one that is guessed.
Code that does not run.

Resources

Foundationstart here

What a verifier looks for

Energy stays constant to a stated accuracy.
The separation grows exponentially, and its rate has an uncertainty from several starting pairs.
Small swings match the normal modes (equal arms and masses: ω² = (2 ± √2) g/l).
The uncertainty is worked out from the data or runs (spread of repeats, instrument limits, fit errors or a range of inputs), not guessed, and it is shown on the result.
Is the physics right, and could someone rerun it from the repo?

You'll sign in first, then come straight back here.

Part of

Physics 7: Classical Mechanics II →

Related outcomes

Milestone map

3 milestones

The Lagrangian

5 weeks (about 12 hrs/week)

Proof required

Link the public repo with 8.223 problem sets 1–2 written up (a typeset PDF, LaTeX source or notebook), and show the self-check: which answers matched and how you fixed the ones that did not.

What gets checked

Generalised coordinates are chosen and justified.
Constraints are handled correctly.
Each set is marked against its solutions.

Common mistakes

Answers copied from the key with no working.
Wrong units or missing signs.

Powstik Guide

A public write-up of 8.223 problem sets 1–2, with the self-check shown.

Steps

Study the linked course and chapters.
Solve the problems on your own first.
Check them against the self-check and rework the misses.
Typeset the write-up (LaTeX or a notebook) in a public repo.
Ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II, unit 1: The Lagrangian
Problems:
Self-check result (matched / reworked):
What I got wrong first and why:

What gets sent back

No working shown.
The self-check result is missing.

Resources

Foundationstart here

Depthgo deeper

What a verifier looks for

Generalised coordinates are chosen and justified.
Constraints are handled correctly.
Each set is marked against its solutions.
Is the physics right, not just the final number?

You'll sign in first, then come straight back here.

Hamiltonians, phase space and symmetry

5 weeks (about 12 hrs/week)

Work through the 8.09 notes on Hamilton's equations, Poisson brackets and Noether's theorem. Then solve 8.223 problem sets 3–4. Self-check: MIT 8.223 problem sets with solutions.

Proof required

Link the public repo with 8.223 problem sets 3–4 written up (a typeset PDF, LaTeX source or notebook), and show the self-check: which answers matched and how you fixed the ones that did not.

What gets checked

Each Hamiltonian comes from its Lagrangian by a Legendre transform.
Every conserved quantity is tied to a symmetry.
Each set is marked against its solutions.

Common mistakes

Answers copied from the key with no working.
Wrong units or missing signs.

Powstik Guide

A public write-up of 8.223 problem sets 3–4, with the self-check shown.

Steps

Study the linked course and chapters.
Solve the problems on your own first.
Check them against the self-check and rework the misses.
Typeset the write-up (LaTeX or a notebook) in a public repo.
Ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II, unit 2: Hamiltonians, phase space and symmetry
Problems:
Self-check result (matched / reworked):
What I got wrong first and why:

What gets sent back

No working shown.
The self-check result is missing.

Resources

Foundationstart here

Depthgo deeper

What a verifier looks for

Each Hamiltonian comes from its Lagrangian by a Legendre transform.
Every conserved quantity is tied to a symmetry.
Each set is marked against its solutions.
Is the physics right, not just the final number?

You'll sign in first, then come straight back here.

Simulate a double pendulum from its Lagrangian

5 weeks (about 12 hrs/week)

Proof required

Link the public repo with the notebook, the result with its uncertainty, and a README that says what you did and what limits the result.

What gets checked

Energy stays constant to a stated accuracy.
The separation grows exponentially, and its rate has an uncertainty from several starting pairs.
Small swings match the normal modes (equal arms and masses: ω² = (2 ± √2) g/l).

Common mistakes

A result with no uncertainty.
Code or data that cannot be rerun from the repo.

Powstik Guide

A notebook that derives the double pendulum's equations of motion from its Lagrangian, integrates them, and shows chaos, in a public repo.

Steps

Write down what you expect before you start.
Integrate two runs from large angles (both arms at 120°) that start 10⁻⁸ rad apart, plot the log of their separation over time, and fit its slope.
Work out the uncertainty from your own data.
Compare with the accepted value and explain any gap.
Write the README and ask someone who knows physics to check it.

Template

README
Physics 7: Classical Mechanics II — Simulate a double pendulum from its Lagrangian
What I expected:
Method:
Result (value ± uncertainty):
Accepted value and comparison:
What limits the result:

What gets sent back

No uncertainty, or one that is guessed.
Code that does not run.

Resources

Foundationstart here

What a verifier looks for

Energy stays constant to a stated accuracy.
The separation grows exponentially, and its rate has an uncertainty from several starting pairs.
Small swings match the normal modes (equal arms and masses: ω² = (2 ± √2) g/l).
The uncertainty is worked out from the data or runs (spread of repeats, instrument limits, fit errors or a range of inputs), not guessed, and it is shown on the result.
Is the physics right, and could someone rerun it from the repo?

You'll sign in first, then come straight back here.