Syllabus coverage · Classical Mechanics

Unit 1 · Constrained motion

The source is the uploaded official syllabus image, headed Part A, together with the expanded teaching requirements supplied in the request. “Covered” means the topic has substantive teaching content, examples, and related practice in this edition.

Official topic	Location	Status
Constraints: definition	Chapter 1, Section 2	Covered
Classification and examples	Chapter 1, Sections 3–6	Covered
Degrees of freedom	Chapter 2, Sections 1–3	Covered
Configuration space	Chapter 2, Section 4	Covered
Constrained system	Chapter 1, Sections 1–2	Covered
Forces of constraint	Chapter 1, Section 5; Chapter 5, Section 3	Covered
Constrained motion	Chapter 1 examples; Chapter 6 applications	Covered
Generalized coordinates	Chapter 3, Sections 1–3	Covered
Transformation equations	Chapter 3, Section 2	Covered
Generalized notation and relations	Chapter 3 notation; Chapter 4 identities	Covered
Principle of virtual work	Chapter 5, Section 4	Covered with derivation
D’Alembert's principle	Chapter 6, Sections 2–4	Covered with derivation and applications

Supporting material included

Supporting topic	Location
Vectors, dot products, Newton's law, work and potential	Prerequisites P1–P2
Partial and total derivatives, chain rule	Prerequisites P3; Chapter 4
Polar coordinate velocity	Prerequisites P4; Chapter 4 Example 3
Simple differential equations and initial conditions	Prerequisites P5; Chapter 6
Elementary equilibrium stability	Prerequisites P6; Chapter 5
Redundant constraints and coordinate singularities	Chapters 2–3
Unilateral contact and string/rod distinction	Chapters 1 and 5
Non-holonomic qualifications	Chapters 1, 2, and 4
Moving ideal constraints and actual work	Chapter 4

Unit 2 · Lagrangian formalism

Official syllabus topic	Location in notes	Status
Lagrangian for conservative systems	The Lagrangian and equations of motion	Covered
Lagrangian for non-conservative systems	Applied forces and dissipation	Covered; remaining forces and dissipation
Lagrange equations: statement, meaning, applications	The Lagrangian and equations of motion	Covered; full derivation separately optional
Comparison of Newtonian and Lagrangian formulations	The Lagrangian and equations of motion	Covered
Cyclic coordinates	Momentum, symmetry, and conservation	Covered with force qualification
Conservation laws and associated proofs	Momentum, symmetry, and conservation	Covered: translation, rotation, energy
Properties of kinetic-energy function	The kinetic-energy function	Covered with proofs
Examples of Lagrangian formulation	Moving constraints and coupled motion	Covered; additional examples in Chapters 7–10

Unit 3 · Hamiltonian formalism

Official syllabus topic	Location in notes	Status
Phase space	Phase space and Hamilton’s equations	Covered with phase portraits
Hamiltonian for conservative systems	From the Lagrangian to the Hamiltonian	Covered
Hamiltonian for non-conservative systems	Energy and forces in Hamiltonian mechanics	Covered through forced Hamilton equations
Physical significance of Hamiltonian	Energy and forces in Hamiltonian mechanics	Covered, including exceptions to H = E
Hamilton equations: statement, meaning, applications	Phase space and Hamilton’s equations	Covered; full derivation separately optional
Comparison of Lagrangian and Hamiltonian formulations	Hamiltonian applications and comparisons	Covered alongside Newtonian mechanics
Cyclic coordinates	Energy and forces in Hamiltonian mechanics	Covered
Construction of Hamiltonian from Lagrangian	From the Lagrangian to the Hamiltonian	Covered step by step
Examples of Hamiltonian formulation	Hamiltonian applications and comparisons	Covered; additional examples in Chapters 12–14

Unit 4 · Central-force motion

Official syllabus topic	Location in notes	Status
Definition of central force	Central forces and two-body motion	Covered
Properties of central force with proofs	Central forces and two-body motion	Covered: torque, angular momentum, plane, area, energy
Equation of motion	Radial motion and effective potential	Covered; polar acceleration and radial dynamics
Differential equation of orbit	The differential equation of the orbit	Covered with full Binet derivation
Bound and unbound orbits	Radial motion and effective potential	Covered with accessible-region tests
Stable and unstable orbits	Orbit stability, closure, and Bertrand’s theorem	Covered; marginal cases distinguished
Closed and open orbits	Orbit stability, closure, and Bertrand’s theorem	Covered; bounded non-closed paths distinguished
Bertrand theorem	Orbit stability, closure, and Bertrand’s theorem	Statement, scope, local argument, sufficiency examples; full necessity proof explicitly beyond scope
Inverse-square force motion	Inverse-square attraction and conic orbits	Covered: conics, energy, angular momentum, escape
Derivation of Kepler laws	Kepler’s laws	All three derived
Laplace–Runge–Lenz vector	The Laplace–Runge–Lenz vector	Definition and conservation/property proofs
Applications of Runge–Lenz vector	The Laplace–Runge–Lenz vector	Covered: orientation, eccentricity, orbit reconstruction, encounters

Course-wide resources

The formula sheet, derivation index, definition index, connection map, problem bank, and final revision span all four units. Chapter exercises and separate solutions are included throughout.

The only deliberately limited proof is the full global necessity proof of Bertrand's theorem; the theorem, its implications, the near-circular argument, and verification for the two permitted potentials are taught explicitly. Full derivations of Lagrange's and Hamilton's equations are available in the optional appendix and are not presented as compulsory material.