Derivation index
Proofs and constructions across the four units.
Proofs and constructions
“Core proof” identifies an argument supporting an official syllabus topic. “Construction” identifies a calculation that should be reproducible for a given system. “Supporting argument” identifies material needed for interpretation. These labels describe the role of the mathematics; they are not claims about past examination frequency.
| Derivation or proof | Unit | Location | Role in written work |
|---|---|---|---|
| Degrees-of-freedom count with regular independent constraints | 1 | Chapter 2 | Supporting argument; state regularity |
| Velocity transformation and fixed-time virtual displacement | 1 | Chapter 4 | Core relation |
| Generalized force from virtual work | 1–2 | Chapter 5, Chapter 8 | Core construction |
| Principle of virtual work | 1 | Chapter 5 | Core derivation with ideality |
| D’Alembert's principle | 1 | Chapter 6 | Core derivation |
| Lagrange's equations from D’Alembert | 2 | Optional appendix | Optional; explicitly excluded as a required derivation |
| Cyclic-coordinate momentum conservation | 2 | Chapter 9 | Core proof |
| Translation and total linear momentum | 2 | Chapter 9 | Core conservation proof |
| Rotation and angular momentum | 2 | Chapter 9 | Core conservation proof |
| Energy-function balance | 2 | Chapter 9 | Core proof |
| Kinetic-energy quadratic expansion | 2 | Chapter 10 | Core kinetic-energy property |
| Symmetry and positivity of inertia matrix | 2 | Chapter 10 | Core property with regularity |
| Euler homogeneity and | 2 | Chapter 10 | Core property and energy interpretation |
| Kinetic-energy derivative identity | 2 | Chapter 10 | Core supporting identity |
| Pendulum, Atwood, moving-support and coupled equations | 2 | Chapter 7, Chapter 11 | Applications; show coordinates, energies, derivatives |
| Rayleigh power relation | 2 | Chapter 8 | Supporting force treatment |
| Legendre construction | 3 | Chapter 12 | Core construction; eliminate velocities |
| Hamilton's equations from the Legendre differential | 3 | Optional appendix | Optional; explicitly excluded as a required derivation |
| Hamiltonian conservation and forced balance | 3 | Chapter 14 | Core interpretation |
| Central torque and angular-momentum conservation | 4 | Chapter 16 | Required central-force property proof |
| Planarity | 4 | Chapter 16 | Required property; separate |
| Constant areal velocity | 4 | Chapter 16 | Required property and Kepler proof |
| Potential and energy conservation | 4 | Chapter 16 | Required property; state time independence |
| Center-of-mass and reduced-mass decomposition | 4 | Chapter 16 | Supporting derivation |
| Polar acceleration | 4 | Chapter 17 | Prerequisite derivation |
| Radial equation and effective potential | 4 | Chapter 17 | Core derivation |
| Circular stability and radial frequency | 4 | Chapter 17 | Core orbit classification |
| Binet differential equation | 4 | Chapter 18 | Core orbit derivation |
| Near-circular apsidal frequency ratio | 4 | Chapter 19 | Supporting argument; not a full Bertrand proof |
| Oscillator closure | 4 | Chapter 19 | Sufficiency check for one Bertrand potential |
| Inverse-square conic and focus geometry | 4 | Chapter 20 | Core derivation |
| Energy–eccentricity and ellipse energy | 4 | Chapter 20 | Core orbit relations |
| Kepler's first, second, and third laws | 4 | Chapter 21 | Required derivations |
| Runge–Lenz conservation | 4 | Chapter 22 | Core proof |
| LRL direction, eccentricity, and magnitude | 4 | Chapter 22 | Core properties and applications |
| Attractive scattering deflection | 4 | Chapter 22 | Application |
| Kepler's anomaly equation | 4 | Chapter 21 | Optional supporting time parametrization |
Writing a derivation from understanding
Start with the physical assumptions and a clear coordinate or vector convention. State the principle being used, carry out the derivatives or substitutions, and identify the result. A short dimensional or limiting check often detects a missing length factor or sign. The main chapters show the intermediate algebra; a compact written version can omit repeated explanatory prose while retaining each mathematical step that makes the result follow.