Foundations

Choose what belongs to an investigation, describe its condition at a stated time, and identify interactions with what lies outside its boundary.

Choose what a model represents, make its simplifications explicit, and check whether its predictions answer the physical question you actually asked.

Identify what a scientific statement does, examine its evidence and conditions, and avoid treating its name as a score for certainty.

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

Use a defined system and a named quantity to track transfers, distinguish changing parts from conserved totals, and check the conditions of energy and momentum balances.

Choose a descriptive level, connect collective quantities with component behavior, and check when a simpler physical description answers a specified question.

Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.

Choose one body, identify the forces acting on it, and use their vector sum to distinguish acceleration, balanced motion and mutual interactions.

Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.

An inverse-square force models attraction between point masses.

Relate a periodic wave's spatial and temporal repetition.

Interaction forces act on different bodies in equal and opposite pairs.

Torque changes angular momentum about a specified origin.

External force controls the mass-weighted centre's acceleration.

A tensor relates rotation to angular momentum.

Body-axis angular acceleration includes rotation of the axes.

Stationary action supplies coordinate equations of motion.

Canonical coordinates and momenta evolve from the Hamiltonian.

A linear restoring force, damping and periodic drive determine response.

For two Newtonian particles with equal opposite internal force and no external force, the centre of mass moves uniformly.

Near a stationary equilibrium, a conservative system with quadratic kinetic matrix M and potential Hessian K has Mq̈+Kq=0.

For a conservative central interaction U(r), relative motion is planar and angular momentum L=μr²θ̇ is conserved.

For a frame rotating at constant Ω about a common inertial origin, a_inertial=a_rot+2Ω×v_rot+Ω×(Ω×r).

For ideal independent holonomic constraints f_α(q,t)=0, constrained Lagrange equations can retain redundant coordinates through d(∂L/∂q̇_i)/dt−∂L/∂q_i=Σ_α λ_α∂_i f_α.

In canonical coordinates, {A,B}=Σ_i[(∂_q_iA)(∂_p_iB)−(∂_p_iA)(∂_q_iB)] and dA/dt={A,H}+∂_tA.

A canonical transformation changes phase-space coordinates and the Hamiltonian while preserving Hamilton's description of the same physical trajectories.

A smooth type-2 generating function F₂(q,P,t) defines p_i=∂F₂/∂q_i and Q_i=∂F₂/∂P_i.

Hamilton's principal function satisfies ∂_tS+H(q,∂_qS,t)=0.

For one regular periodic Hamiltonian orbit, I=(1/2π)∮p dq and its conjugate angle θ advances at ω=∂H/∂I; equivalently dI/dE=T/(2π).