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For a regular Lagrangian, $p=\partial L/\partial \dot{q}$ and $H=p\dot{q}-L$. Regularity means the velocity Hessian is invertible.

The derivative of angular momentum about the moving centre of mass equals the total external torque about that centre, because total relative momentum vanishes and internal central torques cancel.

For a round body with $I=\gamma M{a}^2$ rolling without slipping down an incline, static friction gives

Mass measures inertia and acts as the source of Newtonian gravity.

Displacement is the change in position between two points.

An inverse-square central force has magnitude proportional to the reciprocal square of the separation.

Uniform circular motion of radius $a$ and angular frequency $\omega$ has centripetal acceleration of magnitude $a{\omega }^2$ toward the centre.

A force is an interaction that changes momentum; in Newtonian mechanics it satisfies $F=ma$ for constant mass.

The period is the time required for a periodic motion to complete one full cycle.

Energy is a conserved scalar quantity associated with time-translation symmetry.

For a nonrelativistic point mass of fixed mass $m>0$, measured in an inertial frame, its momentum isMomentum is a vector pointing along the velocity, with International System of Units unit $\mathrm{kg}m/s$. Doubling either mass or velocity doubles momentum. In one dimension, a chosen positive direction makes $p=mv$ a signed quantity: a $2\mathrm{kg}$ body moving right at $3m/s$ has $p=+6\mathrm{kg}m/s$; moving left at the same speed gives $p=-6\mathrm{kg}m/s$. ^[ref]

For a collection, total momentum is the vector sum $\mathbf{P}={\sum }_i{m}_i{\mathbf{v}}_i$. Two equal masses moving with opposite velocities can have zero total momentum while both have positive kinetic energy. Direction is therefore essential when using momentum conservation; adding speeds instead of signed velocities gives the wrong balance.

Newton's second law connects the net force on one fixed-mass point mass to its momentum through ${\mathbf{F}}_{\mathrm{net}}=d\mathbf{p}/dt$. Integrating that change gives impulse. Here $\mathbf{p}$ denotes mechanical momentum; when canonical momentum differs from mass times velocity, the distinction in mechanical momentum below applies. The nonrelativistic formula also has a limited regime: special relativity is needed at speeds comparable with the speed of light.

Position specifies the location of a particle relative to a coordinate system.

A degree of freedom is an independent coordinate needed to specify a mechanical configuration after all mechanical constraints have been imposed.

The Newtonian gravitational field is the gravitational force per unit mass of a test particle. For a point mass $M$ at the origin,The test particle's mass cancels: different small masses have the same gravitational acceleration at the same position. In SI, the Newtonian gravitational field has units $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}=\mathrm{m}{\mathrm{s}}^{-2}$. The same exterior expression holds for a spherically symmetric source, by the spherical shell theorem.

Choose a reference radius $r_0>0$, set $g_0=GM/r_0^2$, and use nondimensionalization with $z=r/r_0$. Then $g/g_0=z^{-2}$: at $z=1,2,4$, the magnitudes are $1,1/4,1/16$. This makes the spatial variation visible without specifying a planet. ^[ref]

The order of a unit modulo a product of coprime moduli is the least common multiple of the orders of its components.

An astrophysical black hole is a black hole studied through its formation, accretion, mergers, dynamical environment, and electromagnetic or gravitational-wave signatures.

A galaxy is a gravitationally bound system of stars, interstellar gas and dust, stellar remnants, and dark matter.

An astrophysical disk is a flattened rotating distribution of gas, dust, stars, or other matter whose orbital motion strongly influences its structure and evolution.

Stellar astrophysics studies the structure, formation, energy production, observable properties, and evolution of stars.

Planetary science studies planets, moons, small bodies, planetary systems, and the processes that form and evolve them.