An Etymological Dictionary of Astronomy and Astrophysics
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Of all the possible paths along which a → dynamical system can move from one configuration to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the → Lagrangian function. Hamilton's principle is often mathematically expressed as δ∫Ldt = 0, where L is the Lagrangian function, the integral summed from t₁ to t₂, and δ denotes the virtual operator of Lagrangian dynamics and the → calculus of variations.

→ Hamiltonian function; → principle.

The study of → dynamical systems in terms of the → Hamilton's equations.

→ Hamiltonian function; → dynamics.

A reformulation of classical mechanics that predicts the same outcomes as classical mechanics. → Hamiltonian dynamics.

→ Hamiltonian; → mechanics.

A function that describes the motion of a → dynamical system in terms of the → Lagrangian function, → generalized coordinates, → generalized momenta, and time. For a → holonomic system having n degrees of freedom, the Hamiltonian function is of the form: H = Σp_iq^._i - L(q_i,q^._i,t) (summed from i = 1 to n), where L is the Lagrangian function. If L does not depend explicitly on time, the system is said to be → conservative and H is the total energy of the system. The Hamiltonian function plays a major role in the study of mechanical systems. Also called → Hamiltonian.

Introduced in 1835 by the Irish mathematician and physicist William Rowan Hamilton (1805-1865); → function.

The dynamical operator in → quantum mechanics that corresponds to the → Hamiltonian function in classical mechanics.

→ Hamiltonian function; → operator.

(adj.) Of, pertaining to, or noting a series of oscillations in which each oscillation has a frequency that is an integral multiple of the same basic frequency.
(n.) A wave motion, superimposed on a fundamental wave, having a frequency which is an integral multiple of the fundamental frequency. → overtone.

From L. harmonicus, from Gk. harmonikos "harmonic, musical," from harmonia "agreement, concord of sounds," related to harmos "joint," arariskein "to join together;" PIE base *ar- "to fit together."

Hamâhang, "harmonious, concordant," from ham- "together, with; same, equally, even" (Mid.Pers. ham-, like L. com- and Gk. syn- with neither of which it is cognate. O.Pers./Av. ham-; Skt. sam-; also O.Pers./Av. hama- "one and the same," Skt. sama-; Gk. homos-; originally identical with PIE numeral *sam- "one," from *som-) + âhang "melody, pitch, tune; harmony, concord," from Proto-Iranian *āhang-, from prefix ā- + *hang-, from PIE base *seng^wh- "to sing, make an incantation;" cf. O.H.G. singan; Ger. singen; Goth. siggwan; Swed. sjunga; O.E. singan "to chant, sing, tell in song;" maybe cognate with Gk. omphe "voice; oracle."

A number whose reciprocal is the → arithmetic mean of the reciprocals of a set of numbers. Denoted by H, it may be written in the discrete case for n quantities x₁, ..., x_n, as: 1/H = (1/n) Σ(1/x_i), summing from i = 1 to n. For example, the harmonic mean between 3 and 4 is 24/7 (reciprocal of 3: 1/3, reciprocal of 4: 1/4, arithmetic mean between them 7/24). The harmonic mean applies more accurately to certain situations involving rates. For example, if a car travels a certain distance at a speed speed 60 km/h and then the same distance again at a speed 40 km/h, then its average speed is the harmonic mean of 48 km/h, and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the car travels for a certain amount of time at a speed v and then the same amount of time at a speed u, then its average speed is the arithmetic mean of v and u, which in the above example is 50 km/h.

→ harmonic; → mean.

A motion that repeats itself in equal intervals of time (also called periodic motion).

→ harmonic; → motion.

Any oscillating particle in harmonic motion.

→ harmonic; → oscillator.

Math.: Any ordered set of numbers, the reciprocals of which have a constant difference between them. For example 1, ½, 1/3, ¼, ..., 1/n. Also called → harmonic sequence.

→ harmonic; progression.

→ harmonic progression.

→ harmonic; → sequence.

Overtones whose frequencies are integral multiples of the → fundamental frequency. The fundamental frequency is the first harmonic.

→ harmonic; → series.

A classification of stellar spectra published in the Henry Draper catalogue, which was prepared in the early twentieth century by E. C. Pickering and Miss Annie Canon. It is based on the characteristic lines and bands of the chemical elements. The most important classes in order of decreasing temperatures are as follows: O, B, A, F, G, K, M.

Harvard, named for John Harvard (1607-1638), the English colonist, principal benefactor of Harvard College, now Harvard University. → classification

The → full moon that appears closest in time to the → autumnal equinox.

→ harvest; → moon.

The radiation produced by a → black hole when → quantum mechanical effects are taken into account. According to quantum physics, large fluctuations in the → vacuum energy occurs for brief moments of time. Thereby virtual particle-antiparticle pairs are created from vacuum and annihilated. If → pair production happens just outside the → event horizon of a black hole, as soon as these particles are formed they would both experience drastically different → gravitational attractions due to the sharp gradient of force close to the black hole. One particle will accelerate toward the black hole and its partner will escape into space. The black hole used some of its → gravitational energy to produce these two particles, so it loses some of its mass if a particle escapes. This gradual loss of mass over time means the black hole eventually evaporates out of existence. See also → Bekenstein formula, → Hawking temperature.

Named after the British physicist Stephen Hawking (1942-2018), who provided the theoretical argument for the existence of the radiation in 1974; → radiation.

The region to the right the → Hayashi track, representing objects that cannot be in → hydrostatic equilibrium. Energy transport in these objects would take place with a → superadiabatic temperature gradient.

→ Hayashi track; → forbidden; → zone.

An early → B-type star showing helium lines with abnormally large equivalent widths. The surface → chemical abundances of He-strong stars are influenced by the presence of a strong → magnetic field, resulting in a He overabundance that typically varies in strength over the stellar surface. Examples include HR 735, HD 184927, and CPD-62°2124.

→ helium; → strong; → line.

A type of → heat transfer by means of molecular agitation within a material without any motion of the material as a whole.

→ heat; → conduction.

A type of → heat transfer involving mass motion of a fluid such as air or water when the heated fluid is caused to move away from the source of heat, carrying energy with it.

→ heat; → convection.

The amount of heat energy required to transform an amount of a substance from the liquid phase to the gas phase. → molar heat of vaporization.

→ heat; → vaporization.