Initial conditionIn mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
Stability theoryIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
Equations of motionIn physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system.
Q factorIn physics and engineering, the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results.
Tautochrone curveA tautochrone curve or isochrone curve (from Greek prefixes tauto- meaning same or iso- equal, and chrono time) is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
Kater's pendulumA Kater's pendulum is a reversible free swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity. Its advantage is that, unlike previous pendulum gravimeters, the pendulum's centre of gravity and center of oscillation do not have to be determined, allowing a greater accuracy. For about a century, until the 1930s, Kater's pendulum and its various refinements remained the standard method for measuring the strength of the Earth's gravity during geodetic surveys.
French Geodesic Mission to the EquatorThe French Geodesic Mission to the Equator (Expédition géodésique française en Équateur), also called the French Geodesic Mission to Peru and the Spanish-French Geodesic Mission, was an 18th-century expedition to what is now Ecuador carried out for the purpose of performing an arc measurement, measuring the length of a degree of latitude near the Equator, by which the Earth radius can be inferred. The mission was one of the first geodesic (or geodetic) missions carried out under modern scientific principles, and the first major international scientific expedition.
Giovanni Battista RiccioliGiovanni Battista Riccioli, SJ (17 April 1598 – 25 June 1671) was an Italian astronomer and a Catholic priest in the Jesuit order. He is known, among other things, for his experiments with pendulums and with falling bodies, for his discussion of 126 arguments concerning the motion of the Earth, and for introducing the current scheme of lunar nomenclature. He is also widely known for discovering the first double star. He argued that the rotation of the Earth should reveal itself because on a rotating Earth, the ground moves at different speeds at different times.
Generalized coordinatesIn analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to Cartesian coordinates.
ClockworkClockwork refers to the inner workings of either mechanical devices called clocks and watches (where it is also called the movement) or other mechanisms that work similarly, using a series of gears driven by a spring or weight. A clockwork mechanism is often powered by a clockwork motor consisting of a mainspring, a spiral torsion spring of metal ribbon. Energy is stored in the mainspring manually by winding it up, turning a key attached to a ratchet which twists the mainspring tighter.
