![<span class="mw-page-title-main">Electroweak interaction</span> Unified description of electromagnetism and the weak interaction](https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Standard_Model_of_Elementary_Particles.svg/320px-Standard_Model_of_Elementary_Particles.svg.png)
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 246 GeV, they would merge into a single force. Thus, if the temperature is high enough – approximately 1015 K – then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch (shortly after the Big Bang), the electroweak force split into the electromagnetic and weak force. It is thought that the required temperature of 1015 K has not been seen widely throughout the universe since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around 5.5x1012 K (from the Large Hadron Collider).
![<span class="mw-page-title-main">Kepler's laws of planetary motion</span> Laws describing the motion of planets](https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kepler_laws_diagram.svg/320px-Kepler_laws_diagram.svg.png)
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:
- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
![<span class="mw-page-title-main">Orbit</span> Curved path of an object around a point](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Animation_of_C-2018_Y1_orbit_1600-2500.gif/320px-Animation_of_C-2018_Y1_orbit_1600-2500.gif)
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.
![<span class="mw-page-title-main">Projectile motion</span> Motion of launched objects due to gravity](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/ParabolicWaterTrajectory.jpg/320px-ParabolicWaterTrajectory.jpg)
Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive and negligible. The curved path of objects in projectile motion was shown by Galileo to be a parabola, but may also be a straight line in the special case when it is thrown directly upward or downward. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards the Earth’s center of mass. Because of the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics.
In celestial mechanics, the specific relative angular momentum of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question.
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.
The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.
![<span class="mw-page-title-main">Spacecraft flight dynamics</span> Application of mechanical dynamics to model the flight of space vehicles](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Apollo-Moon-mission-profile.png/320px-Apollo-Moon-mission-profile.png)
Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
![<span class="mw-page-title-main">Hopf bifurcation</span> Critical point where a periodic solution arises](https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Hopfeigenvalues.png/320px-Hopfeigenvalues.png)
In the mathematical theory of bifurcations, a Hopfbifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis as a parameter crosses a threshold value. Under reasonably generic assumptions about the dynamical system, the fixed point becomes a small-amplitude limit cycle as the parameter changes.
A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
In fluid dynamics, Taylor scraping flow is a type of two-dimensional corner flow occurring when one of the wall is sliding over the other with constant velocity, named after G. I. Taylor.
In fluid dynamics, Landau–Squire jet or Submerged Landau jet describes a round submerged jet issued from a point source of momentum into an infinite fluid medium of the same kind. This is an exact solution to the incompressible form of the Navier-Stokes equations, which was first discovered by Lev Landau in 1944 and later by Herbert Squire in 1951. The self-similar equation was in fact first derived by N. A. Slezkin in 1934, but never applied to the jet. Following Landau's work, V. I. Yatseyev obtained the general solution of the equation in 1950.
In plasma physics and magnetic confinement fusion, neoclassical transport or neoclassical diffusion is a theoretical description of collisional transport in toroidal plasmas, usually found in tokamaks or stellerators. It is a modification of classical diffusion adding in effects of non-uniform magnetic fields due to the toroidal geometry, which give rise to new diffusion effects.
Chandrasekhar–Page equations describe the wave function of the spin-½ massive particles, that resulted by seeking a separable solution to the Dirac equation in Kerr metric or Kerr–Newman metric. In 1976, Subrahmanyan Chandrasekhar showed that a separable solution can be obtained from the Dirac equation in Kerr metric. Later, Don Page extended this work to Kerr–Newman metric, that is applicable to charged black holes. In his paper, Page notices that N. Toop also derived his results independently, as informed to him by Chandrasekhar.