|
Effective Potential
The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the "opposing" centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets (both Newtonian and relativistic) and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions. Definition The basic form of potential U_\text is defined as U_\text(\mathbf) = \frac + U(\mathbf), where : ''L'' is the angular momentum, : ''r'' is the distance between the two masses, : ''μ'' is the reduced mass of the two bodies (approximately equal to the mass of the orbiting body if one mass is much larger than the other), : ''U''(''r'') is the general form of the potential. The effective force, then, is the negative gradient of the effective potential: \begin \mathbf_\text &= -\nabla U_\text(\mathbf) \\ &= \fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple release of energy by objects to the realization of abilities in people. The philosopher Aristotle incorporated this concept into his theory of potentiality and actuality (in Greek, ''dynamis'' and ''energeia''), translated into Latin as ''potentia'' and ''actualitas'' (earlier also ''possibilitas'' and ''efficacia''). a pair of closely connected principles which he used to analyze motion, causality, ethics, and physiology in his ''Physics'', ''Metaphysics'', ''Nicomachean Ethics'', and '' De Anima'', which is about the human psyche. That which is potential can theoretically be made actual by taking the right action; for example, a boulder on the edge of a cliff has potential to fall that could be actualized by pushing it over the edge. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a Hamilton–Jacobi equation, link between classical and quantum mechanics. Overview Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' Let (M, \mathcal L) be a Lagrangian mechanics, mechanical system with configuration space (physics), configuration space M and smooth Lagrangian_mechanics#Lagrangian, Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geopotential
Geopotential (symbol ''W'') is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation. In addition to the actual potential (the geopotential), a theoretical normal potential (symbol ''U'') and their difference, the disturbing potential (), can also be defined. Concepts For geophysical applications, gravity is distinguished from gravitation. Gravity is defined as the resultant force of gravitation and the centrifugal force caused by the Earth's rotation. Likewise, the respective scalar potentials, gravitational potential and centrifugal potential, can be added to form an effective potential called the geopotential, W. The surfaces of constant geopotential or isosurfaces of the geopotential are called ''equigeopotential surfaces'' (sometimes a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physics), work needed to move a test charge from a reference point to a specific point in a static electric field. The test charge used is small enough that disturbance to the field is unnoticeable, and its motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is Earth (electricity), earth or a point at infinity, although any point can be used. In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar (physics), scalar quantity denoted by or occasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916. According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum (non-rotating). A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanical Equilibrium
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero. In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent. * In terms of momentum, a system is in equilibrium if the momentum of its parts is all constant. * In terms of velocity, the system is in equilibrium if velocity is constant. * In a rotational mechanical equilibrium the angular momentum of the object is conserved and the net torque is zero. More generally in conservative systems, equilibrium is established at a point in Configuration space (physics), configuration space where the gradient of the potential energy with respect to the generalized coordinates is zero. If a particle in equilibrium has zero velocity, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It is also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant, denoted by the capital letter . In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse-square law, inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor). The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately The modern notation of Newton's law involving was introduced i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates (spins or revolves) around an axis of rotation and how fast the axis itself changes direction. The magnitude of the pseudovector, \omega=\, \boldsymbol\, , represents the '' angular speed'' (or ''angular frequency''), the angular rate at which the object rotates (spins or revolves). The pseudovector direction \hat\boldsymbol=\boldsymbol/\omega is normal to the instantaneous plane of rotation or angular displacement. There are two types of angular velocity: * Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. * Spin angular velocity refers to how fast a rigid body rotates around a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and light. Energy is a Conservation law, conserved quantity—the law of conservation of energy states that energy can be Energy transformation, converted in form, but not created or destroyed. The unit of measurement for energy in the International System of Units (SI) is the joule (J). Forms of energy include the kinetic energy of a moving object, the potential energy stored by an object (for instance due to its position in a Classical field theory, field), the elastic energy stored in a solid object, chemical energy associated with chemical reactions, the radiant energy carried by electromagnetic radiation, the internal energy contained within a thermodynamic system, and rest energy associated with an object's rest mass. These are not mutual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Points Equipotential
Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set ** Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem * Lagrangian, a functional whose extrema are to be determined in the calculus of variations * Lagrangian submanifold, a class of submanifolds in symplectic geometry * Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density Physics * Lagrangian mechanics, a formulation of classical mechanics * Lagrangian (field theory), a formalism in classical field theory * Lagrangian point, a position in an orbital configuration of two large bodies * Lagrangian coordinates, a way of describing the motions of particles of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restricted Three-Body Problem - Energy Potential Analysis
Restricted may refer to: * R rating (other), list of subjects where "R" stands for "Restricted" *18 rating, media rating designation sometimes called "Restricted" * Restricted (country club), historical use of the term in country clubs in the United States * Restricted airspace, airspace for which air traffic is restricted or prohibited for safety or security concerns * Restricted area, several uses *Restricted free agent, a type of free agent in various professional sports * Restricted list, a roster status in Major League Baseball *Restricted stock Restricted stock, also known as restricted securities, is stock of a company that is not fully transferable (from the stock-issuing company to the person receiving the stock award) until certain conditions (restrictions) have been met. Upon satisfa ..., stock of a company that is not fully transferable See also * * Restrict, keyword in the C programming language * Restriction (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
