Faster access than browser!
69 relations: Action (physics), Action-angle coordinates, Amplitude, Ansatz, Calculus of variations, Canonical transformation, Carl Gustav Jacob Jacobi, Circular polarization, Classical mechanics, Classical Mechanics (Goldstein book), Complex number, Conserved quantity, Constant of motion, Covariance and contravariance of vectors, Differential of a function, Dot product, Dynamical system, Einstein field equations, Electromagnetic four-potential, Ellipse, Elliptic cylindrical coordinates, Energy–momentum relation, Euler–Lagrange equation, Focus (geometry), Four-gradient, Four-momentum, Generalized coordinates, Generating function (physics), Geodesic, Geometry, Gravitational field, Hamilton's principle, Hamilton–Jacobi–Bellman equation, Hamilton–Jacobi–Einstein equation, Hamiltonian mechanics, Hamiltonian vector field, Invariant mass, Isosurface, Johann Bernoulli, Lagrangian mechanics, Lev Landau, Mathematics, Metric tensor, Newton's laws of motion, Nonlinear system, Notation for differentiation, Ordinary differential equation, Orthogonal coordinates, Parabolic cylindrical coordinates, Partial differential equation, ..., Phase (waves), Phase space, Physics, Planck constant, Quadratic function, Quantum chaos, Quantum mechanics, Riemannian geometry, Riemannian manifold, Schrödinger equation, Separation of variables, Speed of light, Spherical coordinate system, Symplectic geometry, Time derivative, Wave, Wave equation, William Rowan Hamilton, WKB approximation. Expand index (19 more) »
Action (physics)
In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived.
New!!: Hamilton–Jacobi equation and Action (physics) · See more »
Action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems.
New!!: Hamilton–Jacobi equation and Action-angle coordinates · See more »
Amplitude
The amplitude of a periodic variable is a measure of its change over a single period (such as time or spatial period).
New!!: Hamilton–Jacobi equation and Amplitude · See more »
Ansatz
In physics and mathematics, an ansatz (meaning: "initial placement of a tool at a work piece", plural ansätze; or ansatzes) is an educated guessIn his book on "The Nature of Mathematical Modelling", Neil Gershenfeld introduces ansatz, with interpretation "a trial answer", to be an important technique for solving differential equations.
New!!: Hamilton–Jacobi equation and Ansatz · See more »
Calculus of variations
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
New!!: Hamilton–Jacobi equation and Calculus of variations · See more »
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations.
New!!: Hamilton–Jacobi equation and Canonical transformation · See more »
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
New!!: Hamilton–Jacobi equation and Carl Gustav Jacob Jacobi · See more »
Circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electric field of the wave has a constant magnitude but its direction rotates with time at a steady rate in a plane perpendicular to the direction of the wave.
New!!: Hamilton–Jacobi equation and Circular polarization · See more »
Classical mechanics
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
New!!: Hamilton–Jacobi equation and Classical mechanics · See more »
Classical Mechanics (Goldstein book)
Classical Mechanics is a textbook about the subject of that name written by Herbert Goldstein.
New!!: Hamilton–Jacobi equation and Classical Mechanics (Goldstein book) · See more »
Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
New!!: Hamilton–Jacobi equation and Complex number · See more »
Conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.
New!!: Hamilton–Jacobi equation and Conserved quantity · See more »
Constant of motion
In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion.
New!!: Hamilton–Jacobi equation and Constant of motion · See more »
Covariance and contravariance of vectors
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.
New!!: Hamilton–Jacobi equation and Covariance and contravariance of vectors · See more »
Differential of a function
In calculus, the differential represents the principal part of the change in a function y.
New!!: Hamilton–Jacobi equation and Differential of a function · See more »
Dot product
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
New!!: Hamilton–Jacobi equation and Dot product · See more »
Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
New!!: Hamilton–Jacobi equation and Dynamical system · See more »
Einstein field equations
The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
New!!: Hamilton–Jacobi equation and Einstein field equations · See more »
Electromagnetic four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived.
New!!: Hamilton–Jacobi equation and Electromagnetic four-potential · See more »
Ellipse
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
New!!: Hamilton–Jacobi equation and Ellipse · See more »
Elliptic cylindrical coordinates
Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction.
New!!: Hamilton–Jacobi equation and Elliptic cylindrical coordinates · See more »
Energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum: holds for a system, such as a particle or macroscopic body, having intrinsic rest mass, total energy, and a momentum of magnitude, where the constant is the speed of light, assuming the special relativity case of flat spacetime.
New!!: Hamilton–Jacobi equation and Energy–momentum relation · See more »
Euler–Lagrange equation
In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.
New!!: Hamilton–Jacobi equation and Euler–Lagrange equation · See more »
Focus (geometry)
In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed.
New!!: Hamilton–Jacobi equation and Focus (geometry) · See more »
Four-gradient
In differential geometry, the four-gradient (or 4-gradient) \mathbf is the four-vector analogue of the gradient \vec from Gibbs–Heaviside vector calculus.
New!!: Hamilton–Jacobi equation and Four-gradient · See more »
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.
New!!: Hamilton–Jacobi equation and Four-momentum · See more »
Generalized coordinates
In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration.
New!!: Hamilton–Jacobi equation and Generalized coordinates · See more »
Generating function (physics)
Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics.
New!!: Hamilton–Jacobi equation and Generating function (physics) · See more »
Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
New!!: Hamilton–Jacobi equation and Geodesic · See more »
Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
New!!: Hamilton–Jacobi equation and Geometry · See more »
Gravitational field
In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body.
New!!: Hamilton–Jacobi equation and Gravitational field · See more »
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action (see that article for historical formulations).
New!!: Hamilton–Jacobi equation and Hamilton's principle · See more »
Hamilton–Jacobi–Bellman equation
The Hamilton–Jacobi–Bellman (HJB) equation is a partial differential equation which is central to optimal control theory.
New!!: Hamilton–Jacobi equation and Hamilton–Jacobi–Bellman equation · See more »
Hamilton–Jacobi–Einstein equation
In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics in superspace, cast in the "geometrodynamics era" around the 1960s, by Asher Peres in 1962 and others.
New!!: Hamilton–Jacobi equation and Hamilton–Jacobi–Einstein equation · See more »
Hamiltonian mechanics
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
New!!: Hamilton–Jacobi equation and Hamiltonian mechanics · See more »
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian.
New!!: Hamilton–Jacobi equation and Hamiltonian vector field · See more »
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system.
New!!: Hamilton–Jacobi equation and Invariant mass · See more »
Isosurface
An isosurface is a three-dimensional analog of an isoline.
New!!: Hamilton–Jacobi equation and Isosurface · See more »
Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.
New!!: Hamilton–Jacobi equation and Johann Bernoulli · See more »
Lagrangian mechanics
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.
New!!: Hamilton–Jacobi equation and Lagrangian mechanics · See more »
Lev Landau
Lev Davidovich Landau (22 January 1908 - April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics.
New!!: Hamilton–Jacobi equation and Lev Landau · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Hamilton–Jacobi equation and Mathematics · See more »
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
New!!: Hamilton–Jacobi equation and Metric tensor · See more »
Newton's laws of motion
Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics.
New!!: Hamilton–Jacobi equation and Newton's laws of motion · See more »
Nonlinear system
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.
New!!: Hamilton–Jacobi equation and Nonlinear system · See more »
Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation.
New!!: Hamilton–Jacobi equation and Notation for differentiation · See more »
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.
New!!: Hamilton–Jacobi equation and Ordinary differential equation · See more »
Orthogonal coordinates
In mathematics, orthogonal coordinates are defined as a set of d coordinates q.
New!!: Hamilton–Jacobi equation and Orthogonal coordinates · See more »
Parabolic cylindrical coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular z-direction.
New!!: Hamilton–Jacobi equation and Parabolic cylindrical coordinates · See more »
Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
New!!: Hamilton–Jacobi equation and Partial differential equation · See more »
Phase (waves)
Phase is the position of a point in time (an instant) on a waveform cycle.
New!!: Hamilton–Jacobi equation and Phase (waves) · See more »
Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
New!!: Hamilton–Jacobi equation and Phase space · See more »
Physics
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
New!!: Hamilton–Jacobi equation and Physics · See more »
Planck constant
The Planck constant (denoted, also called Planck's constant) is a physical constant that is the quantum of action, central in quantum mechanics.
New!!: Hamilton–Jacobi equation and Planck constant · See more »
Quadratic function
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.
New!!: Hamilton–Jacobi equation and Quadratic function · See more »
Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory.
New!!: Hamilton–Jacobi equation and Quantum chaos · See more »
Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
New!!: Hamilton–Jacobi equation and Quantum mechanics · See more »
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.
New!!: Hamilton–Jacobi equation and Riemannian geometry · See more »
Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
New!!: Hamilton–Jacobi equation and Riemannian manifold · See more »
Schrödinger equation
In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.
New!!: Hamilton–Jacobi equation and Schrödinger equation · See more »
Separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
New!!: Hamilton–Jacobi equation and Separation of variables · See more »
Speed of light
The speed of light in vacuum, commonly denoted, is a universal physical constant important in many areas of physics.
New!!: Hamilton–Jacobi equation and Speed of light · See more »
Spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
New!!: Hamilton–Jacobi equation and Spherical coordinate system · See more »
Symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
New!!: Hamilton–Jacobi equation and Symplectic geometry · See more »
Time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function.
New!!: Hamilton–Jacobi equation and Time derivative · See more »
Wave
In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport.
New!!: Hamilton–Jacobi equation and Wave · See more »
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.
New!!: Hamilton–Jacobi equation and Wave equation · See more »
William Rowan Hamilton
Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.
New!!: Hamilton–Jacobi equation and William Rowan Hamilton · See more »
WKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients.
New!!: Hamilton–Jacobi equation and WKB approximation · See more »
Redirects here:
HJE, Hamilton equation, Hamilton jacobi equation, Hamilton's Characteristic Function, Hamilton's characteristic function, Hamilton's principal function, Hamilton's principle function, Hamilton-Jacobi Equation, Hamilton-Jacobi Equations, Hamilton-Jacobi equation, Hamilton-Jacobi equations, Hamilton-Jacobi equations of motion, Hamilton-Jacobi theory, Hamilton–Jacobi equations, Hamilton–Jacobi theory, Staeckel conditions.
References
[1] https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation
