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Scientific principles enabling the use of the calculus of variations
concept of a variational principle emerged from earlier work like Fermat's principle for optics in 1662. The first application of the variational technique
Variational_principle
In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions
Ekeland's variational principle
Ekeland's_variational_principle
Principle in mathematical physics
Herglotz's variational principle, named after German mathematician and physicist Gustav Herglotz, is an extension of the Hamilton's principle, where the
Herglotz's variational principle
Herglotz's_variational_principle
Mathematics of surface waves on a fluid
In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface
Luke's_variational_principle
Equation of stability for a star
In astrophysics, Chandrasekhar's variational principle provides the stability criterion for a static barotropic star, subjected to radial perturbation
Chandrasekhar's variational principle
Chandrasekhar's_variational_principle
Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions
Schwinger variational principle
Schwinger_variational_principle
Differential calculus on function spaces
principle Luke's variational principle Mountain pass theorem Category:Variational analysts Measures of central tendency as solutions to variational problems Stampacchia
Calculus_of_variations
In physics, a variational principle is a method for describing the state or dynamics of a physical system, by identifying it as an extremum (minimum,
History of variational principles in physics
History_of_variational_principles_in_physics
Fundamental mechanical principles
Feynman's integral method was not a variational principle but reduces to the classical least action principle; it led to his Feynman diagrams. Schwinger's
Action_principles
Formulation of the principle of stationary action
It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian,
Hamilton's_principle
Concept in Hlibert spaces mathematics
In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator
Trace_inequality
Hypothesis in neuroscience
parameterise the variational density (for a fixed entropy variational density). This relates free energy minimization to the principle of minimum redundancy
Free_energy_principle
Quantum chemistry algorithm
the Lagrangian and the McLanchlan Variational Principle turn into the Dirac-Frenkel Variational Principle: ⟨ δ Ψ | i ∂ ∂ t − H | Ψ ⟩ = 0 {\displaystyle
Multi-configuration time-dependent Hartree
Multi-configuration_time-dependent_Hartree
Approximating method in quantum mechanics
wavefunctions such as molecular orbitals. The basis for this method is the variational principle. The method consists of choosing a "trial wavefunction" depending
Variational method (quantum mechanics)
Variational_method_(quantum_mechanics)
engineering thermodynamics. A variational principle in physics, such as the principle of least action or Fermat's principle in optics, allows one to describe
Gouy–Stodola_theorem
Candidate unified theory of physics
via a variational principle, the so-called causal action principle. Since one works with different basic objects, the causal action principle has a novel
Causal_fermion_systems
Principle relating to fluid dynamics
Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics. Bernoulli developed his principle from observations
Bernoulli's_principle
Conceptual parallel between optics and classical mechanics
wavefronts characteristic of a full wave equation, resulting from the variational principle, leads to the corresponding differential equations. The propagation
Hamilton's optical-mechanical analogy
Hamilton's_optical-mechanical_analogy
Approach to quantum theory
Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory. This theory was introduced by Julian Schwinger
Schwinger's quantum action principle
Schwinger's_quantum_action_principle
Physical quantity of dimension energy × time
the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are
Action_(physics)
Statistical principle about ratio of effects to causes
The Pareto principle (also known as the 80:20 rule, the law of the vital few and the principle of factor sparsity) states that, for many outcomes, roughly
Pareto_principle
French mathematician and lawyer (1601–1665)
Fermat showed mathematically why this was the case. The first variational principle in physics was articulated by Euclid in his Catoptrica. It says
Pierre_de_Fermat
Light rays follow quickest paths
laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics
Fermat's_principle
Debate about credit for general relativity
Hilbert's almost simultaneous derivation of the theory using an elegant variational principle, during a period when the two corresponded frequently, has led to
General relativity priority dispute
General_relativity_priority_dispute
compact set attains its minimum. Results from variational analysis such as Ekeland's variational principle allow us to extend this result of lower semicontinuous
Variational_analysis
French mathematician (born 1944)
analysis, particularly to variational calculus and mathematical optimization. In mathematical analysis, Ekeland's variational principle, discovered by Ivar
Ivar_Ekeland
Hypothesis that inertial and gravitational masses are equivalent
The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form
Equivalence_principle
mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of surface gravity waves
Clebsch_representation
Principle to predict effects of a change in conditions on a chemical equilibrium
include Chatelier's principle, Braun–Le Chatelier principle, Le Chatelier–Braun principle or the equilibrium law. The principle is named after French
Le_Chatelier's_principle
Theorem in quantum mechanics
to some extent trivial, consequence of the variational principle (the Rayleigh–Ritz variational principle) from which the Schrödinger equation may be
Hellmann–Feynman_theorem
Method of hydrodynamics simulation
standard equation of state pressure with a density constraint and apply a variational time integrator R. Hoetzlein, 2012, develops efficient GPU-based SPH
Smoothed-particle hydrodynamics
Smoothed-particle_hydrodynamics
Belgian quantum physicist (born 1972)
structure to construct a variational ansatz for elementary excitations on top of an MPS and a time-dependent variational principle that unifies real-time
Frank_Verstraete
Statement in classical mechanics
The above equation is often called d'Alembert's principle, but it was first written in this variational form by Joseph Louis Lagrange. D'Alembert's contribution
D'Alembert's_principle
Mathematical methods used in Bayesian inference and machine learning
set of samples, variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. Variational Bayes can be seen
Variational_Bayesian_methods
This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction. Action (physics) Averaged
List_of_variational_topics
Numerical method for solving physical or engineering problems
a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation
Finite_element_method
Japanese mathematician (1932–2022)
states in thermodynamic equilibrium with, on the other hand, the variational principle for quantum mechanical spin systems on lattices. With Yanase he
Huzihiro_Araki
Problem of finding the best feasible solution
(complexity) – Type of computational problem Design Optimization Ekeland's variational principle Function problem – Type of computational problem Glove problem –
Optimization_problem
Approximation method in quantum physics
could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle
Hartree–Fock_method
1998 science fiction novella by Ted Chiang
inspiration for "Story of Your Life" came from his fascination in the variational principle in physics. In the early 1990s, after seeing American actor Paul
Story_of_Your_Life
in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action ∫ V e [ 1 2 ε T C ε − σ T ε +
Hu–Washizu_principle
Configurations of a system that do or do not satisfy classical equations of motion
instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell
On_shell_and_off_shell
Concept in object-oriented programming
In object-oriented programming, the open–closed principle (OCP) states "software entities (classes, modules, functions, etc.) should be open for extension
Open–closed_principle
Physics phenomenon and formula
sea bed at z = − h ( x , y ) , {\displaystyle z=-h(x,y),} Luke's variational principle δ L = 0 {\displaystyle \delta {\mathcal {L}}=0} uses the Lagrangian
Mild-slope_equation
Foundational principle in quantum physics
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is
Uncertainty_principle
Design principle preferring simplicity
with aircraft engineer Kelly Johnson. The term "KISS principle" was in popular use by 1970. Variations on the phrase (usually as some euphemism for the more
KISS_principle
an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally
Time-dependent variational Monte Carlo
Time-dependent_variational_Monte_Carlo
filtering scheme for nonlinear state-space models. It is based on a variational principle of least action, formulated in generalized coordinates of motion
Generalized_filtering
Unified field theory
gravity in free space can be obtained from an action, by applying the variational principle to a certain action. Let M be a (pseudo-)Riemannian manifold, which
Kaluza–Klein_theory
Equations for correlation functions in QFT
diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively
Schwinger–Dyson_equation
Method for describing the electronic structure of molecules using quantum mechanics
equation into the Schrödinger equation and applying the variational principle. The variational principle is a mathematical technique used in quantum mechanics
Molecular_orbital_theory
American theoretical physicist (1918–1988)
"Difficulties in Applying the Variational Principle to Quantum Field Theories", in Polley, L.; Pottinger, D. E. L. (eds.), Variational Calculations in Quantum
Richard_Feynman
Principle used in information theory
In information theory, the principle of minimum Fisher information (MFI) is a variational principle which, when applied with the proper constraints needed
Minimum_Fisher_information
fixed-point theorem modifies the ε {\displaystyle \varepsilon } -variational principle of Ekeland (1974, 1979). The conclusion of Caristi's theorem is
Caristi_fixed-point_theorem
Number representing system complexity
number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy
Topological_entropy
Hungarian-American mathematician (1893–1974)
electromagnetism in terms of quaternions and applied a relativistic variational principle. He sent a copy of his thesis to Albert Einstein, who replied, "I
Cornelius_Lanczos
Statistical principle
Ken A. (2015-08-06). "Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics". The Journal of Chemical
Principle_of_maximum_caliber
Theoretical attempts to unify the forces of nature
equations (fundamental laws of physics) were generally derived from a variational principle expressed in terms of the Riemann curvature tensor for the presumed
Classical unified field theories
Classical_unified_field_theories
Textbook by David J. Griffiths
Chapter 7: Time-independent Perturbation Theory Chapter 8: The Variational Principle Chapter 9: The WKB Approximation Chapter 10: Scattering Chapter
Introduction to Quantum Mechanics (book)
Introduction_to_Quantum_Mechanics_(book)
British neuroscientist
brain. Mathematical contributions include Variational Laplace and generalized filtering, which use variational Bayesian methods for time-series analysis
Karl_J._Friston
Method of numerical integration
derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving and symplectic. Consider a mechanical
Variational_integrator
Soviet mathematician (1899–1981)
in topology and differential geometry, to which he applied the variational principle. Using the theory he introduced, together with Lev Schnirelmann
Lazar_Lyusternik
Proposed theories of gravity
one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are where N j i = 1 2 γ α β ( g
Bimetric_gravity
Concept in general relativity
proposed by David Hilbert in 1915 as part of his application of the variational principle to a combination of gravity and electromagnetism. Deriving equations
Einstein–Hilbert_action
Buoyancy principle in fluid dynamics
Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the
Archimedes'_principle
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
Heine–Borel theorem Completeness of the real numbers Ekeland's variational principle Bartle and Sherbert 2000, p. 78 (for R). Fitzpatrick 2006, p. 52
Bolzano–Weierstrass_theorem
Computational quantum mechanical modelling method to investigate electronic structure
results agree (reasonably) well on comparison to experiment. A variational principle is used to determine the equilibrium density. It can be shown that
Density_functional_theory
Result in dynamical systems
2017, retrieved March 29, 2017 Percival, I C (1979-03-01). "A variational principle for invariant tori of fixed frequency". Journal of Physics A: Mathematical
Kolmogorov–Arnold–Moser theorem
Kolmogorov–Arnold–Moser_theorem
virial equations Chandrasekhar's X- and Y-function Chandrasekhar's Variational Principle Chandra X-ray Observatory 1958 Chandra Himalayan Chandra Telescope
List of things named after Subrahmanyan Chandrasekhar
List_of_things_named_after_Subrahmanyan_Chandrasekhar
Experimental technology level
solved in polynomial time on quantum devices. VQE operates on the variational principle of quantum mechanics, which states that the expectation value of
Noisy intermediate-scale quantum computing
Noisy_intermediate-scale_quantum_computing
Fundamental principle of physics
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli
Superposition_principle
Theorem in complex analysis
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles
Argument_principle
Statement based on repeated empirical observations that describes some natural phenomenon
optics is based on a variational principle: light travels from one point in space to another in the shortest time. Fermat's principle In geometric optics
Scientific_law
Principle of least length in physics
a special case of the more generally stated principle of least action. Using the calculus of variations, it results in an integral equation formulation
Maupertuis's_principle
Formulation in classical mechanics
The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other
Gauss's principle of least constraint
Gauss's_principle_of_least_constraint
Smallest convex set containing a given set
Subderivative Main results (list) Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality
Convex_hull
Lower bound on the log-likelihood of some observed data
In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound or negative variational
Evidence_lower_bound
representation Schwinger reversed-phase coupler Schwinger variational principle Schwinger's quantum action principle Schwinger–Dyson equation Schwinger–Tomonaga equation
List of things named after Julian Schwinger
List_of_things_named_after_Julian_Schwinger
Concept in physics and mathematics
are an example of this. Calculus of variations Mountain pass theorem Principle of least action Variational principle Gradient descent Natural lines of drift
Path_of_least_resistance
Geometric algebra approach to gravity
identical to the Einstein field equations are derivable from a variational principle. A spin tensor can also be supported in a manner similar to
Gauge_theory_gravity
Metric geometry
topological vector space – Structure in functional analysis Ekeland's variational principle Knaster–Tarski theorem – Theorem in order and lattice theory Spherical
Complete_metric_space
Description of the dynamics of magnetization in a solid
from exchange, anisotropy, and magnetoelastic effects. Using the variational principle, the equations of motion can be recast into a Lagrangian form. This
Landau–Lifshitz–Gilbert equation
Landau–Lifshitz–Gilbert_equation
Physics theory
equation. The choice of principle to derive the equations of the CMT. Either the reciprocity theorem or the variational principle have been used. The choice
Coupled_mode_theory
Light motion in curved spacetime
particle and the gravitational field. Standard variational procedure according to Hamilton's principle is applied to action S = ∫ μ b μ a L d μ = − ∫
Fermat's and energy variation principles in field theory
Fermat's_and_energy_variation_principles_in_field_theory
Statement relating differentiable symmetries to conserved quantities
It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies
Noether's_theorem
2014 puzzle video game
The Talos Principle is a 2014 puzzle video game developed by Croteam and published by Devolver Digital. It was simultaneously released on Linux, OS X
The_Talos_Principle
Hypothesis about sapient life and the universe
In cosmology and philosophy of science, the anthropic principle (also known as the observation selection effect) is the proposition that the range of
Anthropic_principle
Music theory of harmony
emphasizing principle chord tones so as not to disrupt the chord progression of the song. Homophonic parallelism is also affected by this variation principle. With
Traditional sub-Saharan African harmony
Traditional_sub-Saharan_African_harmony
Specific sequence of moves in a game
A variation can refer to a specific sequence of successive moves in a turn-based game, often used to specify a hypothetical future state of a game that
Variation_(game_tree)
Luke's variational principle Hamiltonian field theory Nevir & Blender 1993 Blender & Badin 2015 Badin, Gualtiero; Crisciani, Fulvio (2018). Variational Formulation
Hamiltonian_fluid_mechanics
Solution with prescribed norm
these problems. For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer
Normalized solution (mathematics)
Normalized_solution_(mathematics)
Value representing energy density of space
"self-gauging", and Erwin Schrödinger's pure-affine theory using a simple variational principle produced the field equation with a cosmological term. In 1990s,
Cosmological_constant
Interaction between electrons, often complicating physical calculations
the weighting factors c I {\displaystyle c_{I}} according to the Variational Principle. When taking all possible excited determinants, one speaks of Full-CI
Electronic_correlation
Atom of helium
{1}{3}}} of the electric charge. To obtain a more accurate energy the variational principle can be applied to the electron-electron potential Vee using the
Helium_atom
Type of partial differential equation
{\displaystyle -\nabla ^{2}u=f,\qquad u|_{\partial \Omega }=g} satisfy a variational principle, that is to say they minimize the functional E [ u ] = 1 2 ∫ Ω |
Free_boundary_problem
Proposed theories of gravity
theories described here have: an 'action' (see the principle of least action, a variational principle based on the concept of action) a Lagrangian density
Alternatives to general relativity
Alternatives_to_general_relativity
Torus-shaped vortex in a fluid
phenomenon was provided in terms of energy maximisation invoking a variational principle first reported by Kelvin and later proven by Benjamin (1976), or
Vortex_ring
Philosophical problem-solving principle
problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. It is also known as the principle of parsimony
Occam's_razor
Work done by a force to move a particle along a virtual displacement
K. Variational Methods in Elasticity and Plasticity, Pergamon Pr, 1982. ISBN 0-08-026723-8 Wunderlich, W. Mechanics of Structures: Variational and Computational
Virtual_work
Canadian-American mathematician (1925–2020)
in textbooks. In 1991, Brezis and Nirenberg showed how Ekeland's variational principle could be applied to extend the mountain pass theorem, with the effect
Louis_Nirenberg
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