Energy functional
Encyclopedia
Energy functional is total energy of a certain system, as a function
of the system's state.
In the energy methods
of simulating the dynamics of complex structures, a state of the system is often described as an element of an appropriate function space
. To be in this state, the system pays a certain cost in terms of energy required by the state. This energy is a scalar quantity, a function of the state, hence the term functional. The system tends to develop from the state with higher energy (higher cost) to the state with lower energy, thus local minima of this functional are usually related to the stable stationary states. Studying such states is part of the optimization problems
, where the terms energy functional or cost functional are often used to describe the objective function.
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
of the system's state.
In the energy methods
Energy principles in structural mechanics
Energy principles in structural mechanics express the relationships between stresses, strains or deformations, displacements, material properties, and external effects in the form of energy or work done by internal and external forces...
of simulating the dynamics of complex structures, a state of the system is often described as an element of an appropriate function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
. To be in this state, the system pays a certain cost in terms of energy required by the state. This energy is a scalar quantity, a function of the state, hence the term functional. The system tends to develop from the state with higher energy (higher cost) to the state with lower energy, thus local minima of this functional are usually related to the stable stationary states. Studying such states is part of the optimization problems
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, where the terms energy functional or cost functional are often used to describe the objective function.
Examples
- In Hamiltonian systemHamiltonian systemIn physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s, the energy functional is given by the hamiltonianHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
.
See also
- Action (physics)Action (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
- Density functional theoryDensity functional theoryDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
- Hamilton's principleHamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
- History of variational principles in physicsHistory of variational principles in physicsA variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum of a function or functional...
- Potential wellPotential wellA potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy because it is captured in the local minimum of a potential well...
External links
- Energy methods in elasticity, WikiversityWikiversityWikiversity is a Wikimedia Foundation project, which supports learning communities, their learning materials, and resulting activities. It differs from more structured projects such as Wikipedia in that it instead offers a series of tutorials, or courses, for the fostering of learning, rather than...
.