Symplectic manifoldIn differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds.
Covariant formulation of classical electromagnetismThe covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another.
Lagrangian mechanicsIn physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space and a smooth function within that space called a Lagrangian. For many systems, where and are the kinetic and potential energy of the system, respectively.
Lagrangian (field theory)Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
Electromagnetic tensorIn electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below.
Hamiltonian field theoryIn theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.
Symplectic integratorIn mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
States' rightsIn American political discourse, states' rights are political powers held for the state governments rather than the federal government according to the United States Constitution, reflecting especially the enumerated powers of Congress and the Tenth Amendment. The enumerated powers that are listed in the Constitution include exclusive federal powers, as well as concurrent powers that are shared with the states, and all of those powers are contrasted with the reserved powers—also called states' rights—that only the states possess.
Noether's theoremNoether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.
Verlet integrationVerlet integration (vɛʁˈlɛ) is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C.
