Analytical mechanics

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Analytical mechanics

Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics

Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton

Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with analytical mechanics. This distinction makes sense because analytical mechanics uses two scalar properties of motion, the kinetic and potential energies, instead of vector forces, to analyze the motion.

The subject has two parts: Lagrangian mechanics

Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italy mathematician Lagrange in 1788....
and Hamiltonian mechanics

Hamiltonian mechanics

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Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics

Classical mechanics

Isaac Newton

Lagrangian mechanics

Hamiltonian mechanics

Hamiltonian 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 recourse to Lagrangian mechanics using sym...
. The Lagrangian formulation identifies the actual path followed by the motion as a selection of the path over which the time integral of kinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit. The Hamiltonian formulation is more general, allowing time-varying energy, identifying the path followed to be the one with least action
Principle of least action

In physics, the principle of least action or more accurately principle of stationary action is a variational principle which, when applied to the action of a mechanics system, can be used to obtain the equations of motion for that system....
(the integral over the path of the difference between kinetic and potential energies), holding the departure and arrival times fixed. These approaches underlie the path integral formulation

Path integral formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a probability amplitude....
of quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
.

It began with d'Alembert's principle

D'Alembert's principle

D'Alembert's principle, also known as the Lagrange-D'Alembert principle, is a statement of the fundamental classical physics laws of motion....
. By analogy with Fermat's principle

Fermat's principle

In optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light....
, which is the variational principle

Variational principle

A variational principle is a principle in physics whichis expressed in terms of the calculus of variations.According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is Self-adjoint_operator....
in geometric optics, Maupertuis' principle

Maupertuis' principle

In classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path....
was discovered in classical mechanics.

Using generalized coordinates, we obtain Lagrange's equations. Using the Legendre transformation

Legendre transformation

In mathematics, it is often desirable to express a functional relationship as a different function, whose argument is the derivative of f , rather than x ....
, we obtain generalized momentum and the Hamiltonian

Hamiltonian mechanics

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential equation and integral equations, and some problems may be formulated either way....
, while Lagrange's equation provides differential equations. Finally we may derive the Hamilton–Jacobi equation.

The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a Closed and exact differential forms, nondegenerate form, differential form, ?, called the symplectic form....
s and symplectic topology

Symplectic topology

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, nondegenerate form differential form....
. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curve

Integral curve

In mathematics, an integral curve for a vector field defined on a manifold is a curve in the manifold whose tangent vector at each point along the curve is the vector field itself at that point....
s of Hamiltonian vector field

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....
s.

Analytical mechanics

See also