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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is ''vectorial mechanics''. By contrast, analytical mechanics uses ''scalar'' properties of motion representing the system as a whole—usually its total kinetic energy and its potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Analytical mechanics takes advantage of a system's ''constraints'' to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows mechanical problems to be solved with greater efficiency than fully vectorial methods. Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities), and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities, and momenta, therefore both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the very general and deep result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics compared to Newtonian mechanics. Rather it is a collection of equivalent formalisms which are very general. In fact the generality is such that the same principles and formalisms can be used in relativistic mechanics and general relativity, and with some modification, quantum mechanics and quantum field theory also. Analytical mechanics is used widely, from fundamental physics to applied mathematics, particularly chaos theory. The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics. ==Intrinsic motion== ;Generalized coordinates and constraints In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a body's position during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as ''generalized coordinates'', denoted ''qi'' (''i'' = 1, 2, 3...).〔''The Road to Reality'', Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1〕 Difference between curvillinear and generalized coordinates Generalized coordinates incorporate constraints on the system. There is one generalized coordinate ''qi'' for each degree of freedom (for convenience labelled by an index ''i'' = 1, 2...''N''), i.e. each way the system can change its configuration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of ''curvilinear'' coordinates equals the dimension of the position space in question (usually 3 for 3d space), while the number of ''generalized'' coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:〔''Analytical Mechanics'', L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0〕 :''(of position space (usually 3) ) × (of constituents of system ("particles") ) − (number of constraints)'' :''= (number of degrees of freedom) = (number of generalized coordinates)'' For a system with ''N'' degrees of freedom, the generalized coordinates can be collected into an ''N''-tuple: : and the time derivative (here denoted by an overdot) of this tuple give the ''generalized velocities'': :. ;D'Alembert's principle The foundation which the subject is built on is ''D'Alembert's principle''. This principle states that infinitesimal ''virtual work'' done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful - since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is: : where : are the generalized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and q are the generalized coordinates. This leads to the generalized form of Newton's laws in the language of analytical mechanics: : is a useful shorthand (see matrix calculus for this notation). Holonomic constraints If the curvilinear coordinate system is defined by the standard position vector r, and if the position vector can be written in terms of the generalized coordinates q and time ''t'' in the form: : and this relation holds for all times ''t'', then q are called ''Holonomic constraints''.〔McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3〕 Vector r is explicitly dependent on ''t'' in cases when the constraints vary with time, not just because of q(''t''). For time-independent situations, the constraints are also called scleronomic, for time-dependent cases they are called rheonomic.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「analytical mechanics」の詳細全文を読む スポンサード リンク
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