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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lagrangian mechanics ： ウィキペディア英語版 Lagrangian mechanics Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the ''Lagrange equations of the first kind'', which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the ''Lagrange equations of the second kind'', which incorporate the constraints directly by judicious choice of generalized coordinates.〔 In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system. No new physics is introduced by Lagrangian mechanics; it is actually less general than Newtonian mechanics. Newton's laws can include non-conservative forces like friction, however they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces, and some (not all) non-conservative forces, in any coordinate system. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which considerably simplifies describing the dynamics of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, although only as a special case of Noether's theorem. The theory connects with the principle of stationary action, although Lagrangian mechanics is less general because it is restricted to equilibrium problems.〔http://williamhoover.info/Scans1990s/1995-10.pdf〕 Also, Lagrangian mechanics can only be applied to systems with holonomic constraints, because the formulation does not work for nonholonomic constraints. Three examples are when the constraint equations are nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may revert to Newtonian mechanics, or use other methods. The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, ''Hamilton's principle'' that can be used to derive the Lagrange equation was later recognized to be applicable to much of theoretical physics as well. In quantum mechanics, action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either special relativity or general relativity. The action principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system. Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system. Lagrangian mechanics is widely used to solve mechanical problems in physics and engineering when Newton's formulation of classical mechanics is not convenient. Lagrange's equations are also used in optimisation problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. ==Introduction== The strength of Lagrangian mechanics is its ability to handle constrained mechanical systems. The following examples motivate the need for the concepts and terminology used to handle such systems. For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form ''f''(r) = 0, where the position of the bead can be written r = (''x''(''s''), ''y''(''s'')), in which ''s'' is a parameter, the arc length ''s'' along the curve from some point on the wire. Only ''one'' coordinate is needed instead of two, because the position of the bead can be parameterized by one number, ''s'', and the constraint equation connects the two coordinates ''x'' and ''y''; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead. Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively :$f(\backslash mathbf,\; t)\; =\; 0\; \backslash ,,\backslash quad\; \backslash mathbf\; =\; (x(s,\; t),\; y(s,\; t))$ which now both depend on time ''t'' due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates ''and'' explicitly in the constraint equations. Another interesting 2d example is the chaotic double pendulum, again subject to gravity. The length of one pendulum is ''L''1 and the length of the other is ''L''2. Each pendulum bob has a constraint equation, :$f(\backslash mathbf\_1)\; =\; x\_1^2\; +\; y\_1^2\; -\; L\_1^2\; =\; 0\; \backslash ,,\backslash quad\; f(\backslash mathbf\_2)\; =\; x\_2^2\; +\; y\_2^2\; -\; L\_2^2\; =\; 0\; \backslash ,,$ where the positions of the bobs are :$\backslash mathbf\_1\; =\; (x\_1(\backslash theta\_1),\; y\_1(\backslash theta\_1))\; \backslash ,,\backslash quad\; \backslash mathbf\_2\; =\; (x\_2(\backslash theta\_2),\; y\_2(\backslash theta\_2))\; \backslash ,,$ and ''θ''1 is the angle of pendulum 1 from some reference direction, likewise for pendulum 2. Each pendulum can be described by one coordinate since the constraint equation for each connects the two spatial coordinates. For a 3d example, a spherical pendulum with constant length ''l'' free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form :$f(\backslash mathbf)\; =\; x^2\; +\; y^2\; +\; z^2\; -\; l^2\; =\; 0\; \backslash ,,$ where the position of the pendulum bob can be written :$\backslash mathbf\; =\; (x(\backslash theta,\backslash phi),y(\backslash theta,\backslash phi),z(\backslash theta,\backslash phi))\; \backslash ,,$ in which (''θ'', ''φ'') are the spherical polar angles because the bob moves in the surface of a sphere. A logical choice of variables to describe the motion are the angles (''θ'', ''φ''). Notice only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates ''x'', ''y'', ''z'' so any one of them is determined from the other two. For analyzing the small oscillations of multiple coupled simple harmonic oscillators, Lagrangian mechanics is especially natural, since the kinetic and potential energies of the system take a simple form despite the fact there are many particles, and the equations of motion can be derived immediately. For ''N'' particles in 3d space, the position vector of each particle can written as a 3-tuple in Cartesian coordinates :$\backslash mathbf\_1\; =\; (x\_1,y\_1,z\_1)\; \backslash ,,\; \backslash quad\; \backslash mathbf\_2\; =\; (x\_2,y\_2,z\_2)\; \backslash ,,\; \backslash ldots\; \backslash ,,\; \backslash mathbf\_N\; =\; (x\_N,y\_N,z\_N)\backslash ,,$ so overall, there are 3''N'' coordinates to define the configuration of the system. These are all specific points in space to locate the particles, a general point in space is written r = (''x'', ''y'', ''z''). If any or all of the particles are subject to a holonomic constraint, described by a constraint equation of the form ''f''(r, ''t'') = 0, then at any instant of time the position coordinates of those particles are linked together and not independent. If there are ''C'' constraints in the system, then each has a constraint equation, :$f\_1(\backslash mathbf,\; t)\; =\; 0\backslash ,,\backslash quad\; f\_2(\backslash mathbf,\; t)\; =\; 0\; \backslash ,,\; \backslash ldots\; \backslash ,,\; f\_C(\backslash mathbf,\; t)\; =\; 0\; \backslash ,,$ and one coordinate can be eliminated from each constraint equation. The number of independent coordinates is therefore ''n'' = 3''N'' − ''C''. We can transform each position vector to a common set of ''n'' generalized coordinates, conveniently written as an ''n''-tuple q = (''q''1, ''q''2, ... ''qn''), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time, :$\backslash mathbf\_k\; =\; \backslash mathbf\_k(\backslash mathbf,\; t)\; =\; (x\_k(\backslash mathbf,\; t),\; y\_k(\backslash mathbf,\; t),\; z\_k(\backslash mathbf,\; t),\; t)\backslash ,.$ The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is :$\backslash mathbf\_k\; =\; \backslash dot\_k\}\; =\; \backslash left(\backslash frac,\backslash frac,\backslash frac\backslash right)\backslash ,,\; \backslash quad\; \backslash dot\_j\; =\; \backslash frac\; \backslash ,,\; \backslash quad\; \backslash mathbf\_k\; =\; \backslash sum\_^n\; \backslash frac\backslash dot\_j\; +\backslash frac\backslash ,,$ (each overdot indicates a time derivative). In the previous examples, if one tracks each of the massive objects as a particle (bead, pendulum bob, etc.), calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rods). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of ''independent'' generalized coordinates that completely characterize the possible motion of the particle. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. 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「Lagrangian mechanics」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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