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In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. The Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta. The Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, and can be done to simplify the problem. It also has the consequence that the Routhian equations are exactly the Hamiltonian equations for some coordinates and corresponding momenta, and the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations. Often the Routhian approach may offer no new advantage, but one notable case where this is useful is when a system has cyclic coordinates (also called "ignorable coordinates"), by definition those coordinates which do not appear in the original Lagrangian. The Lagrangian equations are powerful results, used frequently in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. Nevertheless, the Hamiltonian equations are perfectly suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates. The Routhian approach has the best of both approaches, because cyclic coordinates can be split off to the Hamiltonian equations and eliminated, leaving behind the non cyclic coordinates to be solved from the Lagrangian equations. Overall fewer equations need to be solved compared to the Lagrangian approach. Moreover, the Routhian method directly makes clearer the physical interpretations of the constants associated with the cyclic coordinates, in the Lagrangian approach the constants are less obvious. As with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, and introduces no new physics. It offers an alternative way to solve mechanical problems. ==Definitions== In the case of Lagrangian mechanics, the generalized coordinates , ... and the corresponding velocities , and possibly time〔The coordinates are functions of time, so the Lagrangian always has implicit time-dependence via the coordinates. If the Lagrangian changes with time irrespective of the coordinates, usually due to some time-dependent potential, then the Lagrangian is said to have "explicit" time-dependence. Similarly for the Hamiltonian and Routhian functions.〕 , enter the Lagrangian, : where the overdots denote time derivatives. In Hamiltonian mechanics, the generalized coordinates and the corresponding generalized momenta and possibly time, enter the Hamiltonian, : where the second equation is the definition of the generalized momentum corresponding to the coordinate (partial derivatives are denoted using ). The velocities are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, is said to be the momentum "canonically conjugate" to . The Routhian is intermediate between and ; some coordinates are chosen to have corresponding generalized momenta , the rest of the coordinates to have generalized velocities , and time may appear explicitly; where again the generalized velocity is to be expressed as a function of generalized momentum via its defining relation. The choice of which coordinates are to have corresponding momenta, out of the coordinates, is arbitrary. The above is used by Landau and Lifshitz, and Goldstien. Some authors may define the Routhian to be the negative of the above definition. Given the length of the general definition, a more compact notation is to use boldface for tuples (or vectors) of the variables, thus , , , and , so that : where · is the dot product defined on the tuples, for the specific example appearing here: : 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「Routhian mechanics」の詳細全文を読む スポンサード リンク
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