|
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ==Overview== In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates , where each component of the coordinate is indexed to the frame of reference of the system. The time evolution of the system is uniquely defined by Hamilton's equations: = -\frac}\\ & \frac} = +\frac} \end |cellpadding= 5 |border |border colour = #0073CF |background colour=#F5FFFA}} where is the Hamiltonian, which often corresponds to the total energy of the system.〔 For a closed system, it is the sum of the kinetic and potential energy in the system. In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it in the Hamiltonian equations. This approach is equivalent to the one used in Lagrangian mechanics. In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems. While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The more degrees of freedom the system has, the more complicated its time evolution is and, in most cases, it becomes chaotic. 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「hamiltonian mechanics」の詳細全文を読む スポンサード リンク
|