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In the case of time-independent and , i.e. / = / = , Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from ...
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. [4] [5] [6] The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. [7] In discrete-time problems, the analogous difference equation is usually referred to as the ...
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.
Together, the state and costate equations describe the Hamiltonian dynamical system (again analogous to but distinct from the Hamiltonian system in physics), the solution of which involves a two-point boundary value problem, given that there are boundary conditions involving two different points in time, the initial time (the differential ...
Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read ˙ = and ˙ =, where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation , which is one of the reasons H {\displaystyle H} is also called the Hamiltonian. Given the state at some initial time ( t = 0 {\displaystyle t=0} ), we can solve it to obtain the state at any subsequent time.