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The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T ∗ E t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between ...
It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. [1] Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. [2]
The Hamiltonian is defined by = = ˙ ˙ and can be obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables canonically conjugate to the original variables. For example, given a set of generalized coordinates, the variables canonically conjugate are the generalized momenta.
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.
Following are overlapping properties between the Lagrangian and Hamiltonian functions. [5] [8] All the individual generalized coordinates q i (t), velocities q̇ i (t) and momenta p i (t) for every degree of freedom are mutually independent.
The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar, while the Hamiltonian is the time component of a four-vector. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum ...
Hamiltonian mechanics results if the conjugate momenta are introduced in place of ˙ by a Legendre transformation of the Lagrangian into the Hamiltonian defined by (,,) = ˙ (, ˙,). The Hamiltonian is the total energy of the system: H = T + U . {\displaystyle H=T+U.}
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.