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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.
The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. . The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction ...
The analysis of the ... the parameter t corresponds to length or energy, ... The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton ...
By a limiting procedure, Richard Schoen and Shing-Tung Yau used Hamilton's theorem to prove that any finite-energy map from a complete Riemannian manifold to a closed Riemannian manifold of nonpositive curvature can be deformed into a finite-energy harmonic map. [26]
Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations, ˙ = ˙ = ˙ = ˙ = Momentum , which corresponds to the vertical component of angular momentum = ˙ , is a constant of motion. That is a consequence of the rotational symmetry of the ...
for a system of particles at coordinates .The function is the system's Hamiltonian giving the system's energy. The solution of this equation is the action, , called Hamilton's principal function.
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.
Cayley–Hamilton theorem (Linear algebra) Cayley–Salmon theorem (algebraic surfaces) Cayley's theorem (group theory) Central limit theorem (probability) Cesàro's theorem (real analysis) Ceva's theorem ; Chasles's theorems; Chebotarev's density theorem (number theory) Chen's theorem (number theory)