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This inequality is known as Hamilton's rule after W. D. Hamilton who in 1964 published the first formal quantitative treatment of kin selection. [ 2 ] [ 3 ] The relatedness parameter ( r ) in Hamilton's rule was introduced in 1922 by Sewall Wright as a coefficient of relationship that gives the probability that at a random locus , the alleles ...
In work prior to Nowak et al. (2010), various authors derived different versions of a formula for , all designed to preserve Hamilton's rule. [ 34 ] [ 38 ] [ 39 ] Orlove noted that if a formula for r {\displaystyle r} is defined so as to ensure that Hamilton's rule is preserved, then the approach is by definition ad hoc.
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 ...
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.
Price had originally come to Hamilton after deriving the Price equation, and thus rederiving Hamilton's rule. Maynard Smith later peer reviewed one of Price's papers, and drew inspiration from it. Maynard Smith later peer reviewed one of Price's papers, and drew inspiration from it.
In addition, in canonical coordinates (with {,} = {,} = and {,} =), Hamilton's equations for the time evolution of the system follow immediately from this formula. It also follows from (1) that the Poisson bracket is a derivation ; that is, it satisfies a non-commutative version of Leibniz's product rule :
Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action S {\displaystyle S} in Hamilton's principle is the Legendre transformation of the action in Maupertuis' principle.
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.