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Additional resources for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

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Even though we are going to extend this to problems for which no Hamiltonian exists, it may still be of interest to describe the Hamiltonian from our point of view, when one does exist. Let us consider the average of any functional F (q i ). To calculate the rate of change with the time of this quantity we may use a relation analogous to (51). Another method is as follows; suppose the variables q i which appear in F are limited to indices between the times t l and tl (l > l ). That is to say, F is a functional of only the variables q l−1 to ql +1 .

F is zero for all t is that function The function q(σ) for which δq(t) for which F is an extremum. For example, in classical mechanics the action, A = L(q(σ), ˙ q(σ))dσ (5) is a functional of q(σ). Its functional derivative is, d δA =− δq(t) dt ∂L(q(t), ˙ q(t)) ∂ q˙ + ∂L(q(t), ˙ q(t)) . ∂q (6) If A is an extremum the right hand side is zero. 5 A. S. Eddington, “The Mathematical Theory of Relativity” (1923) p. 139. Editor’s note: We have changed Eddington’s symbol for the functional derivative to that now commonly in use.

That is to say, we must have, δA δy(t) d δA =− δy(t) dt ∂L ∂ y˙ + ∂L ∂Iy + ∂y t ∂y · x(t) . (31) t We seek the solution of this expression for each expression we may write for x(t). Now an equation such as (31) (which is really an infinite set of equations, one for each value of t) does not always have a soδ δA = lution. One of the necessary requirements is, since δy(s) δy(t) δ δy(t) δA δy(s) , that, d δ − δy(s) dt = ∂Ly ∂ y˙ d δ − δy(t) ds + ∂Ly ∂y ∂Ly ∂ y˙ + t + ∂Iy ∂y ∂Ly ∂y · x(t) t + s ∂Iy ∂y · x(s) .

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