# Get The first book of Jerome Cardan's De Subtilitate, translated PDF

By Jerome Cardan (Girolamo Cardano) ; Myrtle Marguerite Cass (translator)

Read Online or Download The first book of Jerome Cardan's De Subtilitate, translated from original Latin with text introduction and commentary by Myrtle Marguerite Cass PDF

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Extra resources for The first book of Jerome Cardan's De Subtilitate, translated from original Latin with text introduction and commentary by Myrtle Marguerite Cass

Example text

The variables r, θ, φ are then called separated variables. The 1-form α restricted on Mf is then obviously closed. 8 The Euler top and have simple time evolution with respective frequencies (1, 0, 0) by eq. 4). Hence ψJ 2 and ψJ3 remain constant, while ψH = t − t0 . This gives the standard formula for the Kepler motion: t − t0 = r dr 2 H − V (r) − J2 r2 Note that the constancy of ψJ3 implies: J3 φ˙ = 2 sin θ J2 − J32 sin2 θ θ˙ This, in turn, implies the conservation of J1 , J2 : J1 = −J3 cot θ cos φ − sin φ J2 − J32 sin2 θ J2 = −J3 cot θ sin φ + cos φ J2 − J32 sin2 θ so that the motion takes place in the plane perpendicular to J, as expected.

Example. Let us give an example of this construction in the simple example of the harmonic oscillator. The Lax matrix L is given in eq. 8) and we introduce the action–angle coordinates ρ, θ as in eq. 3). In these coordinates the matrix L is diagonalized by: U = U −1 = cos 2θ sin 2θ sin 2θ − cos 2θ Since {U1 , U2 } = 0, r12 = q12 , which is easily computed to be: r12 = ω 2ρ2 0 −1 1 0 ⊗L It is easy to verify that this r-matrix indeed satisﬁes eq. 10). Let us notice that it is a dynamical r-matrix, which means that it depends explicitly on the dynamical variables.

20). The Lax matrix can thus be interpreted as belonging to the coadjoint orbit of the element A− (λ) of G ∗ under the loop group G. 21) This shows that the equation of motion is a ﬂow on the coadjoint orbit. 3 Coadjoint orbits and Hamiltonian formalism 43 Coadjoint orbits in G ∗ are equipped with the canonical Kostant–Kirillov symplectic structure. Choosing two linear functions h1 (Ξ) = Ξ(X) and h2 (Ξ) = Ξ(Y ) with X, Y ∈ G, so that dh1 = X and dh2 = Y , the Kostant–Kirillov Poisson bracket reads: {Ξ(X), Ξ(Y )} = Ξ([X, Y ]) where the right-hand side is the linear function Ξ → Ξ([X, Y ]).