Download PDF by Smirnov V.I.: A course of higher mathematics, vol. 2

By Smirnov V.I.

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For example, pay attention to the direction of the front cover of the book after each toss. Hint: Look for invariant quadric surfaces; that is, manifolds deﬁned as level sets of quadratic polynomials (ﬁrst integrals) in the variables (M1 , M2 , M3 ). For example, show that the kinetic energy given by 12 AΩ, Ω is constant along orbits. The total angular momentum (length of the angular momentum) is also conserved. For a complete mathematical description of rigid body motion, see [10]. For a mathematical description of the observed “twist” in the rotation of the tossed book, see [16].

To deﬁne this important class of diﬀerential equations, let H : Rn × Rn → R be a smooth function given by (q1 , . . , qn , p1 , . . , pn ) → H(q1 , . . , qn , p1 , . . , pn ), and deﬁne the associated Hamiltonian system on R2n with Hamiltonian H by q˙i = ∂H , ∂pi p˙i = − ∂H , ∂qi i = 1, . . , n. Let us note that the dimension of the phase space of a Hamiltonian system is required to be even. The reason for this restriction will soon be made clear. As a prototypical example of a Hamiltonian system, let H : R2 → R be given by H(x, y) := 12 (y 2 + ω 2 x2 ).

Solve the same problems for the second order diﬀerential equation given by x ¨ + (x2 − 1)x˙ + ω 2 x − λx3 = 0. 6 Stability and the Direct Method of Lyapunov Let us consider a rest point x0 for the autonomous diﬀerential equation x˙ = f (x), x ∈ Rn . 15) at x0 provided that 24 1. Introduction to Ordinary Diﬀerential Equations (i) V (x0 ) = 0, (ii) V (x) > 0 for x ∈ U − {x0 }, (iii) the function x → grad V (x) is continuous for x ∈ U − {x0 }, and, on this set, V˙ (x) := grad V (x) · f (x) ≤ 0. If, in addition, (iv ) V˙ (x) < 0 for x ∈ U − {x0 }, then V is called a strict Lyapunov function.