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By John Casey

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Read or Download A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples PDF

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Extra info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

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The Lagrangian considered as a function of velocity increases at infinity faster than a linear function: L(x) ---} +oo lxl as lxl---} oo. Here, I · I is a norm on TxM. Indeed, in this case, the graph of L(x) is a convex surface, and the function y·x-L(x) has exactly one critical point for each y (see Fig. 15). This point is just the solution of Eq. 12). We can therefore introduce the Hamilton function (Hamiltonian) H(x, y, t) = y · x- L(x, x, t) 1. x-+y . 13) For fixed t, the Hamiltonian is a smooth function on the 2n-dimensional manifold that is the union of cotangent spaces T*M= U T;M.

Xm, z} = JRm+l. The vector y = {Yl, ... , Ym, 1} =1- 0 is orthogonal to the integral surface relative to the standard Euclidean metric in JRm+l. A hyperplane in JRm+l passing through the point (x, z) orthogonally to the vector y, whose components satisfy Eq. 6), is said to be admissible. 7), admissible planes passing through the same point of JRm+l form a smooth (m-1)-dimensional family. Therefore, solving Eq. 6) can be treated as finding hypersurfaces z = z(x) such that all their tangent planes are admissible.

The 2n+2 variables x, y, t, and h (h =-H) are not independent: one of them (namely, h) is a known function of the other variables. The equation 1i = 0 can be solved with respect to each of these 2n+2 variables, and the specific choice of such a variable is inessential. " §6. Action in the Phase Space and the Poincar6-Cartan Invariant 51 Fig. 16. 3. The Poincare-Cartan integral invariant, which is very important in the theory of Hamiltonian systems, can be ded~ced from Theorem 9. Let 'Yo be a closed path in the extended phase space P; it can be represented by a periodic function z = zo(a), amod27r.

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