A new integration theory for conservative two degree-of-freedom systems

Details A new integration theory for conservative two degree-of-freedom systems A new integration theory for conservative two degree-of-freedom systems

Mar 1988

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1988cemec..44....1h&link_type=abstract

Celestial Mechanics, Volume 44, Issue 1-2, pp. 1-29

Computer Science

6

Scientific paper

A two degree-of-freedom, conservative system is reduced to a single degree-of-freedom, kinematic system with Hamiltonian integral under the change of independent variable: dt = ζ dt (ζ = \upsilon _x - \upsilon _y ) where ζ is the curl (or vorticity) of the velocity field with cartesian inertial componentsu(x, y, t) andv(x, y, t). In the autonomous case whenu t=v t=0, orbits are globally represented by the level curves of an autonomous Hamiltonian functionH(x,y) satisfying a second-order quasilinear partial differential equation (Szebehely's Equation): 2(H + U)left( {H_{xx} H_y^2 - 2H_{xy} H_x H_y + H_{yy} H_x^2 } right) + (H_x U_x + H_y U_y )left( {H_x^2 + H_y^2 } right) = 0 whereU(x, y) is the autonomous potential function. An inversion of dependent and independent variables reduces this equation to a second-order, ordinary differential equation for a function specifying the orbital curve. The true time variable is recovered by evaluating a quadrature. Fundamental differences exist between this approach and Hamilton-Jacobi theory.

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