Geometrical Dynamics of Certain Nonlinear Conservative Systems with Two Degrees of Freedom |
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approximation Arbitrary Surface ax² b'N² B₁ Centers of Gravitation coefficients complementary solution configuration plane constants conventional dynamics corresponding cose cosh degrees of freedom derived determine differential equation differentiation with respect double pendulum dynamical systems energy expressions energy level equations of motion equipotential curve Euclidean plane Euler's Problem FIGURE fourth order fruit-stem system geometric equation geometrical dynamics given Golden Delicious Golden Delicious apple holonomic Hoop-Pendulum System in-phase initial slope integration isometric kinetic energy Lagrangian Legendre polynomials linear normal mode linearized system m₂ metric modal line mode shapes mode x nonlinear normal modes nonlinear systems nonlinear vibrations nonsimilar normal modal curves normal mode vibrations obtains orbital equation out-of-phase P₂ parameters periodic orbits periodic solutions Perturbation Method perturbational procedure perturbational solution potential energy primes denote differentiation quadratic form Riemannian manifold right-hand side Rosenberg scleronomous Substituting tensor Three Centers tion torsion springs two-degree-of-freedom systems unit mass zero