By Albert R. Baswell
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Additional resources for Advances in Mathematics Research, Vol. 10
Conservation laws for perturbations to the RH-wave are obtained and used to divide all the perturbations into four invariant non-intersecting sets: M−n , M+n , Hn and M0n − Hn (Theorem 2). The main conservation law asserts that any perturbation evolves in such a way that its energy and enstrophy decrease, remain constant or increase simultaneously (Theorem 1). Besides, the energy cascade of growing (or decaying) perturbations has opposite directions in the sets M−n and M+n (Theorem 3). A geometric interpretation of variations in the perturbation kinetic energy is given, according to which the energy of a perturbation ψ ′ (t, x) to the RH Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave 23 wave f (t, x) changes only if the projection of ψ ′ (t, x) onto the main RH-wave component fn (t, x) changes (Theorem 4).
Consequently, the stretch stimulus causes the antagonists to relax so that they cannot resists the shortening of the stretched muscle caused by the main reflex arc. Similarly, firing of the Golgi tendon receptors causes inhibition of the muscle contracting too strong and simultaneous reciprocal activation of its antagonist. 2. Cerebellum and Muscular Synergy The cerebellum is a brain region anatomically located at the bottom rear of the head (the hindbrain), directly above the brainstem, which is important for a number of subconscious 42 Vladimir G.
Synovial joint dynamics, giving the first stabilizing effect to the conservative skeleton dynamics, is described by the (x, x)–form ˙ of the Rayleigh – Van der Pol’s dissipation function n 1 R= (x˙ i )2 [αi + βi (xi )2 ], 2 i=1 where αi and βi denote dissipation parameters. Its partial derivatives give rise to the viscous–damping torques and forces in the joints Fijoint = ∂R/∂ x˙ i , which are linear in x˙ i and quadratic in xi . 2. Muscular dynamics, giving the driving torques and forces Fimuscle = muscle Fi (t, x, x) ˙ with (i = 1, .
Advances in Mathematics Research, Vol. 10 by Albert R. Baswell