The local descriptors characterizing the temporal shape variations of action are then obtained by using the temporal self-similarities defined on the fuzzy log-polar histograms.
Design values are enhanced due to variations of actions in the ensemble of typical realizations.
(Hamilton) In a general dynamical space, the following two trajectories are equivalent: 1. Trajectory that gives the extremal of variation of the action: δ I = 0, 2.
Trajectory that gives the extremal of variation of the action: δ I = 0, Trajectory that gives the maximum probability in Eq. (15).
By variation of the action (1), one encounters a total derivative which gives rise to a surface integral involving the normal derivative of (delta g_{mu nu }).
The variation of the action, on the other hand, can be written as δ I = ∫ d t δ L, which yields δ I = 0 ⇒ δ L = 0.
The following theorem should then naturally hold: (The most probable path) The following two trajectories are equivalent: 1. Trajectory that gives the extremal of variation of the action: δ I = 0, 2.
The first variation of the action functional (18) with respect to φ r,t) and its derivatives has the form begin{aligned} delta S [varphi]& = int dt int d^{N}x, delta mathcal{L} = int dt, int d^{N}x, left[{vphantom{sum_{df}}}frac{partialmathcal{L}}{partial varphi} delta varphiright.
end{aligned} (2.13)We then obtain the variation of the action for matter fields with respect to (Phi) as begin{aligned} delta S_m=-intleft[ T+left( 2-frac{1}{2V}frac{mathrm{d}V}{mathrm{d}Phi }right) E+ nabla _lambda C^lambda right] delta Phi sqrt{-q}mathrm{d}^4x, end{aligned} (2.14 where begin{aligned} nabla _lambda C^lambda equiv frac{1}{sqrt{-q}}(partial _lambda sqrt{-q},C^lambda ).
Trajectory that gives the extremal of variation of the action: δ I = 0, Trajectory that satisfies the characteristic equations: d γ i d t = ∂ H ∂ π i, d π i d t = - ∂ H ∂ γ i, d H d t = ∂ H ∂ t. 1 ⇒ 2 : Applying a variational operator to the action, we obtain δ I = δ ∫ γ ω = ∫ δ γ ω = 0, implying that the integration of the characteristic 1-form is independent of δ γ and satisfies Principle 1.
