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First Order Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
then f is a homogeneous function of degree 0. Since the ratio y/x is the tangent of the argument in the polar system of coordinates, the function f(y/x) is sometimes called “homogeneous-polar.” We avoid the trivial case when f(x,y)=y/x. Let us consider the differential equation with homogeneous right-hand side function: dydx=Fyx,x≠0.
The N—body Problem
Published in G.A. Gurzadyan, Theory of Interplanetary Flights, 2020
In some problems of celestial mechanics one has to deal with the so-called moment of inertia of a celestial body Ji, i.e. the product of the mass mi and the squared distance from the center of the coordinate system ri: Ji=miri2. The Lagrange–Jacobi formula relates the moment of inertia J of the n-body system to its potential energy U. Given by Eq.(4), U is a homogeneous function of the coordinates. Such functions satisfy the well-known Euler theorem: () ∑0n−1(ξi∂U∂ξi+ηi∂U∂ηi+ζi∂U∂ζi)=−U.
Interlude of Physics II: Thermodynamics
Published in Franco Battaglia, Thomas F. George, Understanding Molecules, 2018
Franco Battaglia, Thomas F. George
At this point it is appropriate to recall from mathematics that a homogeneous function satisfies Euler’s formula: If f (x1,x2,x3,...) is a homogeneous function with degree k in the indicated variables, then ∑jxj∂f∂xj=kf.(9.7)
A new nonsingular integral terminal sliding mode control for robot manipulators
Published in International Journal of Systems Science, 2020
(Homogeneity): A function is homogeneous of degree with respect to (w.r.t.) the weights , if for any given , . A vector field is homogeneous of degree w.r.t. the weights , if for all , the component is a homogeneous function of degree . The system is homogeneous of degree if the vector field is homogeneous of degree .
Robust feedback stabilisation of homogeneous differential inclusions
Published in International Journal of Control, 2022
K. Zimenko, A. Polyakov, D. Efimov
A special case of homogeneous function is a homogeneous norm (Kawski, 1995; Polyakov, 2019): a continuous positive definite -homogeneous function of degree 1. Define the canonical homogeneous norm as , where is such that . Note that and Let us introduce the result on finite-time stability for -homogeneous set-valued maps.
Ascent and descent cones of ordered median block functions
Published in Optimization, 2018
J. Grzybowski, J. Kalcsics, S. Nickel, D. Pallaschke, R. Urbański
Whereas the topological description of the ascent- and descent cone complex seems to be fairly complicated, it turns out, that the determination of all steepest ascent- and descent directions of an arbitrary ordered median block function is quite easy. Therefore, let be equipped with the Euclidean norm The steepest descent directions of positively homogeneous function at the point are the vectors