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To state this formally, in general an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr dt), and its acceleration (the second derivative of r, a = d2r dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold.
The motion of a particle (a point-like object) along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running a 100-meter dash along a straight track. [2] Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that do not experience any net ...
The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America.
The superposition principle, [1] also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X, and input B produces response Y, then input (A + B) produces ...
The wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions u = u (x, y, z, t) of a ...
A Green's function, G(x,s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space , at a point s, is any solution of. (1) where δ is the Dirac delta function. This property of a Green's function can be exploited to solve differential equations of the form.
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V, where T and V are the kinetic and potential energy of the system, respectively. [3]
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear ...