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In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: =, where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic ...
Separation of variables may be possible in some coordinate systems but not others, [2] and which coordinate systems allow for separation depends on the symmetry properties of the equation. [3] Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in ...
In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODEs). [1] It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique.
Conversely, if k 2 is less than one, F 2 −F 1 is a positive number; the added inverse-cube force is repulsive. If k is an integer such as 3, the orbit of the second particle is said to be a harmonic of the first particle's orbit; by contrast, if k is the inverse of an integer, such as 1 ⁄ 3, the second orbit is said to be a subharmonic of ...
2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship. 2.4 Modified-factorial denominators. 2.5 Binomial coefficients.
The covariance function K X satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λ k, e k (t) of eigenvalues and eigenfunctions of T K X forming an orthonormal basis of L 2 ([a,b]), and K X can be expressed as (,) = = ()
where ω is the angular frequency and k is the wavevector with magnitude | k | = k, equal to the wave number. Divide by ℏ {\displaystyle \hbar } and take the square root. This gives the relativistic frequency dispersion relation : ω ( k ) = k 2 c 2 + ( m 0 c 2 ℏ ) 2 . {\displaystyle \omega (k)={\sqrt {k^{2}c^{2}+\left({\frac {m_{0}c^{2 ...