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Problems in Higher Dimensions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The solutions of the angular parts of the problem are often combined into one function of two variables, as problems with spherical symmetry arise often, leaving the main differences between such problems confined to the radial equation. These functions are referred to as spherical harmonics, Yℓm(θ, φ), which are defined with a special normalization as Yℓm(θ, φ), are the spherical harmonics. Spherical harmonics are important in applications from atomic electron configurations to gravitational fields, planetary magnetic fields, and the cosmic microwave background radiation. () Yℓm(θ,ϕ)=(−1)m2ℓ+14π(ℓ−m)!(ℓ+m)!Pℓm( cos θ)eimϕ.
Orthogonal Expansions in Curvilinear Coordinates
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
When restricted to the surface of a sphere, the result of orthogonal expansions in spherical coordinates is the spherical harmonics. A more general theory of orthogonal expansions on higher dimensional manifolds follows in the same way.
Spherical harmonic expansion of hydrodynamic hull forces
Published in C. Guedes Soares, T.A. Santos, Trends in Maritime Technology and Engineering Volume 1, 2022
V. Ferrari, S. Sutulo, C. Guedes Soares
With this assumption, any real or complex function f defined on the sphere can be expressed as a series (Hobson 1955): f(β,γ)=∑ℓ=0∞∑m=−ℓℓcℓmYℓm(β,γ), where Yℓm is the spherical harmonic of degreeℓ and orderm, and cℓm are the harmonic coefficients. Spherical harmonics can be interpreted as a generalisation of the Fourier series on the sphere. They form a complete set of orthogonal functions and thus form a basis for square-integrable functions defined on the sphere, that is, functions that satisfy Equation (12). They are used in a wide range of applications, such as quantum physics, cosmology, geophysics and medical imaging. In the field of hydrodynamics, spherical harmonics have already been used for defining the velocity potential in spherical coordinates (Lamb 1968).
On free vibration of piezoelectric nanospheres with surface effect
Published in Mechanics of Advanced Materials and Structures, 2018
Bin Wu, Weiqiu Chen, Chuanzeng Zhang
Substituting Eq. (17) into Eq. (2) and then into Eqs. (1) and (3), we obtain the governing equations of the spherical core in terms of ψ, G, w, and Φ, which can be solved by assuming where is the imaginary unit; Smn(θ, ϕ) = Pnm(cos θ)exp (imϕ) are spherical harmonics and Pmn (cos θ) are the associated Legendre polynomials; n and m are integers; ω is the circular frequency; and ξ = r/r0 is the dimensionless radial coordinate. The four dimensionless unknown functions Un(ξ), Vn(ξ), Wn(ξ), and Xn(ξ) in Eq. (18) satisfy where is the dimensionless frequency; a prime denotes the differentiation with respect to ξ; pi, qi, and fi are the dimensionless material constants defined in Ding and Chen [47], which are omitted here for brevity.