u/Key-Essay-4890

We solve the eigenvalue problem to determine the eigenvalues and eigenvectors, and understand their geometric and physical significance.

In Part II of his memoir, Arthur Cayley presents a generalized, algebraic formulation for transforming multiple integrals. The historical context stems from classical potential theory—specifically, calculating the gravitational potential and attraction exerted by a homogeneous solid ellipsoid on an external point. Earlier methods by Isaac Newton, Adrien-Marie Legendre, and Siméon Denis Poisson relied heavily on subtle geometric constructions (such as circumscribed cones) or intricate analytical substitutions that Poisson famously described as inextricable calculations.

Cayley’s objective was to strip away the geometric specificity and construct a unified, n-dimensional algebraic framework using the then-novel theory of determinants developed by Carl Gustav Jacob Jacobi. Generalization to n-Dimensions and Variable Transformations.

Cayley considers a general multiple integral of the form V = integral of F(x, y, ...) dx dy ..., where the number of spatial variables (x, y, ...) is equal to n, and F(x, y, ...) is a homogeneous function of degree mu. To evaluate this integral, Cayley transitions to generalized spherical-like coordinates by setting r^2 = x^2 + y^2 + ... and defining direction cosines x = r*a, y = r*b, ... subject to the hyperspherical constraint a^2 + b^2 + ... = 1. The variables are further constrained by a homogeneous equation psi(a,b, ...) = 0 containing a parameter ω.

By expressing the direction variables (a, b, ...) as functions of ω and n-2 independent angular coordinates (θ, φ, ...), the differential volume element dx dy ... transforms via a Jacobian determinant.

Exploiting the homogeneity of F, the radial integration of r^(mu+n-1) dr is performed analytically, reducing the n-dimensional volume integral to an (n-1)-dimensional directional integral over the parameter omega and angles.

Removal of Constraints and Application of Jacobi’s Determinants

To eliminate the restrictive constraint a^2 + b^2 + ... = 1, Cayley introduces auxiliary variables p, q, ... defined by a = p/r, b= q/r, ... where r^2 = p^2 + q^2 + ... The Jacobian determinant of the transformation, denoted as D, is constructed from partial derivatives with respect to ω, θ, and other angular coordinates.

When the condition ψ(p, q, ...) = 0 represents a quadratic form ψ= 1/2 * (Ap^2 + Bq^2 + ... + 2Hpq + ...), Cayley applies a theorem from Jacobi’s 1841 memoir De Determinantibus Functionalibus.

He demonstrates that the complex determinant D factors cleanly into the discriminant of the quadratic form k = det(A, B, H, ...) and a reduced angular determinant S.

Reduction to Legendre’s Integral and 3D Attraction

Cayley applies this algebraic machinery to the physical problem of gravitational attraction in n dimensions for an integrand with a denominator of the form (x^2 + y^2 + ...)^(i - 1/2) and ellipsoidal boundary limits l(x-a)^2 + m(y-b)^2 + ... = k. By setting n=3 (three-dimensional space) and i=0 for Newtonian gravity, the angular integrations over theta can be evaluated in closed form. The square roots and algebraic determinants reduce through clever substitutions involving confocal surface identities. Ultimately, Cayley arrives at a single, elegant elliptic integral for the potential V.

This demonstrated that Legendre’s result was not a three-dimensional geometric fluke, but a direct consequence of the algebraic invariance of quadratic forms under linear coordinate transformations.

HOW MODERN PHYSICISTS USE THESE TOOLS :

While 19th-century physicists like Cayley and Legendre relied on explicit matrix determinants and algebraic manipulation of coordinate differentials, modern physics has translated these concepts into Differential Geometry, Tensor Calculus, and Spectral Analysis. Ellipsoidal Coordinates and Separation of Variables In modern field theory and fluid dynamics, Cayley’s coordinate parameterization is expressed via orthogonal ellipsoidal coordinates (lambda, mu, nu), defined as the roots of the confocal quadric equation x^2/(a^2 + s) + y^2/(b^2 + s) + z^2/(c^2 + s) =

Modern physicists use these coordinates because the Laplacian operator {grad}^2 completely separates in this system. The partial differential equations governing gravitational, electrostatic, or fluid velocity potentials reduce to ordinary differential equations known as Lamé Differential Equations. Solving these yields Lamé Harmonics, which are the ellipsoidal analogs to Spherical Harmonics.

Cayley’s n-dimensional generalization of ellipsoidal integrals is crucial in modern theoretical physics, particularly in General Relativity and String Theory: Myers-Perry Black Holes: In spacetime dimensions D > 4, rotating black holes do not have spherical event horizons; they possess ellipsoidal topologies described by higher-dimensional quadrics.

Calculating the total energy and gravitational attraction of rotating higher-dimensional branes requires integrating volume forms over ellipsoidal hypersurfaces using the exact n-dimensional Jacobians formulated by Cayley

Modern mathematical physics uses Differential Forms and Stokes' Theorem: integral over M of dw = integral over boundary of M of w. The transformation of volume elements into angular and radial forms is now understood as the pullback of a volume form under a diffeomorphism.

In galactic dynamics and planetary science, the gravitational potentials of triaxial elliptical galaxies, dark matter halos, and non-spherical asteroids are computed using the closed-form elliptic integrals derived in Cayley's work. The property that the internal gravitational field of an ellipsoidal shell (homoeoid) cancels out identically remains a cornerstone for modeling stellar interiors and planetary gravity fields.

Physics are getting harder and harder, but we must try to see forward. Even though the ex's are tough we get to know concepts and ideas that help us understand our world.

I am excited CERN has some webinars this week their full week program: https://home.cern/events/

The following link is about the search of axions dark matter for Thursday.

https://indico.cern.ch/event/1708711/

I tried replying to the earlier Pillars of Creation post, but Reddit wouldn’t let me attach an image.
Am I the only one who sees a dog inside the red circle? And inside the green circle, I see what looks like a skeleton with its back turned toward us.
(Yes, I know what pareidolia is. 😂)

{{ The first suggestion of the problem is contained in a memoir of Euler's -"Sur une contradiction apparente dans la doctrine des lignes coubres". }}

Arthur Cayley’s 1847 paper, On the Theory of Involution in Geometry, laid the foundational groundwork for modern elimination theory and algebraic geometry. At its core, the paper introduces a systematic way to count independent polynomial equations under constraints, resolving the issue of "overcounting" redundant relations

. While written purely as a geometric treatise, physics made profound use of this "accounting of constraints" in Lagrangian mechanics and quantum field theory: Isolating Degrees of Freedom: "When physical systems are restricted by non-linear constraints (such as a particle bound to the intersection of complex surfaces), Cayley's formulas help physicists calculate the exact, true degrees of freedom, preventing equations of motion from becoming mathematically singular."

The BRST Formalism: In modern gauge theories (like Quantum Chromodynamics), physicists quantize fields using the BRST formalism. To eliminate unphysical gauge states, they introduce "ghost" and "anti-ghost" fields. This alternating addition and subtraction of ghost states is the physical manifestation of Cayley's alternating series (Equation A), which was designed to eliminate "extraneous factors." Ultimately, Cayley's mathematical accounting tool evolved into a vital framework for keeping gauge field equations physically consistent and computationally stable.

The ex's are simple, if anyone has anything to ask feel free to do so.

PG 1115+080 (also known as the "triple quasar," though it is actually quadruply lensed) is a gravitationally lensed quasar located about 8 billion light-years away in the constellation Leo.

Its light is bent and split into four separate images by an intervening elliptical galaxy roughly 3 billion light-years away.

Credit: Israel Velazquez