2: Contemporary Aspects of the Virial Theorem
- Page ID
- 141446
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: The Tensor Virial Theorem
- This page explores the tensor representation of the virial theorem, highlighting its importance over scalar equations by retaining crucial spatial symmetry information. Originating in early 20th-century research and gaining traction in the mid-1900s, it enhances the analysis of stability in gaseous structures and magnetic fields.
- 2.2: Higher Order Virial Equations
- This page explores higher-order virial equations derived from motion equations, building on the classical virial theorem and Chandrasekhar's contributions. It details how these equations, represented as spatial tensor moments, provide insights into system structure and energy dynamics. While they encapsulate valuable information, their complexity can obscure physical interpretations.
- 2.3: Special Relativity and the Virial Theorem
- This page elaborates on modifications to the virial theorem for special relativity, especially in systems like white dwarfs, emphasizing the kinetic and potential energy relationship and referencing Chandrasekhar’s work on equilibrium. It derives a relativistic form of Lagrange's identity through conservation laws.
- 2.4: General Relativity and the Virial Theorem
- This page examines advancements in quantum theory and general relativity, particularly the virial theorem's formulation in a relativistic context. It highlights the importance of moment analysis versus coordinate-free models while addressing challenges posed by general relativity's non-linearity. Chandrasekhar's work modifies fluid motion equations to incorporate relativistic effects, revealing complex interactions between matter and spacetime.
- 2.5: Complications- Magnetic Fields, Internal Energy, and Rotation
- This page covers the virial theorem's importance in relating potential and kinetic energies, emphasizing the Euler-Lagrange equations for systems in rotation and magnetic fields. It highlights the transformation to rotating frames for analyzing macroscopic motions and discusses assumptions used in simplifying equations of motion in plasmas.
- 2.6: Summary
- This chapter explores the virial theorem, including the tensor version and its modern applications. It discusses the insights gained from higher spatial moments and relativistic impacts, while maintaining a scalar form for simplicity in general relativity. The complexities introduced are noted, but the chapter emphasizes the theorem's relevance in dynamic systems and astrophysics, highlighting various physical factors that influence these systems.
- 2.7: Notes to Chapter 2
- This page covers equations of conservation of mass and motion, tensor components, and applications of Gauss's Law related to kinetic energy density and potential energy. It also examines divergences and curls in electromagnetism, angular velocity in fluid dynamics, and introduces angular momentum in rotating frames through integral expressions.
- 2.8: References
- This page provides a compilation of references in theoretical physics and astrophysics, featuring notable scientists like Chandrasekhar, Einstein, and Parker. It covers a range of topics such as hydrodynamic stability, stellar structure, relativity, and gravitational theories, highlighting significant publications and collaborations that contribute to advancements in these fields.