17: Relativistic Mechanics
- Page ID
- 9669
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 17.1: Introduction to Relativistic Mechanics
- Einstein's Special theory of Relativity (1905) and General Theory of Relativity (1916) are revolutionary advances that have had a profound impact on the evolution and understanding of both classical mechanics and modern physics.
- 17.2: Galilean Invariance
- Space and time are separable.
- 17.3: Special Theory of Relativity
- Einstein's Special Theory of Relativity.
- 17.4: Relativistic Kinematics
- Relative differences between Newtonian and relativistic kinematics.
- 17.5: Geometry of Space-time
- Four dimensional space-time.
- 17.6: Lorentz-Invariant Formulation of Lagrangian Mechanics
- The Lagrangian and Hamiltonian formalisms in classical mechanics are based on the Newtonian concept of absolute time t which serves as the system evolution parameter in Hamilton’s Principle. This approach violates the Special Theory of Relativity. The extended Lagrangian and Hamiltonian formalism is a parametric approach, pioneered by Lanczos, that renders it to a form that is compatible with the Special Theory of Relativity.
- 17.7: Lorentz-invariant formulations of Hamiltonian Mechanics
- Extended canonical formation for relativistic mechanics.
- 17.8: The General Theory of Relativity
- Einstein’s General Theory of Relativity expands the scope of relativistic mechanics to include non-inertial accelerating frames plus a unified theory of gravitation. That is, the General Theory of Relativity incorporates both the Special Theory of Relativity as well as Newton’s Law of Universal Gravitation. It provides a unified theory of gravitation that is a geometric property of space and time. In particular, the curvature of space-time is directly related to the four-momentum of matter and r
- 17.9: Implications of Relativistic Theory to Classical Mechanics
- The Special Theory of Relativity replaces Newton’s Laws of motion; i.e. Newton’s law is only an approximation applicable for low velocities. The General Theory of Relativity replaces Newton’s Law of Gravitation and provides a natural explanation of the equivalence principle. Einstein’s theories of relativity imply a profound and fundamental change in the view of the separation of space, time, and mass, that contradicts the basic tenets that are the foundation of Newtonian mechanics.
Thumbnail: Momenta are conserved within a closed system and the laws of conservation of momenta applies. Consider the special case of identical particles colliding symmetrically. (CC BY-SA; RobinH via Wikipedia)