We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
In this second capstone chapter, we extend some of the classical mechanics from the preceding four chapters into the context of more recent developments in physics. We begin with gravitation, including some of the ideas that led Einstein to go way beyond Newton’s nonrelativistic theory to find a fully relativistic theory of gravitation. After years of strenuous effort, his work finally culminated in his stunningly original and greatest single achievement, the general theory of relativity. He was able to predict three effects that could be measured in the solar system, which he used to check his theory. We will cover all three of these. Then we will introduce so-called “magnetic gravity,” which contains the leading terms in general relativity in a form much like Maxwell’s equations for electromagnetism. Next, we delve just a bit deeper into gauge symmetry in Maxwell’s theory, partly because it deepens our understanding of electromagnetism but also because gauge symmetry has played such a large role in physical theories over the past many decades. Finally, we introduce stochastic forces, which are not fundamental forces but the result of huge numbers of small collisions.
We define general relativity. We first consider intrinsically curved spaces and the notion of metric. Einstein's theory of general relativity is defined, based on the two physical assumptions, that gravity is geometry, and that matter sources gravity, and leading to general coordinate invariance and the equivalence principle. Kinematics, specifically tensors, Christoffel symbol and covariant derivatives, is defined. The motion of a free particle in a gravitational field is calculated.
We consider three point positive masses moving on ${{S}^{2}}$ and ${{H}^{2}}$. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of ${{S}^{2}}$, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classification of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibrium is spectrally stable or unstable. On ${{H}^{2}}$, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.
Recommend this
Email your librarian or administrator to recommend adding this to your organisation's collection.