![]() They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein's general theory of relativity. ![]() He taught us classical mechanics, quantum mechanics, and special relativity. 5.The latest volume in the New York Times bestselling physics series explains Einstein's masterpiece: the general theory of relativity.Variational calculus provides a powerful approach for determining the equations of motion constrained to follow a geodesic. 5.10: Geodesic The geodesic is defined as the shortest path between two fixed points for motion that is constrained to lie on a surface.The general method of Lagrange multipliers for n variables, with m constraints, is best introduced using Bernoulli’s ingenious exploitation of virtual infinitessimal displacements, which Lagrange signified by the symbol δ. 5.9: Lagrange multipliers for Holonomic Constraints The Lagrange multiplier technique provides a powerful, and elegant, way to handle holonomic constraints using Euler’s equations.5.8: Generalized coordinates in Variational Calculus Generalized coordinates allow embedding constraint forces which simplifies the solution.5.7: Constrained Variational Systems Holonomic constraints couple coordinates for the system.5.6: Euler’s Integral Equation An integral form of the Euler differential equation can be written which is useful for cases when the function f does not depend explicitly on the independent variable x.It is more common to have a functional that is dependent upon several independent variables f. 5.5: Functions with Several Independent Variables The discussion has focussed on systems having only a single function y(x) such that the functional is an extremum.The following example of a cylindrically-symmetric soap-bubble surface formed by blowing a soap bubble that stretches between two circular hoops, illustrates the importance of the independent variable. Selecting which variable to use as the independent variable does not change the physics of a problem, but some selections can simplify the mathematics for obtaining an analytic solution. 5.4: Selection of the Independent Variable A wide selection of variables can be chosen as the independent variable for variational calculus.The Brachistochrone problem stimulated the development of the calculus of variations by John Bernoulli and Euler. 5.3: Applications of Euler’s Equation The Brachistochrone problem involves finding the path having the minimum transit time between two points.Variational calculus, developed for classical mechanics, now has become an essential approach to many other disciplines in science, engineering, economics, and medicine. 5.2: Euler’s Differential Equation The calculus of variations, presented here, underlies the powerful variational approaches that were developed for classical mechanics.He solved the brachistochrone problem which involves finding the path for which the transit time between two points is the shortest. 5.1: Introduction to the Calculus of Variations During the 18th century, Bernoulli, who was a student of Leibniz, developed the field of variational calculus which underlies the integral variational approach to mechanics.
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