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(d) Demonstrate that the first integral of a time-independent Lagrangian is a constant of motion.

(d) Demonstrate that the first integral of a time-independent Lagrangian is a constant of motion.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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1. This first question contains a number of unrelated problems that should each be given a short answer. (a) Write down the Euler-Lagrange equations for a system with generalised coordinates (41, 42, ... , AN). (b) What is a symmetry? State the general definition (expressed just in words), and not an example such as translational or rotational symmetry. (c) Give the general definition of the Poisson bracket for a Hamiltonian system described by the scalar parameters (q, p). Write down the three fundamental Poisson brackets and their values. (d) Demonstrate that the first integral of a time-independent Lagrangian is a constant of motion. (e) Starting from the conserved energy E = +V(q) for a scalar q, derive the integral formula for the time t = t(q) as a function of q. (f) Let H(q, p) be a Hamiltonian and (q(t), p(t)) a solution of its equations of motion for a fixed energy E. We consider now a new Hamiltonian H'(q, p) obtained through H'(q, p) = f(H(q,P)), where f is an arbitrary nonzero function. We denote (g'(t), p(t)) the solutions for H' at fixed energy E' = f(E). Show that these trajectories are the same as the previous ones, but are governed by a different time dependence.
Transcribed Image Text:1. This first question contains a number of unrelated problems that should each be given a short answer. (a) Write down the Euler-Lagrange equations for a system with generalised coordinates (41, 42, ... , AN). (b) What is a symmetry? State the general definition (expressed just in words), and not an example such as translational or rotational symmetry. (c) Give the general definition of the Poisson bracket for a Hamiltonian system described by the scalar parameters (q, p). Write down the three fundamental Poisson brackets and their values. (d) Demonstrate that the first integral of a time-independent Lagrangian is a constant of motion. (e) Starting from the conserved energy E = +V(q) for a scalar q, derive the integral formula for the time t = t(q) as a function of q. (f) Let H(q, p) be a Hamiltonian and (q(t), p(t)) a solution of its equations of motion for a fixed energy E. We consider now a new Hamiltonian H'(q, p) obtained through H'(q, p) = f(H(q,P)), where f is an arbitrary nonzero function. We denote (g'(t), p(t)) the solutions for H' at fixed energy E' = f(E). Show that these trajectories are the same as the previous ones, but are governed by a different time dependence.
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