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 (91. 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 (q(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.

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 (91. 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 (q(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.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.32P

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