(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.
(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.
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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.
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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