Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in an inertial system (1, 2), m L = 2 m (ಠ+ i) - w² (x² + ax²) − bx₁ x2,
Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in an inertial system (1, 2), m L = 2 m (ಠ+ i) - w² (x² + ax²) − bx₁ x2,
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Transcribed Image Text:Problem 6
Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in an
inertial system (1, 2),
L = m² (1² + ±²) - m² (x² -
w² (x² + a x²) - bx₁ x2,
2
where a and b are constant numbers.
(i) Write down the Euler-Lagrange equations for the system.
(ii) For generic values of a and b, what are the conserved quantities of the system and what are the
corresponding Noether symmetries? Consider next the case a = 1 and b = 0. Discuss if there are any
additional symmetries and find the corresponding conserved quantities.
(iii) Write down the Hamiltonian of the system and the corresponding Hamilton equations for generic values
of a and b.
(iv) The time evolution of a generic physical observable O(x, p) is described by the equation
where
Ӧ
=
{O,H},
{A, B}
=
Σ
2 ДА ӘВ
дхп дрп
ДА ӘВ
дрп дхп
JB )
n=1
is the Poisson bracket of A with B. x := (x1, x2) and p = (P1, P2) denote the position vector and the
momentum of the particle, respectively. Using this result, calculate the time evolution of p₁ and L3,
i.e. calculate p₁ and Ĺ3, for generic values of a and b. Here L3 = x1p2x2P1.
(v) Consider the case a = 1 and b = -mw². Discuss if there are any additional symmetries, and determine
the corresponding conserved quantities.
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