Question

Classical Mechanics -

Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional oscillator :

?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²)

?(?,?)= ?/2 (?²+?² )+???

where x and y denote, respectively, the cartesian displacements of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of the displacements; m the mass of the oscillator; K the stiffness constant of the oscillator; A is the coupling constant.

1) Using the following coordinate transformations,

?= 1/√2 (?+?)

?= 1/√2 (?−?)

show that the kinetic energy term has a similar form like that with the variables (x,y) but, the potential energy reduces to that of two uncoupled harmonic oscillators with different apparent stiffness constants.

2) With the new coordinates, show that the Lagrangian can be split into two independent Lagrangians with two different angular frequencies.

3) Using the canonical coordinates and momenta with the new coordinates, show that the Hamiltonian can be expressed into two independent hamiltonians.

4) Based on the equation of the dynamical quantity above (1st part) find two independent constants of the motion.

5) If A=0, find a third constant of the motion.

Answer #1

Let us consider a particle of mass M moving in one dimension q
in a potential energy field, V(q), and being retarded by a damping
force −2???̇ proportional to its velocity (?̇).
- Show that the equation of motion can be obtained from the
Lagrangian:
?=?^2?? [ (1/2) ??̇² − ?(?) ]
- show that the Hamiltonian is
?= (?² ?^−2??) / 2? +?(?)?^2??
Where ? = ??̇?^−2?? is the momentum conjugate to q.
Because of the explicit dependence of...

Consider the bead on a rotating circular hoop system that we saw
in lecture. Recall that R is the radius of the hoop, ω is the
angular velocity of rotation, and m is the mass of the bead that is
constrained to be on the hoop as it rotates about the z-axis. We
found that we could solve the constraints in terms of a single
coordinate θ and that the Cartesian coordinates were given by
x = −R sin(ωt)sinθ, y...

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