Question

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 the Hamiltonian, H(q,p,t), on
time it is not a conserved quantity. In what follows our goal is to
perform a variable transformation (q, p) to (Q,P) such that the new
transformed Hamiltonian, K(Q,P), is conserved. We apply the
following method based on the generating function:

?₂ (?,?,?)= ???^??

With : ? = ??₂/?? ,

? = ??₂/?? , and

?(?,?,?)= ?(?,?,?) + ??₂/?? .

- Determine p as a function of P

- Determine Q as a function of q

- Determine the transformed Hamiltonian, K.

Take the case of a harmonic oscillator with potential, ?(?)= 1/2
??²?².

- Express V as a function of Q . That is, V(Q,t)

- Show that

? = ?²2/? + 1/2 ??²?²+ ???

- Show that K is conserved.

Answer #1

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Classical Mechanics -
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