Question

# Let us consider a particle of mass M moving in one dimension q in a potential...

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.