Question

9. Hamiltonian of a particle is H=2mpx2+kx4. If energy of the particle is doubled then area of the phase space trajectory will become (a) double (b) half (c) 23/2 times (d) 23/4 times

Answer 1

The area of the phase space trajectory of a particle with a Hamiltonian given by H = 2m{{p_x}}^2 + kx^4 changes according to Liouville's theorem when its energy is doubled, but the exact factor of change in the area cannot be determined without additional information.

Step-by-step explanation:

The question pertains to the Hamiltonian mechanics of a particle and the effects of doubling its energy on the area of the phase space trajectory. The Hamiltonian given is H = 2m{{p_x}}^2 + kx^4, where {{p_x}} is the momentum in the x-direction. If the energy of the particle is doubled, the area of the phase space trajectory will change according to Liouville's theorem, which states that the phase space volume is conserved during the evolution of a Hamiltonian system. However, to provide the exact change in area without additional information about the specific form of k or m would require further details or assumptions about the system.

Answer 2

The area of the phase space trajectory will become double when the energy of the particle is doubled.

To determine the change in the area of the phase space trajectory when the energy of the particle is doubled, we need to understand the relationship between the energy and the area of the phase space. In Hamiltonian mechanics, the phase space trajectory is represented by the position and momentum coordinates of the particle.

The area of the phase space trajectory can be calculated using the formula:

Area = ∮ p dx

where p is the momentum and x is the position. In this case, the Hamiltonian of the particle is given as: H = 2mpx^2 + kx^4.

When the energy of the particle is doubled, the Hamiltonian will also double. Therefore, we can write:

2H = 2(2mpx^2 + kx^4) = 4mpx^2 + 2kx^4

The change in the area of the phase space trajectory can be calculated by comparing the old Hamiltonian (H) and the new Hamiltonian (2H):

Change in Area = ∮ p dx (2H - H) = ∮ p dx H = H

Since the change in area is equal to H, the area of the phase space trajectory will become double when the energy of the particle is doubled.