GATE PH concept:hamiltonian PYQs & Answers

Questions 1–10 of 14. Imported answers require independent review unless marked reviewed.

The Lagrangian of a particle of mass

and charge

moving in a uniform magnetic field of magnitude

that points in the

direction, is given by

where

are the components of its velocity

. If

denote the conjugate momenta in the

directions and

is the Hamiltonian, which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

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A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the

axis. The rod itself is moving vertically upward along the

direction against gravity with a constant speed, starting from

, and remains horizontal. The conjugate momenta are denoted by

and the Hamiltonian by

. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

is the total energy of the system and is conserved

is the total energy of the system and is not conserved

is not the total energy of the system, but it is conserved

is not the total energy of the system and is not conserved

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In a one-dimensional Hamiltonian system with position

and momentum

, consider the canonical transformation

, where

and

are the new position and momentum, respectively. Which of the following option(s) regarding the generating function

is/are correct?

Your answerSelect all correct options. You can choose more than one.

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The energy of a free, relativistic particle of rest mass

moving along the

axis in one dimension, is denoted by

. When moving in a given potential

, its Hamiltonian is

. In the presence of this potential, its speed is

, conjugate momentum

, and the Lagrangian

. Then, which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

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The Hamiltonian for a one dimensional system with mass

, position

and momentum

, where

is a real function of

. If

, then

. The value of

(in integer) is _____.

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Let

be the phase space density of an ensemble of a system. The Hamiltonian of the system is

. If

denotes the Poisson bracket of

and

, then

implies

Your answerChoose one option.

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A system with time independent Hamiltonian

has two constants of motion

and

. Then which of the following Poisson brackets are always zero?

Your answerSelect all correct options. You can choose more than one.

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In the action-angle variables

, consider the Hamiltonian

and

. Let

. Which of the following are possible plots of the trajectories with different initial conditions in

plane?

Your answerSelect all correct options. You can choose more than one.

Diagram for Physics question 55 (figure 2)

Diagram for Physics question 55 (figure 3)

Diagram for Physics question 55 (figure 4)

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Consider a point charge

of mass

suspended by a mass, inextensible string of length

in free space (permittivity

) as shown in the figure. It is placed at a height

(

) over an infinitely large, grounded conducting plane. The gravitational potential energy is assumed to be zero at the position of the conducting plane and is positive above the plane. If

represents the angular position and

its corresponding canonical momentum, then the correct Hamiltonian of the system is

Your answerChoose one option.

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The time derivative of a differentiable function

is added to a Lagrangian

such that

where

are the generalized coordinates, generalized velocities and time, respectively. Let

be the generalized momentum and

the Hamiltonian associated with

. If

and

are those associated with

, then the correct option(s) is(are)

Your answerSelect all correct options. You can choose more than one.

Both

and

satisfy the Euler-Lagrange’s equations of motion

is conserved, then

is necessarily conserved

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GATE PH concept:hamiltonian PYQs

How to practice concept:hamiltonian

Question 51

Question 52

Question 53

Question 54

Question 62

Question 23

Question 54

Question 55

Question 30

Question 46