CSIR NET/ GATE/ JL/DL/ Asst. Prof /SET Physics Online Class

 Classical Mechanics (Generalized Coordinates, Generalized Velocity, Degree of Freedom & Constraints)

1.The minimum number of independent coordinates required to specify a system completely is called:
A) Constraint
B) Degree of Freedom
C) Generalized Velocity
D) Momentum

✅ Answer: B

2. Generalized coordinates are usually represented by:
A) x, y, z
B) p, q
C) q₁, q₂, q₃
D) v₁, v₂

✅ Answer: C

3. A free particle in three-dimensional space has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 6 DOF

✅ Answer: C

4. The generalized velocity is:
A) dq/dt
B) dq/dx
C) dp/dt
D) dx/dq

✅ Answer: A

5. The generalized velocity corresponding to coordinate qᵢ is:
A) q²
B) q̇ᵢ
C) q⁻¹
D) pᵢ

✅ Answer: B

6. A simple pendulum has:
A) 3 DOF
B) 2 DOF
C) 1 DOF
D) 4 DOF

✅ Answer: C

7. The generalized coordinate for a simple pendulum is:
A) x
B) y
C) z
D) θ

✅ Answer: D

8. The constraint x² + y² = l² is:
A) Non-holonomic
B) Holonomic
C) Unilateral
D) Rheonomous

✅ Answer: B

9. A constraint expressible as f(q,t)=0 is called:
A) Bilateral
B) Holonomic
C) Unilateral
D) Scleronomous

✅ Answer: B

10. The DOF of a particle constrained to move on a plane is:
A) 1
B) 2
C) 3
D) 4

✅ Answer: B

11. A particle constrained to move along a straight line has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 6 DOF

✅ Answer: A

12. The relation z = 0 represents:
A) Two constraints
B) Three constraints
C) One constraint
D) No constraint

✅ Answer: C

13. A rigid body moving freely in space has:
A) 3 DOF
B) 4 DOF
C) 5 DOF
D) 6 DOF

✅ Answer: D

14. A rigid body moving in a plane has:
A) 2 DOF
B) 3 DOF
C) 4 DOF
D) 6 DOF

✅ Answer: B

15. DOF is equal to:
A) Constraints – Coordinates
B) Coordinates + Constraints
C) Coordinates – Constraints
D) Coordinates × Constraints

✅ Answer: C

16. The constraint equation of a particle on a sphere is:
A) x+y+z=R
B) x²+y²+z²=R²
C) xy=R
D) yz=R

✅ Answer: B

17. A particle on a sphere has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: B

18. A rolling wheel without slipping is an example of:
A) Holonomic constraint
B) Bilateral constraint
C) Non-Holonomic constraint
D) Scleronomous constraint

✅ Answer: C

19. Which constraint involves velocity?
A) Holonomic
B) Non-Holonomic
C) Bilateral
D) Scleronomous

✅ Answer: B

20. The equation dx − Rdθ = 0 is:
A) Holonomic
B) Non-Holonomic
C) Bilateral
D) Unilateral

✅ Answer: B

21. A time-independent constraint is called:
A) Rheonomous
B) Non-Holonomic
C) Scleronomous
D) Bilateral

✅ Answer: C

22. A time-dependent constraint is called:
A) Holonomic
B) Rheonomous
C) Bilateral
D) Unilateral

✅ Answer: B

23. Which is an example of a rheonomous constraint?
A) x²+y²=l²
B) x²+y²=(l+at)²
C) z=0
D) y=0

✅ Answer: B

24. The constraint y ≥ 0 represents:
A) Bilateral
B) Holonomic
C) Unilateral
D) Rheonomous

✅ Answer: C

25. A ball resting on a floor illustrates:
A) Bilateral constraint
B) Unilateral constraint
C) Holonomic constraint
D) Rheonomous constraint

✅ Answer: B

26. The number of generalized coordinates equals:
A) Number of constraints
B) Degree of freedom
C) Number of particles
D) Number of forces

✅ Answer: B

27. For N particles in space, total coordinates are:
A) N
B) 2N
C) 3N
D) 6N

✅ Answer: C

28. A system with 12 coordinates and 4 constraints has:
A) 16 DOF
B) 8 DOF
C) 12 DOF
D) 4 DOF

✅ Answer: B

29. A particle constrained to a circle has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: A

30. Generalized coordinates need not be:
A) Independent
B) Cartesian
C) Minimal
D) Unique

✅ Answer: B

31. The coordinate θ for rotational motion is:
A) Cartesian coordinate
B) Generalized coordinate
C) Momentum coordinate
D) Constraint

✅ Answer: B

32. The generalized velocity corresponding to θ is:
A) θ²
B) θ⁻¹
C) θ̇
D) θ̈

✅ Answer: C

33. The generalized velocity is also called:
A) Configuration variable
B) Time derivative of generalized coordinate
C) Constraint force
D) Linear momentum

✅ Answer: B

34. The configuration space dimension equals:
A) Number of constraints
B) DOF
C) Number of particles
D) Number of forces

✅ Answer: B

35. A particle on a circle in a plane is described by:
A) Two independent coordinates
B) One independent coordinate
C) Three coordinates
D) Four coordinates

✅ Answer: B

36. Which constraint reduces the number of DOF?
A) Independent coordinate
B) Generalized velocity
C) Constraint
D) Force

✅ Answer: C

37. Which is NOT a type of constraint?
A) Holonomic
B) Non-Holonomic
C) Scleronomous
D) Conservative

✅ Answer: D

38. The motion of a train on a track has:
A) 3 DOF
B) 2 DOF
C) 1 DOF
D) 6 DOF

✅ Answer: C

39. A particle moving on a fixed surface generally has:
A) One constraint
B) No constraint
C) Infinite constraints
D) Two constraints

✅ Answer: A

40. For a pendulum, the constraint arises because:
A) Mass is constant
B) Length is constant
C) Gravity exists
D) Velocity changes

✅ Answer: B

41. The constraint x²+y²−l²=0 is:
A) Holonomic
B) Non-Holonomic
C) Unilateral
D) Rheonomous

✅ Answer: A

42. The coordinates of a free rigid body are:
A) Three translational only
B) Three rotational only
C) Three translational + Three rotational
D) Two translational + Two rotational

✅ Answer: C

43. A wheel rolling without slipping is:
A) Holonomic
B) Non-Holonomic
C) Scleronomous
D) Unilateral

✅ Answer: B

44. Which quantity determines the configuration of a system?
A) Force
B) Energy
C) Generalized coordinates
D) Momentum

✅ Answer: C

45. The generalized coordinate is:
A) Always linear distance
B) Always angular displacement
C) Any independent variable describing the system
D) Always Cartesian coordinate

✅ Answer: C

46. A particle on a plane surface has generalized coordinates:
A) One
B) Two
C) Three
D) Four

✅ Answer: B

47. The generalized coordinates are especially useful in:
A) Newtonian Mechanics
B) Lagrangian Mechanics
C) Thermodynamics
D) Optics

✅ Answer: B

48. The motion of a particle inside a sphere has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 0 DOF

✅ Answer: C

49. A constraint represented by an equality sign is generally:
A) Bilateral
B) Unilateral
C) Non-Holonomic
D) Rheonomous

✅ Answer: A

50. The foundation of Lagrangian mechanics is based on:
A) Generalized coordinates and constraints
B) Maxwell equations
C) Schrödinger equation
D) Gauss law

✅ Answer: A

Advanced MCQs (51–100)

51. The number of degrees of freedom of a particle constrained to move on the surface of a sphere is:
A) 0
B) 1
C) 2
D) 3

✅ Answer: C

52. A rigid body in three-dimensional space possesses:
A) 3 translational DOF only
B) 3 rotational DOF only
C) 6 DOF
D) 9 DOF

✅ Answer: C

53. A double pendulum has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: B

54. The generalized coordinates are:
A) Always Cartesian coordinates
B) Independent coordinates describing the configuration
C) Velocity components
D) Force components

✅ Answer: B

55. The generalized velocity corresponding to coordinate (q_i) is:
A) (\ddot q_i)
B) (\dot q_i)
C) (q_i^2)
D) (p_i)

✅ Answer: B

56. For N particles moving freely in space, the total degrees of freedom are:
A) N
B) 2N
C) 3N
D) 6N

✅ Answer: C

57. A system of 4 particles connected by rigid rods to form a rigid body has:
A) 12 DOF
B) 6 DOF
C) 3 DOF
D) 9 DOF

✅ Answer: B

58. A particle moving on a circle is best described by:
A) Two generalized coordinates
B) Three generalized coordinates
C) One generalized coordinate
D) Zero generalized coordinates

✅ Answer: C

59. The constraint equation of a simple pendulum is:
A) (x+y=l)
B) (x^2+y^2=l^2)
C) (x-y=l)
D) (xy=l)

✅ Answer: B

60. Which of the following is a holonomic constraint?
A) Rolling without slipping
B) (x^2+y^2=a^2)
C) (dx-Rd\theta=0)
D) (v_x+v_y=0)

✅ Answer: B

61. A rolling disk without slipping is:
A) Holonomic
B) Non-Holonomic
C) Bilateral
D) Rheonomous

✅ Answer: B

62. A particle constrained by (x+y+z=5) has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: B

63. The configuration space dimension equals:
A) Number of particles
B) Number of constraints
C) Degrees of freedom
D) Generalized velocities

✅ Answer: C

64. The generalized coordinates in spherical coordinates are:
A) (r,\theta,\phi)
B) (x,y,z)
C) (r,\theta)
D) (x,y)

✅ Answer: A

65. A particle constrained to move along a helix has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 0 DOF

✅ Answer: A

66.

The number of independent generalized coordinates for a rigid body rotating about a fixed point is:
A) 1
B) 2
C) 3
D) 6

✅ Answer: C

67. The generalized momentum corresponding to (q_i) is:
A) (\partial L/\partial q_i)
B) (\partial L/\partial \dot q_i)
C) (m\dot q_i)
D) (F_i)

✅ Answer: B

68. A constraint involving time explicitly is:
A) Holonomic
B) Bilateral
C) Rheonomous
D) Scleronomous

✅ Answer: C

69. A bead sliding on a fixed wire is an example of:
A) Constrained motion
B) Free motion
C) Random motion
D) Rotational motion

✅ Answer: A

70. The motion of an elevator is:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 6 DOF

✅ Answer: A

71. A rigid body in a plane has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 6 DOF

✅ Answer: C

72. The number of rotational degrees of freedom of a rigid body in space is:
A) 1
B) 2
C) 3
D) 6

✅ Answer: C

73. Which constraint can be expressed as an inequality?
A) Holonomic
B) Bilateral
C) Unilateral
D) Scleronomous

✅ Answer: C

74. The motion of a particle on a cylindrical surface has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: B

75. A pendulum with varying length represents:
A) Scleronomous system
B) Rheonomous system
C) Holonomic system only
D) Bilateral system only

✅ Answer: B

76. For a particle moving on a sphere:
A) One constraint exists
B) Two constraints exist
C) Three constraints exist
D) No constraints exist

✅ Answer: A

77. The generalized coordinate need not represent:
A) Length
B) Angle
C) Time
D) Independent variable

✅ Answer: C

78. A system with 8 coordinates and 3 independent constraints has:
A) 11 DOF
B) 8 DOF
C) 5 DOF
D) 3 DOF

✅ Answer: C

79. Which coordinate system is often convenient for central force problems?
A) Cartesian
B) Cylindrical
C) Spherical
D) Rectangular

✅ Answer: C

80. The minimum number of coordinates needed to describe a rigid body's orientation in space is:
A) 1
B) 2
C) 3
D) 6

✅ Answer: C

Numerical & Conceptual MCQs (81–100)

81. A particle in 3D space subject to two independent constraints has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 5 DOF

✅ Answer: A

82. For N particles and k independent constraints:

DOF =

A) N−k
B) 2N−k
C) 3N−k
D) 3N+k

✅ Answer: C

83. A particle moving on a plane satisfies:
A) x=0
B) y=0
C) z=0
D) xyz=0

✅ Answer: C

84. The constraint (x^2+y^2+z^2-R^2=0) is:
A) Holonomic
B) Non-Holonomic
C) Unilateral
D) Rheonomous

✅ Answer: A

85. Which one is NOT a generalized coordinate?
A) θ
B) φ
C) r
D) Force

✅ Answer: D

86. A train moving on a track has:
A) 3 DOF
B) 2 DOF
C) 1 DOF
D) 6 DOF

✅ Answer: C

87. For a particle moving along x-axis:
A) DOF=1
B) DOF=2
C) DOF=3
D) DOF=0

✅ Answer: A

88. The relation (x=a\cos\theta, y=a\sin\theta) suggests:
A) Two independent coordinates
B) One independent coordinate
C) Three independent coordinates
D) No coordinate

✅ Answer: B

89. The number of Euler angles required to specify orientation is:
A) 1
B) 2
C) 3
D) 4

✅ Answer: C

90. A wheel rolling on a rough surface without slipping is:
A) Scleronomous
B) Holonomic
C) Non-Holonomic
D) Bilateral

✅ Answer: C

91. A rigid rod joining two particles introduces:
A) No constraint
B) One constraint
C) Two constraints
D) Three constraints

✅ Answer: B

92. A free particle moving in a plane has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: B

93. The motion of a bead on a circular wire has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: A

94. Generalized coordinates are chosen to:
A) Increase constraints
B) Simplify equations
C) Increase forces
D) Reduce energy

✅ Answer: B

95. The generalized velocity has dimensions of:
A) Force
B) Coordinate/time
C) Energy
D) Momentum

✅ Answer: B

96. A particle inside a cube without touching walls has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 6 DOF

✅ Answer: C

97. The Lagrangian formulation is based primarily on:
A) Coordinates and momenta
B) Generalized coordinates and velocities
C) Forces only
D) Energies only

✅ Answer: B

98. Which is a bilateral constraint?
A) y ≥ 0
B) x²+y²=a²
C) z ≥ 0
D) x > 0

✅ Answer: B

99. A particle constrained on a parabola has:
A) 1 DOF
B) 2 DOF
C) 3 DOF
D) 4 DOF

✅ Answer: A

100. The concept of generalized coordinates was developed mainly for:
A) Maxwell's equations
B) Lagrangian mechanics
C) Thermodynamics
D) Relativity

✅ Answer: B

Classical Mechanics – 50 MCQs

Generalized Coordinates, Generalized Momentum, Lagrangian and Equations of Motion

The number of generalized coordinates required to describe a system equals its:

A) Mass
B) Momentum
C) Degrees of freedom
D) Energy

Answer: C

2. Generalized coordinates are generally denoted by:

A) (x,y,z)
B) (p,q,r)
C) (q_1,q_2,\dots,q_n)
D) (F_x,F_y,F_z)

Answer: C

3. For a simple pendulum, the generalized coordinate is:

A) (x)
B) (y)
C) (r)
D) (\theta)

Answer: D

4. A particle constrained to move on a circle has how many degrees of freedom?

A) 0
B) 1
C) 2
D) 3

Answer: B

5. The time derivative of a generalized coordinate is called:

A) Generalized force
B) Generalized momentum
C) Generalized velocity
D) Angular momentum

Answer: C

6. Generalized velocity is represented by:

A) (q)
B) (\dot q)
C) (\ddot q)
D) (p)

Answer: B

7. The Lagrangian of a system is:

A) (T+V)
B) (T-V)
C) (V-T)
D) (TV)

Answer: B

8. The kinetic energy of a free particle is:

A) (\frac12 mv)
B) (mv)
C) (\frac12 mv^2)
D) (mv^2)

Answer: C

9.

For a free particle, the potential energy is:

A) Infinite
B) Constant nonzero
C) Zero
D) Negative

Answer: C

10. The generalized momentum is defined as:

A) (\frac{\partial L}{\partial q})
B) (\frac{\partial T}{\partial q})
C) (\frac{\partial L}{\partial \dot q})
D) (m\dot q^2)

Answer: C

11. For a free particle, generalized momentum becomes:

A) (m)
B) (v)
C) (mv)
D) (\frac{m}{v})

Answer: C

12. The generalized momentum conjugate to (\theta) in a pendulum is:

A) (m\dot\theta)
B) (ml\dot\theta)
C) (ml^2\dot\theta)
D) (mgl\dot\theta)

Answer: C

13. The action is defined as:

A) (\int Tdt)
B) (\int Vdt)
C) (\int Ldt)
D) (TV)

Answer: C

14. Hamilton's principle states that:

A) Energy is conserved
B) Momentum is conserved
C) Action is stationary
D) Force is constant

Answer: C

15. The Euler-Lagrange equation is:

A) (\frac{\partial L}{\partial q}=0)

B) (\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right)-\frac{\partial L}{\partial q}=0)

C) (F=ma)

D) (p=mv)

Answer: B

16.

The Euler-Lagrange equation is derived from:

A) Newton's law
B) Hamilton's principle
C) Hooke's law
D) Gauss law

Answer: B

17.

A system with (N) particles in 3D space has total coordinates:

A) (N)
B) (2N)
C) (3N)
D) (4N)

Answer: C

18.

The SI unit of action is:

A) Joule
B) Joule-second
C) Newton
D) Watt

Answer: B

19.

For a particle in polar coordinates, generalized coordinates are:

A) (x,y)
B) (r,\theta)
C) (r,z)
D) (x,z)

Answer: B

20.

The Lagrangian is a function of:

A) (q,\dot q,t)
B) (p,q) only
C) (F,t) only
D) (V) only

Answer: A

21.

The generalized momentum has dimensions of:

A) Force
B) Energy
C) Linear momentum
D) Power

Answer: C

22.

The kinetic energy of a pendulum is:

A) (\frac12 ml\dot\theta^2)

B) (\frac12 ml^2\dot\theta^2)

C) (ml^2\dot\theta)

D) (mgl\theta)

Answer: B

23.

Potential energy of a pendulum is:

A) (mgl\cos\theta)

B) (mgl(1-\cos\theta))

C) (mgl\sin\theta)

D) (mg\theta)

Answer: B

24.

The equation of motion of a pendulum is:

A) (\ddot\theta+\frac{g}{l}\sin\theta=0)

B) (\ddot\theta-\frac{g}{l}\sin\theta=0)

C) (\dot\theta+\frac{g}{l}\theta=0)

D) (\ddot\theta+\theta=0)

Answer: A

25.

For small oscillations:

A) (\sin\theta=\theta^2)

B) (\sin\theta=1)

C) (\sin\theta\approx\theta)

D) (\sin\theta=0)

Answer: C

26.

The SHM equation of a pendulum is:

A) (\ddot\theta+\frac{g}{l}\theta=0)

B) (\ddot\theta-\frac{g}{l}\theta=0)

C) (\dot\theta+\frac{g}{l}\theta=0)

D) (\theta=\frac{g}{l})

Answer: A

27.

For SHM, the potential energy is:

A) (\frac12 kx^2)

B) (kx)

C) (kx^3)

D) (x/k)

Answer: A

28.

The Lagrangian of SHM is:

A) (\frac12 m\dot x^2+\frac12 kx^2)

B) (\frac12 m\dot x^2-\frac12 kx^2)

C) (\frac12 kx^2-\frac12 m\dot x^2)

D) (kx)

Answer: B

29.

Equation of motion for SHM:

A) (m\ddot x-kx=0)

B) (m\ddot x+kx=0)

C) (m\dot x+kx=0)

D) (m\ddot x=0)

Answer: B

30.

The generalized force is associated with:

A) (\frac{\partial L}{\partial q})

B) (\frac{\partial L}{\partial \dot q})

C) (\frac{\partial T}{\partial t})

D) (V)

Answer: A

31.

If (L) does not explicitly depend on (q), then:

A) (q) is cyclic coordinate

B) (q) is constrained

C) (q) is constant

D) (L=0)

Answer: A

32.

For a cyclic coordinate:

A) Energy conserved

B) Momentum conserved

C) Force conserved

D) Velocity conserved

Answer: B

33.

The generalized momentum corresponding to a cyclic coordinate is:

A) Variable

B) Conserved

C) Zero

D) Infinite

Answer: B

34.

The variational principle used in mechanics is:

A) Hamilton's principle

B) Huygens principle

C) Fermat principle

D) Pauli principle

Answer: A

35.

Lagrangian mechanics is particularly useful for:

A) Constrained systems

B) Only free particles

C) Only rigid bodies

D) Only fluids

Answer: A

36.

The coordinates (x,y,z) are called:

A) Generalized coordinates

B) Cartesian coordinates

C) Cyclic coordinates

D) Canonical coordinates

Answer: B

37.

The generalized coordinate must be:

A) Independent

B) Dependent

C) Constant

D) Cyclic only

Answer: A

38.

The number of generalized coordinates equals:

A) Constraints

B) Degrees of freedom

C) Number of particles

D) Energy

Answer: B

39.

For a particle moving on a circle:

A) (q=r)

B) (q=\phi)

C) (q=z)

D) (q=x)

Answer: B

40.

The action has dimensions of:

A) Energy × Time

B) Force × Time

C) Momentum × Time

D) Energy × Length

Answer: A

41.

Which quantity remains stationary in Hamilton's principle?

A) Energy

B) Momentum

C) Action

D) Force

Answer: C

42.

The Euler-Lagrange equation is a:

A) First-order equation

B) Second-order differential equation

C) Algebraic equation

D) Integral equation

Answer: B

43.

For SHM, generalized momentum is:

A) (m\dot x)

B) (kx)

C) (mx)

D) (\dot x)

Answer: A

44.

The Lagrangian formalism avoids direct use of:

A) Coordinates

B) Energy

C) Forces

D) Time

Answer: C

45.

A pendulum has generalized momentum:

A) (ml\dot\theta)

B) (ml^2\dot\theta)

C) (mgl)

D) (m\theta)

Answer: B

46.

If a system has 5 degrees of freedom, it requires:

A) 3 generalized coordinates

B) 4 generalized coordinates

C) 5 generalized coordinates

D) 6 generalized coordinates

Answer: C

47.

The kinetic energy contributes to:

A) Potential term only

B) Lagrangian positively

C) Lagrangian negatively

D) Not at all

Answer: B

48.

The potential energy contributes to:

A) Positive sign in L

B) Negative sign in L

C) Zero sign

D) Variable sign only

Answer: B

49.

The equation (m\ddot x+kx=0) represents:

A) Free particle

B) Damped oscillator

C) Simple harmonic oscillator

D) Projectile

Answer: C

50.

The most fundamental equation of Lagrangian mechanics is:

A) Newton's Second Law

B) Hooke's Law

C) Euler-Lagrange Equation

D) Conservation of Energy

Answer: C

50 MCQs on Cyclic Coordinates and Conservation Laws

1.  A coordinate (q_i) is said to be cyclic if:

A) (\dot q_i=0)

B) (q_i=0)

C) (L) does not explicitly depend on (q_i)

D) (L=0)

Answer: C

2. If (q_i) is a cyclic coordinate, then:

A) Velocity is conserved

B) Energy is conserved

C) Conjugate momentum is conserved

D) Force is conserved

Answer: C

3. The generalized momentum conjugate to (q_i) is:

A) (\frac{\partial L}{\partial q_i})

B) (\frac{\partial V}{\partial q_i})

C) (\frac{\partial L}{\partial \dot q_i})

D) (m\dot q_i^2)

Answer: C

4. For a cyclic coordinate (q_i),

[
\frac{\partial L}{\partial q_i}
]

equals:

A) 1

B) (\infty)

C) Constant

D) 0

Answer: D

5.

If (\theta) is cyclic, then the conserved quantity is:

A) Energy

B) Angular momentum

C) Force

D) Torque

Answer: B

6.

Which theorem connects symmetry and conservation laws?

A) Gauss Theorem

B) Green Theorem

C) Noether's Theorem

D) Stokes Theorem

Answer: C

7.

Translational symmetry leads to conservation of:

A) Energy

B) Angular momentum

C) Linear momentum

D) Charge

Answer: C

8.

Rotational symmetry implies conservation of:

A) Mass

B) Energy

C) Angular momentum

D) Velocity

Answer: C

9.

Time translation symmetry implies conservation of:

A) Energy

B) Charge

C) Angular momentum

D) Position

Answer: A

10.

For a free particle,

[
L=\frac12 m\dot x^2
]

the coordinate (x) is:

A) Cyclic

B) Non-cyclic

C) Generalized force

D) Fixed

Answer: A

11.

For the free particle, conserved momentum is:

A) (m)

B) (x)

C) (m\dot x)

D) (\dot x^2)

Answer: C

12.

A cyclic coordinate indicates:

A) Hidden symmetry

B) Friction

C) Dissipation

D) Constraint violation

Answer: A

13.

For a particle in central force motion, which coordinate is cyclic?

A) (r)

B) (\theta)

C) (t)

D) (V)

Answer: B

14.

Conservation of angular momentum follows from:

A) Rotational symmetry

B) Time symmetry

C) Translational symmetry

D) Reflection symmetry

Answer: A

15.

If (L) does not explicitly depend on time, then:

A) Momentum conserved

B) Energy conserved

C) Force conserved

D) Velocity conserved

Answer: B

16.

The Euler-Lagrange equation for a cyclic coordinate becomes:

A) (\dot q=0)

B) (\ddot q=0)

C) (\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right)=0)

D) (L=0)

Answer: C

17.

A conserved generalized momentum remains:

A) Increasing

B) Decreasing

C) Constant

D) Oscillating

Answer: C

18.

For a particle in a central potential (V(r)), conserved quantity is:

A) Linear momentum

B) Angular momentum

C) Position

D) Acceleration

Answer: B

19.

The Lagrangian

[
L=\frac12 m(\dot r^2+r^2\dot\theta^2)-V(r)
]

contains cyclic coordinate:

A) (r)

B) (\theta)

C) Both

D) None

Answer: B

20.

For Question 19, conserved momentum is:

A) (m\dot r)

B) (r)

C) (mr^2\dot\theta)

D) (V(r))

Answer: C

21.

Noether's theorem relates:

A) Energy and force

B) Symmetry and conservation laws

C) Mass and energy

D) Work and power

Answer: B

22.

Linear momentum conservation arises from:

A) Rotational invariance

B) Time invariance

C) Translational invariance

D) Gauge invariance

Answer: C

23.

Energy conservation arises from:

A) Time invariance

B) Space invariance

C) Rotation invariance

D) Reflection invariance

Answer: A

24.

A cyclic coordinate is also called:

A) Ignorable coordinate

B) Fixed coordinate

C) Cartesian coordinate

D) Independent variable

Answer: A

25.

For an ignorable coordinate:

A) (\partial L/\partial q=0)

B) (\partial L/\partial q=1)

C) (\partial L/\partial q=\infty)

D) (\partial L/\partial q=\dot q)

Answer: A

26.

The generalized momentum of a cyclic coordinate is:

A) Conserved

B) Variable

C) Infinite

D) Zero

Answer: A

27.

In polar coordinates, angular momentum is:

A) (m\dot r)

B) (mr\dot\theta)

C) (mr^2\dot\theta)

D) (r\dot\theta)

Answer: C

28.

If the Lagrangian is independent of (\theta), then:

A) (p_\theta) conserved

B) (r) conserved

C) Energy lost

D) Force conserved

Answer: A

29.

The Hamiltonian is conserved if:

A) (L) independent of time

B) (L) independent of position

C) (V=0)

D) (T=0)

Answer: A

30.

For a free particle in three dimensions, the number of cyclic coordinates is:

A) 0

B) 1

C) 2

D) 3

Answer: D

31.

The conserved quantity associated with x-translation symmetry is:

A) (p_x)

B) (L_z)

C) Energy

D) Charge

Answer: A

32.

Rotational invariance about z-axis conserves:

A) (L_x)

B) (L_y)

C) (L_z)

D) Energy

Answer: C

33.

A cyclic coordinate need not be:

A) Independent

B) Generalized

C) Cartesian

D) Conserved

Answer: C

34.

Which quantity is conserved in a central force field?

A) Angular momentum

B) Linear momentum

C) Torque

D) Potential energy

Answer: A

35.

If (\partial L/\partial t=0), the system possesses:

A) Time symmetry

B) Space symmetry

C) Rotational symmetry

D) Reflection symmetry

Answer: A

36.

For a cyclic coordinate:

A) Force is zero

B) Corresponding momentum conserved

C) Velocity zero

D) Potential energy zero

Answer: B

37.

The angular coordinate in planetary motion is:

A) Non-cyclic

B) Cyclic

C) Fixed

D) Constrained

Answer: B

38.

The quantity

[
mr^2\dot\theta
]

represents:

A) Energy

B) Torque

C) Angular momentum

D) Power

Answer: C

39.

The Euler-Lagrange equation reflects:

A) Newton's laws

B) Conservation laws

C) Variational principle

D) All of the above

Answer: D

40.

Which conservation law arises from rotational symmetry?

A) Energy

B) Charge

C) Angular momentum

D) Mass

Answer: C

41.

The existence of a cyclic coordinate simplifies:

A) Equations of motion

B) Constraints

C) Potential energy

D) Mass

Answer: A

42.

A conserved momentum reduces the effective degrees of freedom by:

A) One

B) Two

C) Three

D) Four

Answer: A

43.

A coordinate is cyclic when:

A) It appears in kinetic energy only

B) It appears in potential energy only

C) It does not appear explicitly in (L)

D) It is constant

Answer: C

44.

Noether's theorem was developed by:

A) Newton

B) Hamilton

C) Emmy Noether

D) Lagrange

Answer: C

45.

The generalized momentum associated with time symmetry corresponds to:

A) Angular momentum

B) Energy

C) Force

D) Torque

Answer: B

46.

For a free particle, translational symmetry exists along:

A) x-axis only

B) y-axis only

C) z-axis only

D) All spatial directions

Answer: D

47.

The conservation of angular momentum implies:

A) Zero force

B) Zero torque

C) Constant energy

D) Constant velocity

Answer: B

48.

Which of the following is not directly a consequence of Noether's theorem?

A) Energy conservation

B) Momentum conservation

C) Angular momentum conservation

D) Hooke's law

Answer: D

49.

The cyclic coordinate concept belongs primarily to:

A) Newtonian mechanics

B) Lagrangian mechanics

C) Fluid mechanics

D) Thermodynamics

Answer: B

50.

The most important statement regarding cyclic coordinates is:

A) They always vanish

B) Their conjugate momenta are conserved

C) They are dependent variables

D) They are constants

Answer: B

Classical Mechanics – Canonical Transformations

50 Multiple Choice Questions (MCQs)

1. A canonical transformation is a transformation that preserves:

A) Newton's equations
B) Lagrange's equations
C) Hamilton's equations
D) Euler's equations

Answer: C

2. Canonical transformations are associated with:

A) Hamiltonian mechanics
B) Fluid mechanics
C) Thermodynamics
D) Optics

Answer: A

3. The variables transformed in a canonical transformation are:

A) Position and velocity
B) Force and momentum
C) Generalized coordinates and momenta
D) Mass and energy

Answer: C

4. A canonical transformation changes:

A) ((q,p)) to ((Q,P))
B) ((q,\dot q)) to ((Q,\dot Q))
C) ((x,y)) to ((u,v))
D) None of these

Answer: A

5. Which theorem is preserved under canonical transformations?

A) Bernoulli theorem
B) Liouville theorem
C) Gauss theorem
D) Stokes theorem

Answer: B

6. The fundamental Poisson bracket relation is:

A) ({Q,P}=0)
B) ({Q,P}=1)
C) ({Q,P}=-1)
D) ({Q,P}=2)

Answer: B

7. For a canonical transformation:

A) ({Q,Q}=1)
B) ({P,P}=1)
C) ({Q,Q}=0)
D) ({Q,P}=2)

Answer: C

8. The relation ({P_i,P_j}) for canonical variables is:

A) 1
B) −1
C) 0
D) δij

Answer: C

9. ({Q_i,P_j}) equals:

A) 0
B) δij
C) −δij
D) 2δij

Answer: B

10. Canonical transformations simplify:

A) Hamiltonian systems
B) Maxwell equations
C) Navier-Stokes equations
D) Wave optics only

Answer: A

11. How many standard generating functions exist?

A) 2
B) 3
C) 4
D) 5

Answer: C

12. Type-1 generating function depends on:

A) (q,Q,t)
B) (q,P,t)
C) (p,Q,t)
D) (p,P,t)

Answer: A

13. Type-2 generating function is:

A) F₁(q,Q,t)
B) F₂(q,P,t)
C) F₃(p,Q,t)
D) F₄(p,P,t)

Answer: B

14. Type-3 generating function depends on:

A) q,Q
B) q,P
C) p,Q
D) p,P

Answer: C

15. Type-4 generating function depends on:

A) q,Q
B) q,P
C) p,Q
D) p,P

Answer: D

16. For (F_2(q,P,t)):

A) (Q=\partial F_2/\partial P)
B) (Q=-\partial F_2/\partial P)
C) (P=\partial F_2/\partial q)
D) None

Answer: A

17. For (F_2(q,P,t)):

A) (p=\partial F_2/\partial q)
B) (p=-\partial F_2/\partial q)
C) (Q=\partial F_2/\partial q)
D) None

Answer: A

18. Canonical transformations preserve:

A) Phase-space structure
B) Only momentum
C) Only energy
D) Only coordinates

Answer: A

19. The mathematical tool used to test canonicality is:

A) Gradient
B) Divergence
C) Poisson bracket
D) Curl

Answer: C

20. Hamilton-Jacobi theory is based on:

A) Canonical transformations
B) Newton's laws
C) Fluid dynamics
D) Electromagnetism

Answer: A

21. The new Hamiltonian after transformation is denoted by:

A) H
B) G
C) K
D) L

Answer: C

22. If (K=0), the transformed variables become:

A) Infinite
B) Constants
C) Periodic
D) Undefined

Answer: B

23. The generator of an infinitesimal canonical transformation is:

A) H
B) G
C) K
D) T

Answer: B

24. Infinitesimal transformations involve:

A) Large parameter
B) Small parameter ε
C) Infinite parameter
D) Zero parameter only

Answer: B

25. Hamilton's equations in new variables are:

A) Not preserved
B) Preserved
C) Reversed
D) Eliminated

Answer: B

26. Canonical transformations are useful in:

A) Action-angle variables
B) Perturbation theory
C) Hamilton-Jacobi theory
D) All of these

Answer: D

27. The phase-space rotation is an example of:

A) Non-canonical transformation
B) Canonical transformation
C) Singular transformation
D) Point transformation

Answer: B

28. Poisson brackets were introduced by:

A) Newton
B) Euler
C) Poisson
D) Hamilton

Answer: C

29. Canonical transformations preserve:

A) Symplectic structure
B) Temperature
C) Pressure
D) Density

Answer: A

30. Canonical variables are:

A) Independent coordinates only
B) Generalized coordinates and momenta
C) Forces only
D) Velocities only

Answer: B

31. The Kronecker delta δij equals:

A) 1 for i=j, 0 otherwise
B) Always 1
C) Always 0
D) −1 for i=j

Answer: A

32. Which generating function uses (q,Q)?

A) F₁
B) F₂
C) F₃
D) F₄

Answer: A

33. Which generating function uses (p,P)?

A) F₁
B) F₂
C) F₃
D) F₄

Answer: D

34. Canonical transformations are central to:

A) Hamiltonian mechanics
B) Hydrodynamics
C) Acoustics
D) Thermodynamics

Answer: A

35. The Hamilton-Jacobi equation involves:

A) Principal function S
B) Entropy S
C) Action F
D) Potential V

Answer: A

36. Canonical transformations preserve:

A) Equations of motion
B) Hamiltonian form
C) Poisson brackets
D) All of these

Answer: D

37. Which is NOT a generating function?

A) F₁
B) F₂
C) F₃
D) F₅

Answer: D

38. The transformation ((q,p)\rightarrow(Q,P)) is canonical if:

A) Poisson brackets are preserved
B) Energy is zero
C) Momentum is constant
D) Force is conserved

Answer: A

39. The quantity generating canonical transformations is called:

A) Potential
B) Generator
C) Action constant
D) Tensor

Answer: B

40. Canonical transformations are also called:

A) Symplectic transformations
B) Orthogonal transformations
C) Galilean transformations
D) Lorentz transformations

Answer: A

41. In Hamiltonian mechanics, coordinates and momenta form:

A) Configuration space
B) Phase space
C) Velocity space
D) Hilbert space

Answer: B

42. A canonical transformation may change:

A) Variables only
B) Physical predictions
C) Equations of motion
D) None

Answer: A

43. Which relation must hold?

A) ({Q_i,Q_j}=0)
B) ({P_i,P_j}=0)
C) ({Q_i,P_j}=\delta_{ij})
D) All of these

Answer: D

44. Canonical transformations are particularly useful for:

A) Solving dynamical systems
B) Calculating density
C) Measuring temperature
D) Finding refractive index

Answer: A

45. The Hamilton-Jacobi method seeks:

A) A suitable generating function
B) Velocity field
C) Electric field
D) Pressure field

Answer: A

46. Canonical transformations preserve:

A) Area in phase space
B) Volume in phase space
C) Liouville theorem
D) All of these

Answer: D

47. A transformation with ({Q,P}=1) is:

A) Non-canonical
B) Canonical
C) Singular
D) Impossible

Answer: B

48. Canonical transformations provide a connection to:

A) Quantum mechanics
B) Thermodynamics only
C) Fluid mechanics only
D) Elasticity only

Answer: A

49. The primary objective of canonical transformations is:

A) Increase complexity
B) Simplify the Hamiltonian
C) Change mass
D) Change force

Answer: B

50. Canonical transformations are a topic in:

A) Hamiltonian formulation of Classical Mechanics
B) Electromagnetic Theory
C) Statistical Mechanics only
D) Optics only

Answer: A