TRB PG Assistant - Generalised coordinates, D'Alembert's principle, Lagrange's equations of motion, Hamiltonian equations, conservative and non-conservative systems, cyclic variables, principle of least action Online Test (MCQ Quiz)

1. In classical mechanics, what is the primary advantage of using generalized coordinates?

A) They simplify the description of constraints in a system. B) They always represent physical distances. C) They increase the number of degrees of freedom. D) They are only applicable to linear motion.

2. D'Alembert's principle extends Newton's second law of motion by introducing which concept?

A) Inertial forces B) Potential energy C) Angular momentum D) Centripetal force

3. What is the fundamental equation of motion derived from D'Alembert's principle?

A) Sum of forces equals mass times acceleration B) Sum of real and inertial forces equals zero C) Sum of potential and kinetic energy is constant D) Sum of impulses equals change in momentum

4. Lagrange's equations of motion are expressed in terms of which quantities?

A) Generalized coordinates and generalized velocities B) Force and acceleration C) Momentum and position D) Energy and work

5. The Lagrangian (L) of a system is defined as the difference between which two quantities?

A) Kinetic Energy (T) and Potential Energy (V) B) Potential Energy (V) and Kinetic Energy (T) C) Total Energy (E) and Kinetic Energy (T) D) Work done (W) and Potential Energy (V)

6. What is the standard form of Lagrange's equation of motion for a system with generalized coordinate q_i?

A) d/dt (∂L/∂(dq_i/dt)) - ∂L/∂q_i = 0 B) d/dt (∂L/∂q_i) - ∂L/∂(dq_i/dt) = 0 C) ∂L/∂q_i + d/dt (∂L/∂(dq_i/dt)) = 0 D) ∂L/∂(dq_i/dt) - d/dt (∂L/∂q_i) = 0

7. The Hamiltonian (H) of a system is typically defined in terms of generalized coordinates and which other quantities?

A) Generalized momenta B) Generalized velocities C) Forces D) Work

8. Hamiltonian mechanics uses canonical variables, which are pairs of generalized coordinates (q_i) and their corresponding what?

A) Generalized momenta (p_i) B) Generalized velocities (dq_i/dt) C) Forces (F_i) D) Energies (E_i)

9. What are the Hamiltonian equations of motion?

A) dq_i/dt = ∂H/∂p_i and dp_i/dt = -∂H/∂q_i B) dq_i/dt = ∂H/∂q_i and dp_i/dt = -∂H/∂p_i C) dq_i/dt = -∂H/∂p_i and dp_i/dt = ∂H/∂q_i D) dq_i/dt = ∂H/∂p_i and dp_i/dt = ∂H/∂q_i

10. For a conservative system, the potential energy (V) depends only on what?

A) Position coordinates B) Velocity coordinates C) Time D) Force

11. In a conservative system, the total mechanical energy (E = T + V) is conserved if the potential energy does not explicitly depend on what?

A) Time B) Position C) Velocity D) Mass

12. Which of the following is characteristic of a non-conservative system?

A) Work done by non-conservative forces depends on the path taken. B) Total mechanical energy is always conserved. C) Potential energy is a function of position only. D) Forces are derivable from a scalar potential.

13. Friction is an example of which type of force?

A) Non-conservative force B) Conservative force C) Central force D) Inertial force

14. A cyclic variable in Lagrangian mechanics is a generalized coordinate whose corresponding term does not appear explicitly in which function?

A) The Lagrangian (L) B) The Hamiltonian (H) C) The kinetic energy (T) D) The potential energy (V)

15. If a generalized coordinate q_i is cyclic, what is conserved according to Lagrange's equations?

A) The corresponding generalized momentum (p_i) B) The generalized velocity (dq_i/dt) C) The generalized force (Q_i) D) The total energy (E)

16. In Hamiltonian mechanics, if a coordinate q_i is cyclic, what is conserved?

A) The conjugate momentum p_i B) The Hamiltonian H C) The velocity dq_i/dt D) The generalized force Q_i

17. Which of the following is a direct consequence of a cyclic variable in the Lagrangian?

A) The corresponding canonical momentum is conserved. B) The total energy is conserved. C) The system's degrees of freedom increase. D) The Lagrangian becomes time-independent.

18. The Principle of Least Action states that the actual path taken by a system between two points in configuration space is the one for which the action is what?

A) Stationary (minimum, maximum, or saddle point) B) Minimum C) Maximum D) Zero

19. The 'action' (S) in the Principle of Least Action is defined as the time integral of which quantity?

A) The Lagrangian (L) B) The Hamiltonian (H) C) The kinetic energy (T) D) The potential energy (V)

20. The Principle of Least Action is a fundamental variational principle that leads to which set of equations of motion?

A) Lagrange's equations B) Hamilton's equations C) Newton's laws D) D'Alembert's principle

21. Consider a system with N degrees of freedom. How many generalized coordinates are needed to describe its configuration?

A) N B) 3N C) 2N D) N-1

22. D'Alembert's principle transforms a dynamic problem into a problem of what?

A) Statics B) Kinematics C) Energy conservation D) Momentum conservation

23. Which quantity is conserved for a system where the Lagrangian does not explicitly depend on time?

A) The Hamiltonian (H) B) The action (S) C) The kinetic energy (T) D) The potential energy (V)

24. If the potential energy V depends on velocity, the system is generally considered:

A) Non-conservative B) Conservative C) Holonomic D) Non-holonomic

25. The transformation from Lagrangian to Hamiltonian mechanics involves replacing generalized velocities with what?

A) Generalized momenta B) Generalized accelerations C) Forces D) Work

26. Which of the following is NOT a generalized coordinate for a particle moving in 3D space?

A) Time (t) B) Cartesian coordinates (x, y, z) C) Spherical coordinates (r, θ, φ) D) Generalized velocity (dq/dt)

27. The equation dp_i/dt = -∂H/∂q_i in Hamiltonian mechanics represents the rate of change of generalized momentum as related to the negative gradient of the Hamiltonian with respect to what?

A) Generalized coordinate B) Generalized velocity C) Time D) Generalized momentum

28. A system is holonomic if its constraints can be expressed as equations relating coordinates and possibly time, but NOT involving what?

A) Velocities B) Positions C) Time D) Mass

29. What is the relationship between the Lagrangian (L) and the Hamiltonian (H) via a Legendre transformation?

A) H = Σ(p_i * dq_i/dt) - L B) H = Σ(p_i * q_i) - L C) H = L - Σ(p_i * dq_i/dt) D) H = Σ(p_i) * L

30. If a system's constraints are independent of time and only involve coordinates, it is called:

A) Scleronomic B) Rheonomic C) Holonomic D) Non-holonomic

31. A constraint that depends explicitly on velocity, such as rolling without slipping, is typically:

A) Non-holonomic B) Holonomic C) Scleronomic D) Integrable

32. The conservation of energy in a conservative system is a direct consequence of the potential energy depending only on what?

A) Position B) Velocity C) Time D) Momentum

33. If the Hamiltonian H = H(q, p, t), it means the system's energy is conserved only if which condition is met?

A) H does not explicitly depend on time (t) B) H depends only on q C) H depends only on p D) p does not explicitly depend on time (t)

34. Which principle forms the foundation for deriving both Lagrangian and Hamiltonian mechanics?

A) Principle of Least Action B) D'Alembert's Principle C) Principle of Superposition D) Principle of Conservation of Energy

35. In the context of generalized coordinates, a system's degrees of freedom is defined as the minimum number of independent coordinates required to specify its configuration. What is the number of degrees of freedom for a single particle moving freely in 3D space?

A) 3 B) 1 C) 2 D) 6

36. The term 'inertial force' in D'Alembert's principle is mathematically equivalent to what?

A) -m * a B) m * a C) Potential energy D) Kinetic energy

37. If a coordinate q_i is not cyclic, its corresponding momentum p_i is generally:

A) Not conserved B) Conserved C) Zero D) Infinite

38. For a system described by the Lagrangian L = T - V, where V is a function of position only, the Hamiltonian H is equal to:

A) T + V B) T - V C) V - T D) T * V

39. Which branch of mechanics offers a more symmetric treatment of coordinates and momenta compared to Lagrangian mechanics?

A) Hamiltonian mechanics B) Newtonian mechanics C) D'Alembert's principle D) Statics

40. The canonical momentum conjugate to a generalized coordinate q_i is defined as:

A) ∂L/∂(dq_i/dt) B) ∂L/∂q_i C) ∂H/∂q_i D) ∂H/∂(dq_i/dt)

41. A system where constraints can be expressed in the form f(q_i, t) = 0 is called:

A) Holonomic B) Non-holonomic C) Rheonomic D) Scleronomic

42. If the Lagrangian depends explicitly on time, i.e., L = L(q, dq/dt, t), then the quantity Σ(p_i * dq_i/dt) - L is conserved if and only if:

A) ∂L/∂t = 0 B) ∂L/∂t ≠ 0 C) ∂L/∂t is maximized D) ∂L/∂t is minimized

43. The mathematical formulation of the Principle of Least Action involves variations of the path and leads to Euler-Lagrange equations. What is the quantity being varied?

A) The action integral S B) The Lagrangian L C) The Hamiltonian H D) The kinetic energy T

44. Consider a simple pendulum with angle θ as the generalized coordinate. If the length of the pendulum is constant and gravity is the only force, is θ a cyclic variable?

A) No, because the Lagrangian depends on θ. B) Yes, because the Lagrangian depends on θ. C) No, because the Lagrangian depends on dθ/dt. D) Yes, because the Lagrangian depends on dθ/dt.

45. What does the term dp_i/dt = -∂H/∂q_i signify in Hamiltonian mechanics?

A) The rate of change of momentum is related to the spatial variation of energy. B) The rate of change of position is related to the temporal variation of energy. C) The rate of change of momentum is related to the temporal variation of energy. D) The rate of change of position is related to the spatial variation of momentum.

46. Which of the following is a fundamental difference between Lagrangian and Hamiltonian mechanics?

A) Lagrangian uses generalized coordinates and velocities; Hamiltonian uses generalized coordinates and momenta. B) Lagrangian uses forces; Hamiltonian uses energy. C) Lagrangian uses time-independent equations; Hamiltonian uses time-dependent equations. D) Lagrangian applies to conservative systems; Hamiltonian applies to non-conservative systems.

47. A system is rheonomic if its constraints depend explicitly on:

A) Time B) Position C) Velocity D) Momentum

48. The quantity ∂L/∂(dq_i/dt) represents the:

A) Generalized momentum conjugate to q_i B) Generalized force acting on q_i C) Generalized velocity D) Time derivative of generalized momentum

49. The Principle of Least Action is a variational principle. What does 'variational' refer to in this context?

A) Finding the path for which a certain quantity (action) is extremum. B) Minimizing the kinetic energy. C) Maximizing the potential energy. D) Considering all possible trajectories instantaneously.

50. In Hamiltonian mechanics, the equation dq_i/dt = ∂H/∂p_i describes the time evolution of:

A) Generalized coordinates B) Generalized momenta C) Generalized velocities D) Hamiltonian

Generalised coordinates, D'Alembert's principle, Lagrange's equations of motion, Hamiltonian equations, conservative and non-conservative systems, cyclic variables, principle of least action - Online Test

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