Generalised coordinates, D'Alembert's principle, Lagrange's equations of motion, Hamiltonian equations, conservative and non-conservative systems, cyclic variables, principle of least action - Question Bank
1. In Hamiltonian mechanics, the equation dq_i/dt = ∂H/∂p_i describes the time evolution of:
2. The Principle of Least Action is a variational principle. What does 'variational' refer to in this context?
3. The quantity ∂L/∂(dq_i/dt) represents the:
4. A system is rheonomic if its constraints depend explicitly on:
5. Which of the following is a fundamental difference between Lagrangian and Hamiltonian mechanics?
6. What does the term dp_i/dt = -∂H/∂q_i signify in Hamiltonian mechanics?
7. 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?
8. 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?
9. 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:
10. A system where constraints can be expressed in the form f(q_i, t) = 0 is called:
11. The canonical momentum conjugate to a generalized coordinate q_i is defined as:
12. Which branch of mechanics offers a more symmetric treatment of coordinates and momenta compared to Lagrangian mechanics?
13. For a system described by the Lagrangian L = T - V, where V is a function of position only, the Hamiltonian H is equal to:
14. If a coordinate q_i is not cyclic, its corresponding momentum p_i is generally:
15. The term 'inertial force' in D'Alembert's principle is mathematically equivalent to what?
16. 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?
17. Which principle forms the foundation for deriving both Lagrangian and Hamiltonian mechanics?
18. If the Hamiltonian H = H(q, p, t), it means the system's energy is conserved only if which condition is met?
19. The conservation of energy in a conservative system is a direct consequence of the potential energy depending only on what?
20. A constraint that depends explicitly on velocity, such as rolling without slipping, is typically:
21. If a system's constraints are independent of time and only involve coordinates, it is called:
22. What is the relationship between the Lagrangian (L) and the Hamiltonian (H) via a Legendre transformation?
23. A system is holonomic if its constraints can be expressed as equations relating coordinates and possibly time, but NOT involving what?
24. 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?
25. Which of the following is NOT a generalized coordinate for a particle moving in 3D space?
26. The transformation from Lagrangian to Hamiltonian mechanics involves replacing generalized velocities with what?
27. If the potential energy V depends on velocity, the system is generally considered:
28. Which quantity is conserved for a system where the Lagrangian does not explicitly depend on time?
29. D'Alembert's principle transforms a dynamic problem into a problem of what?
30. Consider a system with N degrees of freedom. How many generalized coordinates are needed to describe its configuration?
31. The Principle of Least Action is a fundamental variational principle that leads to which set of equations of motion?
32. The 'action' (S) in the Principle of Least Action is defined as the time integral of which quantity?
33. 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?
34. Which of the following is a direct consequence of a cyclic variable in the Lagrangian?
35. In Hamiltonian mechanics, if a coordinate q_i is cyclic, what is conserved?
36. If a generalized coordinate q_i is cyclic, what is conserved according to Lagrange's equations?
37. A cyclic variable in Lagrangian mechanics is a generalized coordinate whose corresponding term does not appear explicitly in which function?
38. Friction is an example of which type of force?
39. Which of the following is characteristic of a non-conservative system?
40. In a conservative system, the total mechanical energy (E = T + V) is conserved if the potential energy does not explicitly depend on what?
41. For a conservative system, the potential energy (V) depends only on what?
42. What are the Hamiltonian equations of motion?
43. Hamiltonian mechanics uses canonical variables, which are pairs of generalized coordinates (q_i) and their corresponding what?
44. The Hamiltonian (H) of a system is typically defined in terms of generalized coordinates and which other quantities?
45. What is the standard form of Lagrange's equation of motion for a system with generalized coordinate q_i?
46. The Lagrangian (L) of a system is defined as the difference between which two quantities?
47. Lagrange's equations of motion are expressed in terms of which quantities?
48. What is the fundamental equation of motion derived from D'Alembert's principle?
49. D'Alembert's principle extends Newton's second law of motion by introducing which concept?
50. In classical mechanics, what is the primary advantage of using generalized coordinates?