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

1. The term 'inertial force' in D'Alembert's principle is mathematically equivalent to what? — -m * a
2. The canonical momentum conjugate to a generalized coordinate q_i is defined as: — ∂L/∂(dq_i/dt)
3. 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: — ∂L/∂t = 0
4. 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? — 3
5. What is the standard form of Lagrange's equation of motion for a system with generalized coordinate q_i? — d/dt (∂L/∂(dq_i/dt)) - ∂L/∂q_i = 0
6. What are the Hamiltonian equations of motion? — dq_i/dt = ∂H/∂p_i and dp_i/dt = -∂H/∂q_i
7. The Principle of Least Action is a variational principle. What does 'variational' refer to in this context? — Finding the path for which a certain quantity (action) is extremum.
8. 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? — Generalized coordinate
9. In Hamiltonian mechanics, the equation dq_i/dt = ∂H/∂p_i describes the time evolution of: — Generalized coordinates
10. Lagrange's equations of motion are expressed in terms of which quantities? — Generalized coordinates and generalized velocities
11. The Hamiltonian (H) of a system is typically defined in terms of generalized coordinates and which other quantities? — Generalized momenta
12. The transformation from Lagrangian to Hamiltonian mechanics involves replacing generalized velocities with what? — Generalized momenta
13. Hamiltonian mechanics uses canonical variables, which are pairs of generalized coordinates (q_i) and their corresponding what? — Generalized momenta (p_i)
14. The quantity ∂L/∂(dq_i/dt) represents the: — Generalized momentum conjugate to q_i
15. What is the relationship between the Lagrangian (L) and the Hamiltonian (H) via a Legendre transformation? — H = Σ(p_i * dq_i/dt) - L
16. If the Hamiltonian H = H(q, p, t), it means the system's energy is conserved only if which condition is met? — H does not explicitly depend on time (t)
17. Which branch of mechanics offers a more symmetric treatment of coordinates and momenta compared to Lagrangian mechanics? — Hamiltonian mechanics
18. A system where constraints can be expressed in the form f(q_i, t) = 0 is called: — Holonomic
19. D'Alembert's principle extends Newton's second law of motion by introducing which concept? — Inertial forces
20. The Lagrangian (L) of a system is defined as the difference between which two quantities? — Kinetic Energy (T) and Potential Energy (V)
21. The Principle of Least Action is a fundamental variational principle that leads to which set of equations of motion? — Lagrange's equations
22. Which of the following is a fundamental difference between Lagrangian and Hamiltonian mechanics? — Lagrangian uses generalized coordinates and velocities; Hamiltonian uses generalized coordinates and momenta.
23. Consider a system with N degrees of freedom. How many generalized coordinates are needed to describe its configuration? — N
24. 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? — No, because the Lagrangian depends on θ.
25. If the potential energy V depends on velocity, the system is generally considered: — Non-conservative
26. Friction is an example of which type of force? — Non-conservative force
27. A constraint that depends explicitly on velocity, such as rolling without slipping, is typically: — Non-holonomic
28. If a coordinate q_i is not cyclic, its corresponding momentum p_i is generally: — Not conserved
29. The conservation of energy in a conservative system is a direct consequence of the potential energy depending only on what? — Position
30. For a conservative system, the potential energy (V) depends only on what? — Position coordinates
31. Which principle forms the foundation for deriving both Lagrangian and Hamiltonian mechanics? — Principle of Least Action
32. If a system's constraints are independent of time and only involve coordinates, it is called: — Scleronomic
33. D'Alembert's principle transforms a dynamic problem into a problem of what? — Statics
34. 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? — Stationary (minimum, maximum, or saddle point)
35. What is the fundamental equation of motion derived from D'Alembert's principle? — Sum of real and inertial forces equals zero
36. For a system described by the Lagrangian L = T - V, where V is a function of position only, the Hamiltonian H is equal to: — T + V
37. 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? — The action integral S
38. In Hamiltonian mechanics, if a coordinate q_i is cyclic, what is conserved? — The conjugate momentum p_i
39. Which of the following is a direct consequence of a cyclic variable in the Lagrangian? — The corresponding canonical momentum is conserved.
40. If a generalized coordinate q_i is cyclic, what is conserved according to Lagrange's equations? — The corresponding generalized momentum (p_i)
41. Which quantity is conserved for a system where the Lagrangian does not explicitly depend on time? — The Hamiltonian (H)
42. A cyclic variable in Lagrangian mechanics is a generalized coordinate whose corresponding term does not appear explicitly in which function? — The Lagrangian (L)
43. The 'action' (S) in the Principle of Least Action is defined as the time integral of which quantity? — The Lagrangian (L)
44. What does the term dp_i/dt = -∂H/∂q_i signify in Hamiltonian mechanics? — The rate of change of momentum is related to the spatial variation of energy.
45. In classical mechanics, what is the primary advantage of using generalized coordinates? — They simplify the description of constraints in a system.
46. In a conservative system, the total mechanical energy (E = T + V) is conserved if the potential energy does not explicitly depend on what? — Time
47. A system is rheonomic if its constraints depend explicitly on: — Time
48. Which of the following is NOT a generalized coordinate for a particle moving in 3D space? — Generalized velocity (dq/dt)
49. A system is holonomic if its constraints can be expressed as equations relating coordinates and possibly time, but NOT involving what? — Velocities
50. Which of the following is characteristic of a non-conservative system? — Work done by non-conservative forces depends on the path taken.
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E-Book 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 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 Generalised coordinates, D'Alembert's principle, Lagrange's equations of motion, Hamiltonian equations, conservative and non-conservative systems, cyclic variables, principle of least action