ADVANCED MECHANICS
COMPREHENSIVE EXAM QUESTION AND
CORRECT ANSWERS (VERIFIED
ANSWERS) PLUS RATIONALES 2026 Q&A
INSTANT DOWNLOAD PDF
1. The generalized coordinates in Lagrangian mechanics are best defined as
A. Cartesian coordinates only
B. Coordinates equal to the number of forces
C. Coordinates equal to the number of particles
D. Coordinates equal to the degrees of freedom
Correct answer: D
Rationale: Generalized coordinates are chosen to match the system’s degrees of
freedom and need not be Cartesian.
2. A system with 𝑛degrees of freedom has how many independent generalized
velocities?
A. 𝑛 − 1
B. 𝑛
C. 2𝑛
D. 3𝑛
Correct answer: B
Rationale: Each generalized coordinate has a corresponding generalized
velocity.
, 3. The Lagrangian 𝐿of a system is defined as
A. 𝑇 + 𝑉
B. 𝑉 − 𝑇
C. 𝑇 − 𝑉
D. 𝑇/𝑉
Correct answer: C
Rationale: The Lagrangian is defined as kinetic energy minus potential energy.
4. If a coordinate does not appear explicitly in the Lagrangian, it is called
A. Constrained
B. Cyclic
C. Holonomic
D. Virtual
Correct answer: B
Rationale: A cyclic (ignorable) coordinate leads to a conserved conjugate
momentum.
5. Conservation of linear momentum arises from invariance under
A. Time translation
B. Space translation
C. Rotation
D. Scaling
Correct answer: B
Rationale: Noether’s theorem links spatial translation symmetry to momentum
conservation.
6. The Euler–Lagrange equation is
𝑑
A. (∂𝐿/ ∂𝑞) = ∂𝐿/ ∂𝑞̇
𝑑𝑡
, B. ∂𝐿/ ∂𝑞 = 0
𝑑
C. (∂𝐿/ ∂𝑞̇ ) − ∂𝐿/ ∂𝑞 = 0
𝑑𝑡
D. ∂𝑇/ ∂𝑞 = ∂𝑉/ ∂𝑞
Correct answer: C
Rationale: This is the fundamental equation governing motion in Lagrangian
mechanics.
7. The Hamiltonian is obtained from the Lagrangian by
A. Legendre transformation
B. Fourier transformation
C. Laplace transformation
D. Canonical substitution
Correct answer: A
Rationale: The Hamiltonian is the Legendre transform of the Lagrangian with
respect to velocities.
8. In classical mechanics, the Hamiltonian usually represents
A. Potential energy
B. Kinetic energy
C. Total energy
D. Work done
Correct answer: C
Rationale: For time-independent systems, the Hamiltonian equals total energy.
9. Canonical equations of motion are
A. Second-order differential equations
B. First-order differential equations
COMPREHENSIVE EXAM QUESTION AND
CORRECT ANSWERS (VERIFIED
ANSWERS) PLUS RATIONALES 2026 Q&A
INSTANT DOWNLOAD PDF
1. The generalized coordinates in Lagrangian mechanics are best defined as
A. Cartesian coordinates only
B. Coordinates equal to the number of forces
C. Coordinates equal to the number of particles
D. Coordinates equal to the degrees of freedom
Correct answer: D
Rationale: Generalized coordinates are chosen to match the system’s degrees of
freedom and need not be Cartesian.
2. A system with 𝑛degrees of freedom has how many independent generalized
velocities?
A. 𝑛 − 1
B. 𝑛
C. 2𝑛
D. 3𝑛
Correct answer: B
Rationale: Each generalized coordinate has a corresponding generalized
velocity.
, 3. The Lagrangian 𝐿of a system is defined as
A. 𝑇 + 𝑉
B. 𝑉 − 𝑇
C. 𝑇 − 𝑉
D. 𝑇/𝑉
Correct answer: C
Rationale: The Lagrangian is defined as kinetic energy minus potential energy.
4. If a coordinate does not appear explicitly in the Lagrangian, it is called
A. Constrained
B. Cyclic
C. Holonomic
D. Virtual
Correct answer: B
Rationale: A cyclic (ignorable) coordinate leads to a conserved conjugate
momentum.
5. Conservation of linear momentum arises from invariance under
A. Time translation
B. Space translation
C. Rotation
D. Scaling
Correct answer: B
Rationale: Noether’s theorem links spatial translation symmetry to momentum
conservation.
6. The Euler–Lagrange equation is
𝑑
A. (∂𝐿/ ∂𝑞) = ∂𝐿/ ∂𝑞̇
𝑑𝑡
, B. ∂𝐿/ ∂𝑞 = 0
𝑑
C. (∂𝐿/ ∂𝑞̇ ) − ∂𝐿/ ∂𝑞 = 0
𝑑𝑡
D. ∂𝑇/ ∂𝑞 = ∂𝑉/ ∂𝑞
Correct answer: C
Rationale: This is the fundamental equation governing motion in Lagrangian
mechanics.
7. The Hamiltonian is obtained from the Lagrangian by
A. Legendre transformation
B. Fourier transformation
C. Laplace transformation
D. Canonical substitution
Correct answer: A
Rationale: The Hamiltonian is the Legendre transform of the Lagrangian with
respect to velocities.
8. In classical mechanics, the Hamiltonian usually represents
A. Potential energy
B. Kinetic energy
C. Total energy
D. Work done
Correct answer: C
Rationale: For time-independent systems, the Hamiltonian equals total energy.
9. Canonical equations of motion are
A. Second-order differential equations
B. First-order differential equations
