CLASSICAL MECHANICS QUALIFYING
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
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1
1. A particle moves in one dimension with Lagrangian 𝐿 = 𝑚𝑥̇ 2 − 𝑘𝑥 2 . The
2
equation of motion is
A. 𝑚𝑥̈ = −𝑘𝑥
B. 𝑚𝑥̈ = −2𝑘𝑥
C. 𝑚𝑥̈ = 𝑘𝑥
D. 𝑚𝑥̇ = −𝑘𝑥
Rationale: The Euler–Lagrange equation gives 𝑚𝑥̈ + 2𝑘𝑥 = 0, so the force
is −2𝑘𝑥.
2. The generalized momentum corresponding to coordinate 𝑞is defined as
A. 𝑝 = ∂𝐿/ ∂𝑞
B. 𝑝 = ∂𝐻/ ∂𝑞
C. 𝑝 = ∂𝐿/ ∂𝑞̇
D. 𝑝 = 𝑞̇ /𝐿
Rationale: In Lagrangian mechanics, generalized momentum is the partial
derivative of the Lagrangian with respect to the generalized velocity.
3. A coordinate is cyclic if
A. It appears only in kinetic energy
B. It does not appear explicitly in the Lagrangian
C. Its velocity is zero
, D. The Hamiltonian is zero
Rationale: Absence of a coordinate in the Lagrangian implies conservation
of its conjugate momentum.
4. For a conservative force, the work done over a closed path is
A. Maximum
B. Minimum
C. Nonzero
D. Zero
Rationale: Conservative forces are path-independent, giving zero net work
over a closed loop.
5. The Hamiltonian of a system is equal to the total energy when
A. Forces are nonconservative
B. Coordinates are constrained
C. The Lagrangian has no explicit time dependence
D. The potential depends on velocity
Rationale: If 𝐿has no explicit time dependence, the Hamiltonian equals
total energy.
6. In central force motion, the conserved vector in an inverse-square law is
A. Angular momentum
B. Energy
C. Runge–Lenz vector
D. Linear momentum
Rationale: The Runge–Lenz vector is uniquely conserved for inverse-square
central forces.
7. The effective potential in radial motion includes
A. Only the true potential
B. Kinetic energy
C. Centrifugal term plus true potential
D. Total energy
Rationale: Effective potential combines the actual potential with the
angular momentum barrier term.
, 8. The Poisson bracket {𝑞 , 𝑝}equals
A. 0
B. –1
C. 1
D. 𝑞𝑝
Rationale: Canonical coordinates satisfy {𝑞, 𝑝} = 1.
9. A canonical transformation preserves
A. Coordinates
B. Momenta
C. Form of Hamilton’s equations
D. Energy
Rationale: Canonical transformations preserve the structure of Hamilton’s
equations.
10.Hamilton’s principal function satisfies
A. Euler–Lagrange equation
B. Newton’s law
C. Hamilton–Jacobi equation
D. Liouville’s theorem
Rationale: Hamilton’s principal function is a solution to the Hamilton–
Jacobi equation.
11.The number of degrees of freedom of a rigid body in 3D space is
A. 3
B. 4
C. 6
D. 9
Rationale: A rigid body has 3 translational and 3 rotational degrees of
freedom.
12.The moment of inertia depends on
A. Velocity
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
DOWNLOAD PDF
1
1. A particle moves in one dimension with Lagrangian 𝐿 = 𝑚𝑥̇ 2 − 𝑘𝑥 2 . The
2
equation of motion is
A. 𝑚𝑥̈ = −𝑘𝑥
B. 𝑚𝑥̈ = −2𝑘𝑥
C. 𝑚𝑥̈ = 𝑘𝑥
D. 𝑚𝑥̇ = −𝑘𝑥
Rationale: The Euler–Lagrange equation gives 𝑚𝑥̈ + 2𝑘𝑥 = 0, so the force
is −2𝑘𝑥.
2. The generalized momentum corresponding to coordinate 𝑞is defined as
A. 𝑝 = ∂𝐿/ ∂𝑞
B. 𝑝 = ∂𝐻/ ∂𝑞
C. 𝑝 = ∂𝐿/ ∂𝑞̇
D. 𝑝 = 𝑞̇ /𝐿
Rationale: In Lagrangian mechanics, generalized momentum is the partial
derivative of the Lagrangian with respect to the generalized velocity.
3. A coordinate is cyclic if
A. It appears only in kinetic energy
B. It does not appear explicitly in the Lagrangian
C. Its velocity is zero
, D. The Hamiltonian is zero
Rationale: Absence of a coordinate in the Lagrangian implies conservation
of its conjugate momentum.
4. For a conservative force, the work done over a closed path is
A. Maximum
B. Minimum
C. Nonzero
D. Zero
Rationale: Conservative forces are path-independent, giving zero net work
over a closed loop.
5. The Hamiltonian of a system is equal to the total energy when
A. Forces are nonconservative
B. Coordinates are constrained
C. The Lagrangian has no explicit time dependence
D. The potential depends on velocity
Rationale: If 𝐿has no explicit time dependence, the Hamiltonian equals
total energy.
6. In central force motion, the conserved vector in an inverse-square law is
A. Angular momentum
B. Energy
C. Runge–Lenz vector
D. Linear momentum
Rationale: The Runge–Lenz vector is uniquely conserved for inverse-square
central forces.
7. The effective potential in radial motion includes
A. Only the true potential
B. Kinetic energy
C. Centrifugal term plus true potential
D. Total energy
Rationale: Effective potential combines the actual potential with the
angular momentum barrier term.
, 8. The Poisson bracket {𝑞 , 𝑝}equals
A. 0
B. –1
C. 1
D. 𝑞𝑝
Rationale: Canonical coordinates satisfy {𝑞, 𝑝} = 1.
9. A canonical transformation preserves
A. Coordinates
B. Momenta
C. Form of Hamilton’s equations
D. Energy
Rationale: Canonical transformations preserve the structure of Hamilton’s
equations.
10.Hamilton’s principal function satisfies
A. Euler–Lagrange equation
B. Newton’s law
C. Hamilton–Jacobi equation
D. Liouville’s theorem
Rationale: Hamilton’s principal function is a solution to the Hamilton–
Jacobi equation.
11.The number of degrees of freedom of a rigid body in 3D space is
A. 3
B. 4
C. 6
D. 9
Rationale: A rigid body has 3 translational and 3 rotational degrees of
freedom.
12.The moment of inertia depends on
A. Velocity
