LAGRANGIAN & HAMILTONIAN
MECHANICS PRACTICE EXAM QUESTION
AND CORRECT ANSWERS (VERIFIED
ANSWERS) PLUS RATIONALES 2026 Q&A
INSTANT DOWNLOAD PDF
1. The Lagrangian of a mechanical system is defined as
A. K + V
B. V − K
C. K/V
D. K − V
Rationale: The Lagrangian is defined as kinetic energy minus potential
energy.
2. Which principle leads directly to Lagrange’s equations?
A. Newton’s second law
B. Conservation of momentum
C. Principle of least action
D. Conservation of energy
Rationale: Lagrange’s equations are derived from the stationary action
principle.
3. The action S is defined as
A. ∫V dt
B. ∫K dt
C. ∫L dt
D. ∫(K+V) dt
Rationale: Action is the time integral of the Lagrangian.
,4. Generalized coordinates are
A. Always Cartesian coordinates
B. Always angles
C. Any independent coordinates describing the system
D. Only spatial coordinates
Rationale: Generalized coordinates are chosen for convenience and
independence.
5. The Euler–Lagrange equation is
A. dL/dq = 0
B. d/dt(∂L/∂q) − ∂L/∂q̇ = 0
C. d/dt(∂L/∂q̇ ) − ∂L/∂q = 0
D. ∂L/∂t = 0
Rationale: This is the fundamental equation of motion in Lagrangian
mechanics.
6. If a coordinate does not appear explicitly in L, it is called
A. Generalized
B. Dependent
C. Cyclic (ignorable)
D. Constrained
Rationale: Absence from L implies conservation of its conjugate
momentum.
7. The conjugate momentum is defined as
A. ∂L/∂q
B. ∂L/∂t
C. ∂L/∂q̇
D. dL/dt
Rationale: This definition generalizes linear momentum.
8. If L has no explicit time dependence, then
A. Momentum is conserved
B. Angular momentum is conserved
C. Energy is conserved
, D. Action is zero
Rationale: Time-translation symmetry implies energy conservation.
9. Holonomic constraints are
A. Velocity dependent
B. Time dependent only
C. Expressible as equations among coordinates and time
D. Inequality constraints
Rationale: Holonomic constraints reduce degrees of freedom via
equations.
10.Non-holonomic constraints depend on
A. Coordinates only
B. Time only
C. Velocities or inequalities
D. Forces only
Rationale: They cannot be integrated into coordinate-only relations.
11.The Hamiltonian is defined as
A. K − V
B. ∑pᵢqᵢ
C. ∑pᵢq̇ ᵢ − L
D. ∂L/∂t
Rationale: This Legendre transform defines the Hamiltonian.
12.For conservative systems, the Hamiltonian equals
A. Potential energy
B. K − V
C. Total energy
D. Action
Rationale: When L has no explicit time dependence, H = E.
13.Hamilton’s equations are
A. Second-order differential equations
MECHANICS PRACTICE EXAM QUESTION
AND CORRECT ANSWERS (VERIFIED
ANSWERS) PLUS RATIONALES 2026 Q&A
INSTANT DOWNLOAD PDF
1. The Lagrangian of a mechanical system is defined as
A. K + V
B. V − K
C. K/V
D. K − V
Rationale: The Lagrangian is defined as kinetic energy minus potential
energy.
2. Which principle leads directly to Lagrange’s equations?
A. Newton’s second law
B. Conservation of momentum
C. Principle of least action
D. Conservation of energy
Rationale: Lagrange’s equations are derived from the stationary action
principle.
3. The action S is defined as
A. ∫V dt
B. ∫K dt
C. ∫L dt
D. ∫(K+V) dt
Rationale: Action is the time integral of the Lagrangian.
,4. Generalized coordinates are
A. Always Cartesian coordinates
B. Always angles
C. Any independent coordinates describing the system
D. Only spatial coordinates
Rationale: Generalized coordinates are chosen for convenience and
independence.
5. The Euler–Lagrange equation is
A. dL/dq = 0
B. d/dt(∂L/∂q) − ∂L/∂q̇ = 0
C. d/dt(∂L/∂q̇ ) − ∂L/∂q = 0
D. ∂L/∂t = 0
Rationale: This is the fundamental equation of motion in Lagrangian
mechanics.
6. If a coordinate does not appear explicitly in L, it is called
A. Generalized
B. Dependent
C. Cyclic (ignorable)
D. Constrained
Rationale: Absence from L implies conservation of its conjugate
momentum.
7. The conjugate momentum is defined as
A. ∂L/∂q
B. ∂L/∂t
C. ∂L/∂q̇
D. dL/dt
Rationale: This definition generalizes linear momentum.
8. If L has no explicit time dependence, then
A. Momentum is conserved
B. Angular momentum is conserved
C. Energy is conserved
, D. Action is zero
Rationale: Time-translation symmetry implies energy conservation.
9. Holonomic constraints are
A. Velocity dependent
B. Time dependent only
C. Expressible as equations among coordinates and time
D. Inequality constraints
Rationale: Holonomic constraints reduce degrees of freedom via
equations.
10.Non-holonomic constraints depend on
A. Coordinates only
B. Time only
C. Velocities or inequalities
D. Forces only
Rationale: They cannot be integrated into coordinate-only relations.
11.The Hamiltonian is defined as
A. K − V
B. ∑pᵢqᵢ
C. ∑pᵢq̇ ᵢ − L
D. ∂L/∂t
Rationale: This Legendre transform defines the Hamiltonian.
12.For conservative systems, the Hamiltonian equals
A. Potential energy
B. K − V
C. Total energy
D. Action
Rationale: When L has no explicit time dependence, H = E.
13.Hamilton’s equations are
A. Second-order differential equations
