Q1. In Hamilton's principle, the action \(S\) is defined as:

\(S=\int T\,dt\)
\(S=\int V\,dt\)
\(S=\int_{t_1}^{t_2} L\,dt\)
\(S=T+V\)

Q2. For a conservative mechanical system, the Lagrangian is:

\(L=T+V\)
\(L=T-V\)
\(L=V-T\)
\(L=TV\)

Q3. Hamilton's principle is mathematically expressed as:

\(S=0\)
\(dS/dt=0\)
\(\delta S=0\)
\(S=\infty\)

Q4. The condition \(\delta S=0\) means that the actual path makes the action:

Always zero
Always maximum
Stationary with respect to small variations of the path
Infinite

Q5. In applying Hamilton's principle, the variations of generalized coordinates at the end points satisfy:

\(\delta q(t_1)=1\)
\(\delta q(t_2)=1\)
\(\delta q(t_1)=\delta q(t_2)=0\)
\(\delta q(t_1)=\delta q(t_2)=\infty\)

Q6. The principle of least action is more accurately described as a principle of:

Maximum force
Minimum kinetic energy
Stationary action
Zero potential energy

Q7. The shortest distance between two points in a plane is:

A parabola
A circle
A straight line
A cycloid

Q8. For a curve \(y(x)\), an infinitesimal arc length is:

\(ds=dx+dy\)
\(ds=\sqrt{1+(dy/dx)^2}\,dx\)
\(ds=(dy/dx)dx\)
\(ds=dx/dy\)

Q9. The functional to be minimized in the shortest-distance problem is:

\(\int y\,dx\)
\(\displaystyle \int\sqrt{1+y'^2}\,dx\)
\(\int y^2\,dx\)
\(\int y'\,dx\)

Q10. The brachistochrone problem asks for the curve along which a particle moves between two points:

With minimum distance
In minimum time under gravity
With constant velocity
With zero acceleration

Q11. The solution of the brachistochrone problem is a:

Straight line
Parabola
Circle
Cycloid

Q12. In the brachistochrone problem, the straight line is generally:

The path of minimum time
The shortest path, but not the path of minimum time
Longer than every possible curve
Impossible for the particle to follow

Q13. The mathematical technique used to find the function that makes an integral stationary is called:

Vector algebra
Matrix mechanics
Calculus of variations
Dimensional analysis

Q14. The Euler-Lagrange equation for a functional involving \(f(y,y',x)\) is:

\(\displaystyle \frac{\partial f}{\partial y}+\frac{d}{dx}\frac{\partial f}{\partial y'}=0\)
\(\displaystyle \frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right)=0\)
\(\displaystyle \frac{df}{dx}=0\)
\(\displaystyle \frac{\partial f}{\partial y'}=0\)

Q15. Applying Hamilton's principle to \(S=\int L\,dt\) gives:

Newton's law of gravitation directly
Lagrange's equations of motion
Kepler's third law only
Hooke's law only

Q16. Lagrange's equation obtained from the variational principle is:

\(\displaystyle \frac{d}{dt}\frac{\partial L}{\partial q_i}-\frac{\partial L}{\partial\dot q_i}=0\)
\(\displaystyle \frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right)-\frac{\partial L}{\partial q_i}=0\)
\(\displaystyle \frac{dL}{dt}=0\)
\(\displaystyle \frac{\partial L}{\partial t}=0\)

Q17. In deriving Lagrange's equation from Hamilton's principle, integration by parts is used on the term containing:

\(\delta q_i\)
\(\delta\dot q_i\)
\(V\)
\(T\) only

Q18. The generalized momentum conjugate to \(q_i\) is:

\(\displaystyle p_i=\frac{\partial L}{\partial q_i}\)
\(\displaystyle p_i=\frac{\partial L}{\partial\dot q_i}\)
\(p_i=mq_i\)
\(p_i=Lq_i\)

Q19. The Hamiltonian is obtained from the Lagrangian through:

Fourier transformation
Coordinate rotation
Legendre transformation
Laplace transformation

Q20. The Hamiltonian is defined by:

\(H=L-\sum_i p_i\dot q_i\)
\(\displaystyle H=\sum_i p_i\dot q_i-L\)
\(H=T-V\)
\(H=LV\)

Q21. For many ordinary conservative mechanical systems, the Hamiltonian is equal to:

\(T-V\)
\(T+V\)
\(T/V\)
\(V-T\)

Q22. Hamilton's first canonical equation is:

\(\displaystyle \dot q_i=-\frac{\partial H}{\partial p_i}\)
\(\displaystyle \dot q_i=\frac{\partial H}{\partial p_i}\)
\(\displaystyle \dot q_i=\frac{\partial H}{\partial q_i}\)
\(\dot q_i=H\)

Q23. Hamilton's second canonical equation is:

\(\displaystyle \dot p_i=\frac{\partial H}{\partial q_i}\)
\(\displaystyle \dot p_i=-\frac{\partial H}{\partial q_i}\)
\(\displaystyle \dot p_i=\frac{\partial H}{\partial p_i}\)
\(\dot p_i=0\) always

Q24. A system with \(n\) degrees of freedom has how many first-order Hamilton equations?

\(n/2\)
\(n\)
\(2n\)
\(n^2\)

Q25. Hamiltonian mechanics describes the state of a system using:

\(q_i\) only
\(p_i\) only
Generalized coordinates \(q_i\) and conjugate momenta \(p_i\)
Forces only

Q26. The space whose coordinates are \((q_i,p_i)\) is called:

Configuration space
Real space
Phase space
Euclidean plane only

Q27. The phase space of a system with 3 degrees of freedom has dimension:

2
3
6
9

Q28. If the Hamiltonian has no explicit time dependence, then:

\(H=0\)
\(H\) is conserved
Momentum must be zero
Position must be constant

Q29. Newtonian mechanics is primarily formulated in terms of:

Forces and accelerations
Action only
Phase space only
Hamiltonian only

Q30. Lagrangian mechanics is primarily based on:

\(F=ma\) alone
The Lagrangian and generalized coordinates
Momentum only
Torque only

Q31. Hamiltonian mechanics replaces \(n\) second-order Lagrange equations by:

One second-order equation
\(2n\) first-order equations
\(n^2\) algebraic equations
Three vector equations

Q32. For a free particle in one dimension, \(H=p^2/(2m)\). Hamilton's first equation gives:

\(\dot x=p^2/m\)
\(\dot x=p/m\)
\(\dot x=-p/m\)
\(\dot x=0\)

Q33. The Hamiltonian of a one-dimensional linear harmonic oscillator is:

\(\displaystyle H=\frac{p^2}{2m}-\frac12kx^2\)
\(\displaystyle H=\frac{p^2}{2m}+\frac12kx^2\)
\(H=px\)
\(H=p^2+kx^2\)

Q34. For \(H=p^2/(2m)+kx^2/2\), Hamilton's first equation gives:

\(\dot x=kx\)
\(\dot x=p/m\)
\(\dot x=-kx\)
\(\dot x=-p/m\)

Q35. For the same harmonic oscillator, Hamilton's second equation gives:

\(\dot p=kx\)
\(\dot p=-kx\)
\(\dot p=p/m\)
\(\dot p=0\)

Q36. Combining the two Hamilton equations for the harmonic oscillator gives:

\(m\ddot x-kx=0\)
\(m\ddot x+kx=0\)
\(m\dot x+kx=0\)
\(m\ddot x=0\)

Q37. For a harmonic oscillator with \(m=2\,\text{kg}\), \(k=8\,\text{N/m}\), its angular frequency is:

\(1\,\text{rad/s}\)
\(2\,\text{rad/s}\)
\(4\,\text{rad/s}\)
\(8\,\text{rad/s}\)

Q38. For a simple pendulum of mass \(m\) and length \(l\), the momentum conjugate to \(\theta\) is:

\(p_\theta=m\dot\theta\)
\(p_\theta=ml\dot\theta\)
\(p_\theta=ml^2\dot\theta\)
\(p_\theta=mgl\)

Q39. Taking the lowest point as zero potential energy, the Hamiltonian of a simple pendulum is:

\(\displaystyle H=\frac{p_\theta^2}{2ml^2}-mgl(1-\cos\theta)\)
\(\displaystyle H=\frac{p_\theta^2}{2ml^2}+mgl(1-\cos\theta)\)
\(H=p_\theta+mgl\)
\(H=p_\theta^2/2m\)

Q40. Hamilton's equation for \(\dot\theta\) of a simple pendulum gives:

\(\dot\theta=p_\theta/m\)
\(\displaystyle \dot\theta=\frac{p_\theta}{ml^2}\)
\(\dot\theta=-mgl\sin\theta\)
\(\dot\theta=ml^2p_\theta\)

Q41. Hamilton's equation for the conjugate momentum of the simple pendulum gives:

\(\dot p_\theta=mgl\sin\theta\)
\(\dot p_\theta=-mgl\sin\theta\)
\(\dot p_\theta=p_\theta/ml^2\)
\(\dot p_\theta=0\)

Q42. Combining Hamilton's equations for a simple pendulum gives:

\(\ddot\theta-\frac{g}{l}\sin\theta=0\)
\(\ddot\theta+\frac{g}{l}\sin\theta=0\)
\(\ddot\theta+\frac{g}{l}\cos\theta=0\)
\(\ddot\theta=0\)

Q43. For small oscillations of a pendulum, the Hamiltonian approximately becomes:

\(\displaystyle H=\frac{p_\theta^2}{2ml^2}-\frac12mgl\theta^2\)
\(\displaystyle H=\frac{p_\theta^2}{2ml^2}+\frac12mgl\theta^2\)
\(H=mgl\theta\)
\(H=p_\theta\theta\)

Q44. For planetary motion in polar coordinates, the kinetic energy of a particle of mass \(m\) is:

\(\frac12m(\dot r^2+r\dot\theta^2)\)
\(\frac12m(\dot r^2+r^2\dot\theta^2)\)
\(\frac12mr^2\dot r^2\)
\(\frac12m\dot\theta^2\)

Q45. The gravitational potential energy for a planet of mass \(m\) moving around a much heavier mass \(M\) is:

\(\displaystyle +\frac{GMm}{r}\)
\(\displaystyle -\frac{GMm}{r}\)
\(GMmr\)
\(-GMmr^2\)

Q46. The Hamiltonian for planetary motion in a plane is:

\(\displaystyle H=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{GMm}{r}\)
\(\displaystyle H=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}-\frac{GMm}{r}\)
\(\displaystyle H=\frac{p_r^2+p_\theta^2}{2m}\)
\(H=p_rp_\theta-GMm/r\)

Q47. In the Hamiltonian for planetary motion, the coordinate \(\theta\) does not explicitly appear. Therefore:

\(r\) is constant
\(p_r\) is always zero
\(p_\theta\) is conserved
The energy is zero

Q48. The conserved momentum \(p_\theta\) in planetary motion represents:

Linear momentum
Radial momentum
Angular momentum
Potential energy

Q49. A particle of mass \(2\,\text{kg}\) has momentum \(6\,\text{kg m/s}\) and moves in a region where \(V=5\,\text{J}\). Its Hamiltonian \(H=p^2/(2m)+V\) is:

\(9\,\text{J}\)
\(11\,\text{J}\)
\(14\,\text{J}\)
\(23\,\text{J}\)

Q50. Which statement best compares Newtonian, Lagrangian and Hamiltonian mechanics?

Only Newtonian mechanics gives correct equations of motion
Lagrangian mechanics is valid only for oscillators
Hamiltonian mechanics is valid only for planetary motion
They are equivalent formulations of classical mechanics, expressed using different variables and methods