Hamilton’s Equations
We will cover following topics
PYQs
Hamilton’s Equations
1) Using Hamilton’s equation, find the acceleration for a sphere rolling down a rough inclined plane, if \(x\) be a distance of the point of contact of the sphere from a fixed point of the plane.
[2019, 15M]
2) The Hamiltonian of a mechanical system is given by,
\(H=p_1q_1-aq^2_1+bq^2_2=p_2q_2\), where \(a\), \(b\) are constants. Solve the Hamiltonian equations and show that \(\dfrac{p_2 - bq_2}{q_1}=constant\).
[2018, 20M]
3) Consider single free particle of mass \(m\), moving in space under no forces. If the particle starts form the origin at \(t=0\) and reaches the position \((x, y, z)\) at time \(\tau\), find the Hamilton’s characteristic function \(S\) as a function of \(x, y, z, \tau\).
[2016, 10M]
4) Solve the plane pendulum problem using the Hamiltonian approach and show that \(H\) is a constant of motion.
[2015, 15M]
5) A Hamiltonian of a system with one degree of freedom has form
\[H=\dfrac{p^{2}}{2 \alpha}- bqpe^{-\alpha t}+\dfrac{b \alpha}{2} q^{2} e^{-\alpha t} \left(\alpha+b e^{-\alpha t}\right)+\dfrac{k}{2} q^{2}\]where \(\alpha\), \(b\), \(k\) are constants, \(q\) is the generalized coordinate and \(p\) is the corresponding generalized momentum.
(i) Find a Lagrangian corresponding to this Hamiltonian.
(ii) Find an equivalent Lagrangian that is not explicitly dependent on time.
[2015, 20M]
6) Find the equation of motion of a compound pendulum using Hamilton’s equations.
[2014, 10M]
7) A sphere of radius \(a\) land mass \(m\) rolles down a rough plane inclined at an angle \(\alpha\) to the horizontal. If \(x\) be the distance of the point of contact of the sphere from a fixed point on the plane, find the acceleration by using Hamilton’s equation.
[2010, 30M]
8) A point mass \(m\) is placed on a frictionless plane that is tangent to the Earth’s surface. Determine Hamilton’s equations by taking \(x\) or \(\theta\) as the generalized coordinate.
[2007, 30M]
9) A particle of mass \(m\) is constrained to move on the surface of a cylinder. The particle is subject to a force directed towards the origin and proportional to the distance to of particle from the origin. Construct the Hamiltonian and Hamilton’s equations of motion.
[2006, 30M]
10) Derive the Hamilton equations of motion from the principle of least action and obtain the same for a particle of mass \(m\) moving in a force field of potential \(V\). Write these equations in spherical coordinates \((r, \theta, \phi)\).
[2004, 30M]