We will cover following topics

Hamilton’s Equations

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,

$H = p_{1} q_{1} - a q_{1}^{2} + b q_{2}^{2} = p_{2} q_{2}$ , where $a$ , $b$ are constants. Solve the Hamiltonian equations and show that $\frac{p_{2} - b q_{2}}{q_{1}} = c o n s t a n t$ .

[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 $τ$ , find the Hamilton’s characteristic function $S$ as a function of $x, y, z, τ$ .

[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 = \frac{p^{2}}{2 α} - b q p e^{- α t} + \frac{b α}{2} q^{2} e^{- α t} (α + b e^{- α t}) + \frac{k}{2} q^{2}

where $α$ , $b$ , $k$ 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 $α$ 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 $θ$ 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 $(r, θ, ϕ)$ .

[2004, 30M]

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