[A] Calculate the value of the action integral between the limits $t=0$ and $t=T$ for a a particle falling under influence of gravity along the following three paths.
(i) for a fictitious motion with path given by $z= at.$ (ii) for a second fictitious path given by $z=bt^3.$ (iii) for the real motion $z={1\over 2} g t^2. $
where the constants $a, b$ must be determined so that the initial and final positions coincide with the rules of variation in the action principle. [B] Check if the action integral has smaller value for the real motion in(c) than the fictitious ones (a) and (b). Discuss the result you have obtained and write conclusions you may draw about the action principle.
Show that the kinetic energy of a uniform rod of mass \(m\) is\[ T = \frac{1}{6}m(\vec{u}.\vec{u} + \vec{u}.\vec{v}+\vec{v}.\vec{v} )\]where \(\vec{u}\) and \(\vec{v}\) are the velocities of the two ends.
Woodhouse
\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)
A point charge moves with velocity \(\vec{v}\) in presence of electric field \(\vec{E}\) and magentic field \(\vec{B}\). The Lorentz force on the charged particle is\[\vec{F}= q\Big\{ \vec{E} + \frac{1}{c}\vec{v}\times\vec{B} \Big\}\]In terms of the scalar potential \(\phi\) and vector potential \(\vec{A}\) related to the electric and magnetic fields by\[\vec{E}= -\nabla \phi- \frac{1}{c} \pp[\vec{A}]{t}, \quad\vec{B}=\nabla\times \vec{A}.\]the Lorentz force becomes \[\vec{F} = q\Big\{ - \nabla\phi -\frac{1}{c} \pp[\vec{A}]{t} +\frac{1}{c} \vec{v}\times(\nabla \times \vec{A})\Big\} \]Determine the generalzed potential for this system and write the Lagrangian.
The molar energy of a monoatomic gas which obeys van der Waal's equation is given by\( E= \frac{3}{2}kT - \frac{a}{v}\),where \(V\) is the volume at temperature \(T\), and \(a\) is a constant. Initially one mole of gas is at temperature \(T_1\) and occupies volume \(V_1\). The gas is allowed to expand adiabatically into a vacuum so that it occpies a total volume \(V_2\). What is the final temperature of the gas?
MANDL