Lagrangian Formulation

In the given problem, we add the time derivative of a function to the Lagrangian , resulting in a new Lagrangian . The goal is to determine the implications of this modification on the generalized momentum and the Hamiltonian .
1. Euler-Lagrange Equations: The Euler-Lagrange equations are given by: If we modify the Lagrangian to , the new Lagrangian still satisfies the Euler-Lagrange equations. The time derivative of does not affect the form of the equations, so both and satisfy the same Euler-Lagrange equations. Thus, option (A) is correct. 2. Generalized Momentum: The generalized momentum is defined as: For the modified Lagrangian , we have: Since depends only on and , we can compute: Therefore, option (B) is correct. 3. Conservation of : If is conserved (i.e., ), it does not necessarily imply that is conserved, since the function may influence the dynamics of . Thus, option (C) is not correct. 4. Modified Hamiltonian: The Hamiltonian associated with is given by: For the modified Lagrangian , the new Hamiltonian is: Substituting , we get: Therefore, option (D) is correct.
Thus, the correct options are (A) and (B).

The system consists of two parts: 1. The hoop of mass and radius , which is rolling without slipping.
2. The point mass , which is sliding along the inner surface of the hoop.
- The hoop has one degree of freedom due to its motion along the straight line.
- The point mass performs small oscillations about the mean position, which contributes one additional degree of freedom.
Thus, the total number of degrees of freedom for the system is the sum of the degrees of freedom of both parts: Therefore, the number of degrees of freedom of the system is 2.

The system described consists of two masses and a string that connects them. The generalized coordinates are (the displacement of the block ) and (the angle of the string relative to the horizontal). For the generalized momenta, we need to compute the partial derivatives of the Lagrangian with respect to the generalized velocities. The Lagrangian is a function of the kinetic and potential energies of the system. The kinetic energy of the system is given by the sum of the energies due to both masses: The generalized momenta are given by: Thus, option (A) is correct for , and option (C) is correct as is conserved (since no external forces are acting in the -direction). Therefore, the correct answers are (A) and (C).

In this problem, we need to write the Hamiltonian for a system consisting of a charged mass suspended by an inextensible string, with a conducting plane below it. The problem involves both electrostatic and gravitational forces. Step 1: General Setup of the System.
- The position of the charge is described by the angular position with respect to the vertical. - The canonical momentum is associated with this angular position, representing the rotational motion of the charge. - The kinetic energy of the system is purely due to the rotation of the charge around the pivot point: where is the length of the string. Step 2: Electrostatic Potential Energy.
The electrostatic potential energy involves the interaction between the point charge and its image charge (due to the grounded conducting plane). The distance between the charge and its image is . The potential energy is given by the formula for point charges: Step 3: Gravitational Potential Energy.
The gravitational potential energy is given by the formula for a point mass in a gravitational field: where is the acceleration due to gravity, and is the height of the charge above the conducting plane. Step 4: Constructing the Hamiltonian.
The Hamiltonian is the sum of the kinetic energy and potential energies: Substituting the expressions for , , and , we get: Thus, the correct Hamiltonian is: which matches Option (D). Final Answer: (D)