Lagrangian and Hamiltonian Mechanics:
UPDATED ACTUAL Exam Questions and
CORRECT Answers
What is Hamilton's principle of least action? State it algebraically. - CORRECT
ANSWER - Given a path between two points A and B in space, the system will explore an
infinite number of paths between the two points A and B. The path chosen will be the one that
extremises the action.
What is the equation for the Action? - CORRECT ANSWER - The integral of the
Lagrangian with respect to time.
What is the equation of the Lagrangian in classical mechanics? - CORRECT ANSWER -
The difference between the kinetic and potential energies of a system.
State the relativistic Lagrangian: - CORRECT ANSWER - The difference between
relativistic kinetic energy the potential.
Write the Lagrangian for a particle subject to a central force. - CORRECT ANSWER - See
image.
What quantity is conserved for a particle subject to a central force. - CORRECT
ANSWER - See image.
Write the equation for a uniform gravitational potential field: - CORRECT ANSWER -
Mass times gravity times height.
Write the equation for an elastic potential field: - CORRECT ANSWER - One half spring
constant extension squared.
, What is the Hamiltonian? - CORRECT ANSWER - The expected value of the Hamiltonian
is the energy of the system. The Hamiltonian is a way of defining the energy of a system.
Write the equation for the electrostatic potential between two charged particle: - CORRECT
ANSWER - See image.
What is the Euler-Lagrange equation? - CORRECT ANSWER - An equation derived from
using variational principles and the principle of least action.
Derive the Euler Lagrange Equation: - CORRECT ANSWER - See image.
What assumptions are required for normal mode theory? - CORRECT ANSWER - 1. The
system can be thought of as a system of coupled harmonic oscillators.
2. The system is close to an equilibrium position and thus can be modelled as sitting in a
quadratic potential.
3. The system's kinetic energy is quadratic in velocity.
Write the equation for the Hamiltonian with respect to the Lagrangian. - CORRECT
ANSWER - See image.
Write the equation for the Hamiltonian in classical form: - CORRECT ANSWER - See
image.
What is the canonical momentum? State it's equation. - CORRECT ANSWER - The
generalised momentum.
What is the canonical force? State it's equation. - CORRECT ANSWER - The generalised
force.
UPDATED ACTUAL Exam Questions and
CORRECT Answers
What is Hamilton's principle of least action? State it algebraically. - CORRECT
ANSWER - Given a path between two points A and B in space, the system will explore an
infinite number of paths between the two points A and B. The path chosen will be the one that
extremises the action.
What is the equation for the Action? - CORRECT ANSWER - The integral of the
Lagrangian with respect to time.
What is the equation of the Lagrangian in classical mechanics? - CORRECT ANSWER -
The difference between the kinetic and potential energies of a system.
State the relativistic Lagrangian: - CORRECT ANSWER - The difference between
relativistic kinetic energy the potential.
Write the Lagrangian for a particle subject to a central force. - CORRECT ANSWER - See
image.
What quantity is conserved for a particle subject to a central force. - CORRECT
ANSWER - See image.
Write the equation for a uniform gravitational potential field: - CORRECT ANSWER -
Mass times gravity times height.
Write the equation for an elastic potential field: - CORRECT ANSWER - One half spring
constant extension squared.
, What is the Hamiltonian? - CORRECT ANSWER - The expected value of the Hamiltonian
is the energy of the system. The Hamiltonian is a way of defining the energy of a system.
Write the equation for the electrostatic potential between two charged particle: - CORRECT
ANSWER - See image.
What is the Euler-Lagrange equation? - CORRECT ANSWER - An equation derived from
using variational principles and the principle of least action.
Derive the Euler Lagrange Equation: - CORRECT ANSWER - See image.
What assumptions are required for normal mode theory? - CORRECT ANSWER - 1. The
system can be thought of as a system of coupled harmonic oscillators.
2. The system is close to an equilibrium position and thus can be modelled as sitting in a
quadratic potential.
3. The system's kinetic energy is quadratic in velocity.
Write the equation for the Hamiltonian with respect to the Lagrangian. - CORRECT
ANSWER - See image.
Write the equation for the Hamiltonian in classical form: - CORRECT ANSWER - See
image.
What is the canonical momentum? State it's equation. - CORRECT ANSWER - The
generalised momentum.
What is the canonical force? State it's equation. - CORRECT ANSWER - The generalised
force.
