Hamiltonian Mechanics

Under what condition is momentum conserved?

Momentum is conserved if dp/dt ​​=0

What is the formula for rotational kinetic energy of a rigid body rotating about an axis?

T_rot​=1/2Iω^2

Define the Inertia Matrix (I​​).

The Inertia Matrix (I​​) relates the angular momentum (M​) to the angular velocity (ω​) by M​=I​​ω​.

What is the Moment of Inertia (I) and its dimensions?

The Moment of Inertia (I) is a scalar quantity that represents the resistance of an object to changes in its rotational motion. Its dimensions are ML^2.

What is Fermat's Law in optics?

Light rays travel along paths that extremize the optical path length. This is why light appears to constructively interfere at the path of minimum optical path length.

What does it mean for a path to be "stationary" or "extremal"?

• The realized path in nature is one where the path length (or other quantity) is stationary with respect to crying paths.

How does General Relativity relate to variational principles?

In General Relativity, trajectories in space-time are those that extremize (maximize in this case) the elapsed proper time.

Lagrangian

L=T−V

What is the Principle of Least Action?

A classical path x(t) is one that extremizes (minimizes) the action

A=\int_{t_0}^{t_1}\!L\,dt

What is a generalized coordinate (q)?

A variable that describes the configuration of a system. It can be Cartesian coordinates (x), angles (θ), or arc length (s).

State the Euler-Lagrange Equation.

\partial L/\partial q-d/dt(\partial L/\partial q˙)=0

How is generalized momentum (p) defined in Lagrangian mechanics, and what does the Euler-Lagrange equation imply about it?

Generalized momentum is defined as p=∂L/∂x˙​. For a Newtonian Lagrangian, p=mx˙. The Euler-Lagrange equation implies that force is the rate of change of momentum.

How is the Euler-Lagrange Equation generalized for a system with 'd' generalized coordinates?

For a system with 'd' generalized sets of coordinates, there are 'd' euler-lagrange equations

How can Newton's Second Law in vector form be recovered from the Lagrangian framework?

The force components are given by the negative partial derivatives of the potential energy, and the Euler-Lagrange equations show that the rate of change of the momentum vector is equal to the force vector: dp​/dt ​= F​=−∇​V.

Define the Hamiltonian (H)

The Hamiltonian (H) is defined as the total energy of the system, H=T+V (Kinetic Energy + Potential Energy). It can be expressed in terms of coordinates and velocities H(x,x˙) or coordinates and momenta H(x,p)

Under what conditions is the Hamiltonian a conserved quantity?

In the absence of friction or explicit time dependence in the Lagrangian (∂L/∂t​=0), the Hamiltonian is a constant of motion, meaning dH​/dt=0. This implies that energy is conserved.

State Noether's Theorem.

For each symmetry of the system, there is a conserved quantity.

How is spatial translation symmetry related to momentum conservation?

If space is homogeneous (meaning the Lagrangian does not explicitly depend on position), then the generalized momenta are conserved.

How is rotational symmetry linked to the conservation of angular momentum?

If space is isotropic (the same in all directions), then rotations do not change the physics, and the Lagrangian is invariant under rotations. This leads to the conservation of angular momentum.

When is angular momentum conserved?

Angular momentum is conserved if space is isotropic, meaning dM​​/dt=0

What are the two fundamental Hamilton's Equations derived from first principles?

• q˙​i​=∂H/∂pi​​

• p˙​i​=−∂H/∂qi​​

Define a "central force."

force where F=−∂V/∂r​ and the potential V depends only on the radial distance r, i.e., V(r).

In polar coordinates, what is the expression for the conjugate momentum pθ​ for a particle under a central force, and why is it significant?

p_{\theta}=mr^2\theta˙

It represents the angular momentum and is a conserved quantity for central forces because the Lagrangian does not depend on θ (i.e., ∂L/∂θ​=0).

What are the three degrees of freedom typically associated with a spinning top or gyroscope?

θ (nutation), ϕ (precession), α (spin).

For a spinning top, which two generalized momenta are conserved, and why?

p​_ϕ and p_α​ are conserved because the Lagrangian does not depend on ϕ or α (due to rotational symmetries).

How are the equations of motion for coupled oscillators typically represented in matrix form?

Using an Inertia (or Mass) matrix M​​ and a Stiffness matrix K​​, the equations are Mx​¨​=−Kx​

Define a "normal mode" in the context of coupled oscillations.

A normal mode represents a small displacement around equilibrium where the response is linear and all parts of the system oscillate with the same frequency and in a fixed phase relationship.

For a symmetric coupled oscillator system (e.g., two identical masses with three springs), what is the "symmetric mode" and its associated frequency?

The symmetric mode occurs when x₁​=x₂​ (masses move together). The central spring is not stretched. The frequency is ω₁​=sqrt(k/m)​​.

What is the "antisymmetric mode" for a symmetric coupled oscillator system, and its associated frequency?

The antisymmetric mode occurs when x₁=−x₂​ (masses move in opposite directions, and the centre of the middle spring is stationary). The frequency is ω₂​=sqrt(3k/m)​​.

How is the Mass (or Inertia) matrix Mij​ defined in terms of kinetic energy?

M_{ij}=\partial^2T/(\partial x_{i}˙\partial x_{j}˙).

What conditions define an "equilibrium point" for a system in terms of forces and potential energy?

Equilibrium points occur where all components of the force vanish (Fi​=−∂V/∂xi​​=0). These are points where the potential energy has a minimum (for stable equilibrium).

What is the fundamental eigenvalue problem used to find the normal frequencies (ω) and corresponding normal modes (x​) of a coupled oscillating system?

(K-\omega^2M)\underline{x}=0.