Exploring Lagrangian and Hamiltonian Mechanics

What is D’Alembert’s principle?

• D'Alembert's principle relates forces and inertial effects in a dynamic system, stating that the sum of the applied forces minus the inertial forces equals zero.

• D'Alembert's principle describes the relationship between velocity and acceleration in a fluid.

• D'Alembert's principle is a method for calculating potential energy in a system.

• D'Alembert's principle states that all forces in a static system are balanced.

Derive the Lagrangian equations of motion for a simple pendulum.

• θ̈ + (g/l) sin(θ) = 0

• θ̈ + (g/l) θ = 0

• θ̈ + (g/l) tan(θ) = 0

• θ̈ + (g/l) cos(θ) = 0

Explain the working of Atwood’s machine using Lagrangian mechanics.

• The Lagrangian method is only applicable to static systems, not dynamic ones like Atwood's machine.

• Atwood's machine uses a spring to measure force instead of a pulley system.

• Atwood's machine demonstrates the principles of Lagrangian mechanics by analyzing the motion of two masses connected by a string over a pulley, using the Lagrangian to derive the equations of motion.

• Atwood's machine consists of three masses instead of two, affecting its motion analysis.

How do Lagrange’s equations apply in the presence of non-conservative forces?

• Non-conservative forces are ignored in Lagrange's equations.

• Lagrange's equations can include non-conservative forces by adding a generalized force term to the equations.

• Lagrange's equations only apply to conservative systems.

• Lagrange's equations cannot be used with non-conservative forces.

Write the Lagrangian for a charged particle in an electromagnetic field.

• L = (1/2)m(v^2) + qφ - q(v·A)

• L = m(v^2) + q(φ + v·A)

• L = (1/2)m(v^2) + qA - q(φ·v)

• L = (1/2)m(v^2) - qφ + q(v×A)

Define phase space in the context of Hamiltonian mechanics.

• Phase space is a fixed point in time for a mechanical system.

• Phase space is a concept used only in quantum mechanics.

• Phase space is a single-dimensional line representing the position of a particle.

• Phase space is a multidimensional space of generalized positions and momenta representing all possible states of a system.

What are generalized momentum and cyclic coordinates?

• Generalized momentum is the product of mass and velocity; cyclic coordinates are those that appear in the Lagrangian.

• Generalized momentum is the total energy of a system; cyclic coordinates are always time-dependent.

• Generalized momentum is the derivative of the Lagrangian with respect to generalized velocity; cyclic coordinates are those that do not appear in the Lagrangian, leading to conserved momenta.

• Generalized momentum is the integral of the Lagrangian; cyclic coordinates are those that vary with time.

Explain the Hamiltonian function and its relation to conservation of energy.

• The Hamiltonian function only applies to quantum mechanics.

• The Hamiltonian function represents the position of a particle.

• The Hamiltonian function is unrelated to energy conservation.

• The Hamiltonian function represents the total energy of a system and is conserved in closed systems, reflecting the conservation of energy.

State Hamilton’s canonical equations of motion.

• \( \dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i} \quad (i = 1, 2, \ldots, n) \)

• $ \dot{q}_i = -\frac{\partial H}{\partial p_i}, \quad \dot{p}_i = \frac{\partial H}{\partial q_i} $

• $ \dot{q}_i = \frac{\partial H}{\partial q_i}, \quad \dot{p}_i = \frac{\partial H}{\partial p_i} $

• $ \dot{q}_i = \frac{\partial p_i}{\partial H}, \quad \dot{p}_i = -\frac{\partial q_i}{\partial H} $

Apply Hamiltonian mechanics to a one-dimensional simple harmonic oscillator.

• H = mv^2/2 + (1/2)kx^2

• H = (1/2)mv^2 + kx^2

• H = p^2/(2m) - (1/2)kx^2

• The Hamiltonian for a one-dimensional simple harmonic oscillator is H = p^2/(2m) + (1/2)kx^2.