Question

Consider a particle of mass \(m\) moving in two dimensions, subject to a force \(\mathbf{F}=-k x \hat{\mathbf{x}}+K \hat{\mathbf{y}}\) where \(k\) and \(K\) are positive constants. Write down the Hamiltonian and Hamilton's equations, using \(x\) and \(y\) as generalized coordinates. Solve the latter and describe the motion.

Short answer

The particle oscillates harmonically in \(x\) and moves linearly in \(y\).

Step-by-step solution

Define the Lagrangian

To find the Hamiltonian, we first need the Lagrangian. The Lagrangian, \(L\), is given by the kinetic energy minus the potential energy. The kinetic energy \(T\) is \(T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2)\). The potential energy \(V\) can be deduced from the force \(\mathbf{F}\). Since \(\mathbf{F} = -abla V\), and \(\mathbf{F} = -kx \hat{\mathbf{x}} + K \hat{\mathbf{y}}\), the potential energy is \(V = \frac{1}{2}kx^2 - Ky\). Thus, \(L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - \left(\frac{1}{2}kx^2 - Ky\right)\).

Derive the Hamiltonian

The Hamiltonian \(H\) is found via the Legendre transformation: \(H = \sum p_i \dot{q_i} - L\). The conjugate momenta are \(p_x = \frac{\partial L}{\partial \dot{x}} = m\dot{x}\) and \(p_y = \frac{\partial L}{\partial \dot{y}} = m\dot{y}\). Substituting these into the Hamiltonian expression, we have \(H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{1}{2}kx^2 - Ky\).

Write Hamilton's Equations

Hamilton's equations are given by \(\dot{q_i} = \frac{\partial H}{\partial p_i}\) and \(\dot{p_i} = -\frac{\partial H}{\partial q_i}\). For \(x\) and \(y\), Hamilton's equations are: \(\dot{x} = \frac{p_x}{m}\), \(\dot{y} = \frac{p_y}{m}\), \(\dot{p_x} = -kx\), and \(\dot{p_y} = 0\).

Solve the Differential Equations

Solving \(\dot{p_x} = -kx\), we get a simple harmonic oscillator equation: \(m\ddot{x} = -kx\), whose solution is \(x(t) = A\cos(\omega t + \phi)\) with \(\omega = \sqrt{\frac{k}{m}}\). The equation \(\dot{p_y} = 0\) implies \(p_y = \text{constant}\). Thus, \(y(t) = \frac{p_y}{m} t + C\), where \(C\) is a constant.

Describe the Motion

The motion of the particle is given by: along the \(x\)-axis, it executes simple harmonic motion with an angular frequency \(\omega = \sqrt{\frac{k}{m}}\); along the \(y\)-axis, it moves with constant velocity \(\dot{y} = \frac{p_y}{m}\). The trajectory in the \(xy\)-plane is a straight line superimposed with oscillations along the \(x\)-axis.

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, which provides a powerful method for analyzing systems. It is based on the Lagrangian function, denoted as \( L \), which is defined as the difference between the kinetic and potential energy of the system. For a particle of mass \( m \) moving in two dimensions, the kinetic energy \( T \) is expressed as \( T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) \), where \( \dot{x} \) and \( \dot{y} \) represent the time derivatives of the particle's position, or its velocities in the \( x \) and \( y \) directions.

The potential energy, \( V \), hinges on the forces acting on the particle. If the force \( \mathbf{F} \) is given by \( \mathbf{F} = -kx \hat{\mathbf{x}} + K \hat{\mathbf{y}} \), the potential energy can be found through the relationship \( \mathbf{F} = - abla V \). This yields \( V = \frac{1}{2}kx^2 - Ky \). Thus, the Lagrangian \( L \) becomes \( L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) - \left(\frac{1}{2}kx^2 - Ky \right) \).

• Lagrangian mechanics simplifies the process of finding the equations of motion, especially for complex systems.
• It does this by using generalized coordinates, which can be customized to suit the specific problem.
• This approach is particularly useful for systems with constraints.

Hamilton's equations

Hamilton's equations are a set of first-order differential equations that provide an efficient way to describe the evolution of a system over time. These equations are derived from a transformation of the Lagrangian, known as the Legendre transformation, which allows us to express the dynamics in terms of conjugate variables, namely the generalized coordinates \( q_i \) and their conjugate momenta \( p_i \).

For our problem with the mass moving under the forces \( -kx \) and \( K \), the conjugate momenta are derived as \( p_x = m\dot{x} \) and \( p_y = m\dot{y} \). The Hamiltonian \( H \) can be calculated as:
\[H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{1}{2}kx^2 - Ky\]
Hamilton's equations then provide:

• \( \dot{x} = \frac{p_x}{m} \)
• \( \dot{y} = \frac{p_y}{m} \)
• \( \dot{p_x} = -kx \)
• \( \dot{p_y} = 0 \)

These equations map out the system's behaviors in terms of the positions and velocities, offering a complete picture of its dynamics. They effectively replace the traditional Newton's second law for many applications. Instead of directly finding accelerations, you solve for velocities and momenta, which can be advantageous in certain analyses.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a fundamental type of periodic motion where the restoring force is directly proportional to the displacement. This principle is exemplified in our exercise by the particle's movement in the \( x \)-direction under the influence of the force \( -kx \).

Solving Hamilton's equation for \( \dot{p_x} = -kx \) yields the differential equation for a harmonic oscillator:
\[ m\ddot{x} = -kx \]
This indicates that the system's response is sinusoidal and can be described by:
\[ x(t) = A\cos(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency \( \omega = \sqrt{\frac{k}{m}} \), and \( \phi \) is the phase constant. This solution describes oscillations with a specific frequency dictated by the system's properties.

• SHM is characterized by a regular, back-and-forth motion around an equilibrium position.
• This type of motion is prevalent in physics, seen in systems such as pendulums and springs.
• In our problem, the particle on the \( x \)-axis oscillates, while on the \( y \)-axis, it moves with a constant velocity.

SHM is a core concept that helps in understanding more complex oscillatory motions encountered in different physical contexts.