Question

Consider a mass \(m\) constrained to move in a vertical line under the influence of gravity. Using the coordinate \(x\) measured vertically down from a convenient origin \(O\), write down the Lagrangian \(\mathcal{L}\) and find the generalized momentum \(p=\partial \mathcal{L} / \partial \dot{x}\). Find the Hamiltonian \(\mathcal{H}\) as a function of \(x\) and \(p\) and write down Hamilton's equations of motion. (It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

Short answer

The Lagrangian is \( \mathcal{L} = \frac{1}{2} m \dot{x}^2 - mgx \), generalized momentum \( p = m \dot{x} \), and Hamiltonian \( \mathcal{H} = \frac{p^2}{2m} + mgx \).

Step-by-step solution

Determine Kinetic and Potential Energy

The kinetic energy (\( T \)) of the mass is given by \( T = \frac{1}{2} m \dot{x}^2 \), where \( \dot{x} \) is the velocity. The potential energy (\( U \)) associated with the gravitational force is \( U = mgx \), where \( g \) is the acceleration due to gravity, and \( x \) is the vertical position.

Formulate the Lagrangian

The Lagrangian (\( \mathcal{L} \)) is defined as the difference between the kinetic and potential energy: \[ \mathcal{L} = T - U = \frac{1}{2} m \dot{x}^2 - mgx \].

Compute Generalized Momentum

The generalized momentum (\( p \)) is given by the partial derivative of the Lagrangian with respect to the velocity: \[ p = \frac{\partial \mathcal{L}}{\partial \dot{x}} = m \dot{x} \].

Express the Hamiltonian

The Hamiltonian (\( \mathcal{H} \)) is expressed as \( \mathcal{H} = \sum p_i \dot{q}_i - \mathcal{L} \). For this system, \[ \mathcal{H} = p \dot{x} - \left(\frac{1}{2} m \dot{x}^2 - mgx\right) = \frac{1}{2m} p^2 + mgx \].

Derive Hamilton's Equations of Motion

Hamilton's equations are given by \( \dot{q} = \frac{\partial \mathcal{H}}{\partial p} \) and \( \dot{p} = -\frac{\partial \mathcal{H}}{\partial q} \). From the Hamiltonian: \( \dot{x} = \frac{\partial \mathcal{H}}{\partial p} = \frac{p}{m} \) and \( \dot{p} = -\frac{\partial \mathcal{H}}{\partial x} = -mg \). These match the expected equations for a particle in a gravitational field: constant acceleration \( g \) and velocity \( \dot{x} \).

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a reformation of classical mechanics introduced by Joseph-Louis Lagrange. Instead of dealing directly with forces, it focuses on energy—a concept that can be easier to apply in complex systems. In this approach, the Lagrangian (\( \mathcal{L} \)) is a function representing the difference between kinetic and potential energy: \[ \mathcal{L} = T - U \], where \( T \) is the kinetic energy and \( U \) is the potential energy.

• The kinetic energy for a mass \( m \) moving with velocity \( \dot{x} \) is given by \( T = \frac{1}{2} m \dot{x}^2 \)
• The potential energy, due to gravity, for a height \( x \) is \( U = mgx \)

By plugging these into the Lagrangian formula, we obtain:\[ \mathcal{L} = \frac{1}{2} m \dot{x}^2 - mgx \].Lagrangian mechanics is useful because it allows derivation of the equations of motion via the Euler-Lagrange equation, which is often simpler in complex systems compared to the traditional Newtonian approach.

generalized momentum

Generalized momentum is derived from the Lagrangian and relates to the velocity of a system. In this context, the generalized momentum (\( p \)) is analogous to classical momentum but extends to more complex, generalized coordinates.To find the generalized momentum, we take the partial derivative of the Lagrangian with respect to the velocity \( \dot{x} \):\[ p = \frac{\partial \mathcal{L}}{\partial \dot{x}} \].
For the example of a mass moving vertically, this derivative results in:\[ p = m \dot{x} \].

• Generalized coordinates: While classical systems are in Cartesian coordinates, generalized coordinates fit systems with constraints.
• Momentum relation: It connects motion's energy properties with generalized forces.

This concept is integral in transforming systems from Lagrangian to Hamiltonian mechanics, bridging positional derivatives to forces and motion constraints.

Hamilton's equations

Hamilton's equations form the cornerstone of Hamiltonian mechanics, a reformulation of classical mechanics that's particularly elegant for systems with many degrees of freedom. These equations offer a set of first-order differential equations that describe the evolution of a system.The Hamiltonian (\( \mathcal{H} \)) is calculated as:\[ \mathcal{H} = \sum p_i \dot{q}_i - \mathcal{L} \], which, for a mass in a gravitational field, simplifies to:\[ \mathcal{H} = \frac{1}{2m} p^2 + mgx \].The core of Hamilton's mechanics lies in the equations:

• \( \dot{q} = \frac{\partial \mathcal{H}}{\partial p} \)
• \( \dot{p} = -\frac{\partial \mathcal{H}}{\partial q} \)

For a mass under gravity, these translate to:

• \( \dot{x} = \frac{p}{m} \)
• \( \dot{p} = -mg \)

Understanding these equations helps describe systems under constant forces and potential energies. Hamilton's equations reveal nature's symmetry and ensure energy conservation in mechanics, providing a powerful tool for modern physics.