Question

Given \(L(q, \dot{q})\) such that \(d L=j \cdot d q+\rho d \dot{q}\), find \(H(\rho, q)\) so that \(d H=\dot{q} d p-\dot{p} d q\). Comments: \(I\) and \(H\) are functions used in mechanics called the Lagrangian and the Hamil. tonian. The quantities \(\dot{q}\) and \(\dot{p}\) are actually time derivatives of \(p\) and 4 , but you make no use of the fact in this problem. Treat \(\hat{p}\) and \(\dot{q}\) as if they were two more variables having nothing to do with \(p\) and \(q\). Hint: Use a Legendre transformation. On your first try you will probably get \(-H\), Look at the text discussion of Legendre transformations and satisfy. yourself that \(g=q y-f\) would have been just as satisfactory as \(g=f-q y\) in \((11.23)\).

Short answer

The Hamiltonian \( H(\rho, q) = \rho \dot{q} - L(q, \dot{q}) \).

Step-by-step solution

Understand the relationship between variables

The given function is the Lagrangian, represented as \( L(q, \dot{q}) \). It is described by the differential form \( dL = j \cdot dq + \rho \cdot d \dot{q} \). We need to find the Hamiltonian \( H(\rho, q) \) such that \( dH = \dot{q} \cdot dp - \dot{p} \cdot dq \).

Apply the Legendre transformation

A Legendre transformation helps convert between the Lagrangian and Hamiltonian formalisms. To find \( H \), we use the relation: \[ H = \rho \dot{q} - L \]where \( \rho = \frac{\partial L}{\partial \dot{q}} \).

Calculate differentials

Calculate the differentials of \( H \) in terms of \( q \) and \( \dot{q} \): \[ dH = d(\rho \dot{q} - L) \]Expanding this, we have:\[ dH = \dot{q} \cdot d\rho + \rho \cdot d\dot{q} - dL \]From the given, recall that \( dL = j \cdot dq + \rho \cdot d\dot{q} \). Replace \( dL \) with this expression:\[ dH = \dot{q} \cdot d\rho + \rho \cdot d\dot{q} - (j \cdot dq + \rho \cdot d\dot{q}) \]Collecting terms, we get:\[ dH = \dot{q} \cdot d\rho - j \cdot dq \]

Relate differentials to given form

To match the form \( dH = \dot{q} \cdot dp - \dot{p} \cdot dq \), identify the corresponding terms:Compare:\[ dH = \dot{q} \cdot d\rho - j \cdot dq \]with:\[ dH = \dot{q} \cdot dp - \dot{p} \cdot dq \]We can see that \( dp = d\rho \) and \( \dot{p} = j \). So,\[ H(\rho, q) = \rho \dot{q} - L(q, \dot{q}) \]

Finalize the Hamiltonian function

Based on the transformations and comparisons, the Hamiltonian \( H \) in terms of \( \rho \) and \( q \) is determined to be:\[ H(\rho, q) = \rho \dot{q} - L(q, \dot{q}) \]

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. Its centerpiece is the Lagrangian function, denoted as \( L(q, \, \dot{q}) \), which is a function of generalized coordinates \( q \) and their time derivatives \( \dot{q} \). It's given by the difference between the kinetic and potential energy of the system.
Here are the key points:

• The principle of least action is fundamental; it states that the actual path taken by a system is the one for which the action integral is stationary (i.e., exhibits a minimum or maximum value).
• The Euler-Lagrange equation, derived from the principle of least action, provides the equations of motion: \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \).

Lagrangian mechanics is especially powerful for dealing with constraints, making it crucial for understanding more complex mechanical systems.

Legendre transformation

The Legendre transformation is a mathematical procedure used to switch between different sets of variables in thermodynamics and mechanics. In the context of mechanics, it converts the Lagrangian \( L \) to the Hamiltonian \( H \).
Here's how:

• Given a function \( f(x) \) and its derivative \( p = f'(x) \), the Legendre transformation introduces a new variable \( p \) and defines a new function \( g(p) = px - f(x) \).
• In mechanics, we start with the Lagrangian \( L(q, \dot{q}) \) and transform it into the Hamiltonian \( H \) using the relation \( H = \rho \dot{q} - L \), where \( \rho = \frac{\partial L}{\partial \dot{q}} \).

This transformation re-expresses the system's dynamics in terms of the new variables \( \rho \) and \( q \), providing a different perspective that can simplify the analysis and solution of the equations of motion.

Differential forms

Differential forms are a sophisticated yet powerful mathematical language used to express multidimensional calculus. In mechanics, they provide a compact and elegant way to write down the equations of motion and related quantities.
Important concepts include:

• A differential form is an object that can be integrated over a manifold. The most common types are 0-forms (functions), 1-forms (linear maps on vectors), and higher forms.
• In mechanics, the differential form \( dL \) represents the variations of the Lagrangian: \( dL = j \cdot dq + \rho \cdot d\dot{q} \).
• To find the Hamiltonian \( H \), the forms are transformed: \( dH = d( \rho \dot{q} - L) \), leading to useful expressions linking \( p \), \( q \), \( \dot{p} \), and \( \dot{q} \).

Differential forms encapsulate changes and variations in the system, making them invaluable for analyzing and solving problems in advanced mechanics.

Hamiltonian function

The Hamiltonian function \( H \) is a central concept in Hamiltonian mechanics. It represents the total energy of the system, re-expressed in terms of generalized coordinates and momenta.
Here's what you need to know:

• The Hamiltonian is defined as \( H = \rho \dot{q} - L \), where \( \rho = \frac{\partial L}{\partial \dot{q}} \).
• It provides a powerful way to describe the dynamics of a system through Hamilton's equations: \( \dot{q} = \frac{\partial H}{\partial p} \) and \( \dot{p} = -\frac{\partial H}{\partial q} \).
• These equations offer a first-order differential representation of the system, often making complex problems more manageable.

In summary, the Hamiltonian function transforms the analysis of mechanical systems from position-velocity space to phase space, allowing for a more versatile and sometimes more straightforward exploration of their dynamics.