Question

Given \(L(q, \dot{q})\) such that \(d L=\dot{p} d q+p d \dot{q},\) find \(H(p, q)\) so that \(d H=\dot{q} d p-\dot{p} d q\) Comments: \(L\) and \(H\) are functions used in mechanics called the Lagrangian and the Hamiltonian. The quantities \(\dot{q}\) and \(\bar{p}\) are actually time derivatives of \(p\) and \(q\), but you make no use of the fact in this problem. Treat \(\dot{p}\) 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 is \(H(p, q) = p \dot{q} - L(q, \dot{q})\).

Step-by-step solution

Define the Legendre Transformation

The Hamiltonian function, denoted as \(H(p, q)\), can be found using a Legendre transformation of the Lagrangian \(L(q, \dot{q})\). The transformation is defined as: \[ H(p, q) = p \dot{q} - L(q, \dot{q}) \]where \(p\) is conjugate to \(q\) and is given by: \[ p = \frac{\partial L}{\partial \dot{q}} \]

Expressing \(dL\)

The given condition says: \[ dL = \dot{p} dq + p d\dot{q} \]Using the total differential, express \(dL\) as:\[ dL = \frac{\partial L}{\partial q} dq + \frac{\partial L}{\partial \dot{q}} d\dot{q} \]

Compare the Differentials

Compare the two expressions for \(dL\):\[ \frac{\partial L}{\partial q} dq + \frac{\partial L}{\partial \dot{q}} d\dot{q} = \dot{p} dq + p d\dot{q} \]By comparing the coefficients of \(dq\) and \(d\dot{q}\), we obtain:\[ \frac{\partial L}{\partial q} = \dot{p} \]\[ \frac{\partial L}{\partial \dot{q}} = p \]

Compute the Differential of \(H\)

By definition, the differential \(dH\) of the Hamiltonian \(H\) is:\[ dH = d(p \dot{q} - L) \]Expanding this, we get:\[ dH = p d\dot{q} + \dot{q} dp - \frac{\partial L}{\partial q} dq - \frac{\partial L}{\partial \dot{q}} d\dot{q} \]

Simplify \(dH\)

Substitute the values \( \frac{\partial L}{\partial q} = \dot{p} \) and \( \frac{\partial L}{\partial \dot{q}} = p \) into the expression for \(dH\):\[ dH = p d\dot{q} + \dot{q} dp - \dot{p} dq - p d\dot{q} \]Notice that the terms \(p d\dot{q}\) cancel out, leaving:\[ dH = \dot{q} dp - \dot{p} dq \]

Verify the Differential

It has been shown that the differential \(d H\) is:\[ dH = \dot{q} dp - \dot{p} dq \]This matches the given differential form in the problem, verifying that the Hamiltonian \(H\) obtained through the Legendre transformation is correct.

Key concepts

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by Joseph-Louis Lagrange in 1788. It provides an alternative to Newtonian mechanics and is widely used in theoretical physics. The core idea is to describe the dynamics of a system by a function called the Lagrangian, denoted by \( L(q, \dot{q}) \). The variables \( q \) and \( \dot{q} \) represent the generalized coordinates and their time derivatives, respectively. The Lagrangian is defined as the difference between the kinetic energy \( T \) and the potential energy \( V \) of the system, i.e., \( L = T - V \).
The equations of motion in Lagrangian mechanics are derived using the **principle of least action**, which states that the path taken by the system between two states is the one for which the action \( S = \int L \, dt \) is stationary (usually a minimum). This leads to the **Euler-Lagrange equations**:
\[ \frac{d}{dt} \frac{\text{\text{∂}}L}{\text{\text{∂}}\text{\text{∂}}\text{\text{∂}}\text{\text{∂}}\text{\text{∂}}\text{\text{q}}} - \frac{\text{\text{\text{∂}}\text{\text{L}}}{\text{\text{q}}}} = 0 \]
This framework is particularly powerful when dealing with complex coordinate transformations and constraints, as it naturally incorporates these into the equations of motion.

Legendre transformation

The Legendre transformation is a mathematical procedure that transforms one function into another, typically converting the Lagrangian \( L(q, \dot{q}) \) into the Hamiltonian \( H(p, q) \). This is instrumental for switching from Lagrangian mechanics to Hamiltonian mechanics. The general form of the Legendre transformation is given by:
\[H(p, q) = p \, \dot{q} - L(q, \dot{q})\]
where \( p \) is the conjugate momentum defined by:

• \[ p = \frac{\text{\text{∂}}\text{ L}}{\text{\∂}\dot{q}} \]
  The Legendre transformation swaps dependence on \( \dot{q} \, \text{(velocity)} \) for \( p \text{(momentum)} \). This transformation is particularly useful because it leads to Hamilton’s equations:
  
  \[ \frac{dq}{dt} = \frac{\text{\text{\text{∂}}\text{H}}}{\text{\∂}p} \, \, \frac{dp}{dt} = -\frac{\text{\text{\text{∂}}\text{H}}}{\text{\∂}q} \]
  These equations describe the time evolution of a system in terms of its coordinates and momenta, thereby offering a different perspective compared to the Lagrangian approach.

Conjugate variables

In mechanics, particularly in the Hamiltonian framework, conjugate variables are pairs of quantities that describe the state of a physical system. The most common pair of conjugate variables are the generalized coordinates \( q \) and their corresponding conjugate momenta \( p \). These variables are related through the Legendre transformation, where the momentum \( p \) is defined by:
\[ p = \frac{\∂ L}{\∂ \dot{q}} \]
Conjugate variables are essential in understanding phase space, a conceptual space where each point represents a possible state of the system. The Hamiltonian \( H(p, q) \) often simplifies the equations of motion, as it is a function of these conjugate pairs.
The time evolution of these variables is governed by Hamilton’s equations:

• \[ \frac{dq}{dt} = \frac{\text{∂} H}{\text{\∂} p} \]
• \[ \frac{dp}{dt} = -\frac{\text{∂} H}{\text{\∂} q} \]

This setup complements the way systems are analyzed in quantum mechanics and statistical mechanics, highlighting the importance of conjugate variables in diverse fields of physics.

Differential forms

Differential forms are a fundamental concept in mathematics and physics, particularly in the fields of calculus on manifolds and theoretical physics. They provide a generalized way to integrate over curves, surfaces, and higher-dimensional manifolds. In the context of Hamiltonian mechanics, differential forms are used to express the relations between different variables in a concise manner.
For example, the differential forms involved in defining the Hamiltonian \( H \) are:

• \( dL = \dot{p} \, dq + p \, d\dot{q} \)
• \( dH = d(p \, \dot{q} - L) \)

By expanding and simplifying these forms, we derive:

• \( dH = \dot{q} \, dp - \dot{p} \, dq \)

This final form indicates how the Hamiltonian preserves the structure of the equations of motion and ensures the consistency of the transformation from Lagrangian to Hamiltonian mechanics.
The use of differential forms provides a compact and elegant way to handle the multi-variable calculus inherent in physics, simplifying the derivation and interpretation of equations.