Chapter 2: Problem 19
Prove that if the Hamiltonian function is independent of \(q_{i}\), then \(p_{Y}\) is a first integral of the Hamilton equations.
Short Answer
Expert verified
Short Answer: If the Hamiltonian function is independent of the generalized coordinates \(q_i\) (i.e., \(\frac{\partial H}{\partial q_i} = 0\)), then \(p_Y\) is a first integral of the Hamilton equations. This is because the condition makes the time derivative of the generalized momentum, \(\dot{p}_Y\), equal to zero, implying that \(p_Y\) remains constant with time.
Step by step solution
01
Hamilton Equations
Hamilton equations are given by the following two equations:
\begin{align}
\dot{q}_i &= \frac{\partial H}{\partial p_i} \\
\dot{p}_i &= -\frac{\partial H}{\partial q_i}
\end{align}
Where \(H(q, p, t)\) is the Hamiltonian function of the system, \(\dot{q}_i\) and \(\dot{p}_i\) are the time derivatives of the generalized coordinates \(q_i\) and momenta \(p_i\), respectively.
02
First Integrals of Hamilton Equations
A function \(F(q, p, t)\) is a first integral of the Hamilton equations if its time derivative along the solutions of the Hamilton equations is zero:
\[\frac{d F}{d t} = 0.\]
Using the chain rule, the total derivative of \(F\) with respect to time is:
\[\frac{d F}{d t} = \frac{\partial F}{\partial q_i} \dot{q}_i + \frac{\partial F}{\partial p_i} \dot{p}_i + \frac{\partial F}{\partial t}.\]
By substituting the Hamilton equations into this expression, we get:
\[\frac{d F}{d t} = \frac{\partial F}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial H}{\partial q_i} + \frac{\partial F}{\partial t}.\]
03
Analyzing the Given Condition
We are given that the Hamiltonian function is independent of \(q_i\), which means:
\[\frac{\partial H}{\partial q_i} = 0.\]
It is important to notice that this condition does not apply to any particular coordinate, but rather applies to all generalized coordinates of the system.
04
Applying the Condition to the First Integral
Now let's assume that \(F\) is a first integral of the form \(F=p_Y\). Taking the derivative of \(F\) with respect to time, we have:
\[\frac{dF}{dt} = \frac{d p_Y}{d t} = \dot{p}_Y.\]
From the Hamilton equations, we know that \(\dot{p}_Y = -\frac{\partial H}{\partial q_Y}\). Due to the given condition that the Hamiltonian function is independent of \(q_i\), this means:
\[-\frac{\partial H}{\partial q_Y} = 0.\]
Thus, we have \(\dot{p}_Y = 0\), meaning that the first integral function we assumed, \(F=p_Y\), remains constant with time. This proves that if the Hamiltonian function is independent of \(q_i\), then \(p_Y\) is a first integral of the Hamilton equations.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hamilton's Equations
Hamilton's Equations are fundamental in the realm of Hamiltonian mechanics, offering a robust framework for analyzing dynamic systems. These equations arise from a reformation of classical mechanics, shifting from Newtonian mechanics to a new approach by using energy functions. In essence, Hamilton's equations describe the evolution of a physical system over time through its generalized coordinates and momenta.
The two primary equations are:
These subtle shifts can help solve complex problems, as all components of the system remain accounted for, making it easier to observe conservation laws.
The two primary equations are:
- \( \dot{q}_i = \frac{\partial H}{\partial p_i} \)
- \( \dot{p}_i = -\frac{\partial H}{\partial q_i} \)
These subtle shifts can help solve complex problems, as all components of the system remain accounted for, making it easier to observe conservation laws.
First Integral
In the study of Hamiltonian mechanics, a First Integral is a special type of function that remains constant over time for a given dynamic system. This constancy arises when specific symmetries are present in the system, often linked to conservation laws.
Mathematically, a function \( F(q, p, t) \) is considered a first integral if its time derivative is zero, i.e., \[ \frac{d F}{d t} = 0.\]The derivative, when expanded, takes the form: \[ \frac{d F}{d t} = \frac{\partial F}{\partial q_i} \dot{q}_i + \frac{\partial F}{\partial p_i} \dot{p}_i + \frac{\partial F}{\partial t}.\]Substituting the Hamilton's equations into this expression reveals important behaviors and conservation within the system.
First integrals provide insight into the conservation properties of systems, such as conservation of energy, linear momentum, or angular momentum. Recognizing a first integral is crucial for simplifying the analysis and solution of large mechanical systems.
Mathematically, a function \( F(q, p, t) \) is considered a first integral if its time derivative is zero, i.e., \[ \frac{d F}{d t} = 0.\]The derivative, when expanded, takes the form: \[ \frac{d F}{d t} = \frac{\partial F}{\partial q_i} \dot{q}_i + \frac{\partial F}{\partial p_i} \dot{p}_i + \frac{\partial F}{\partial t}.\]Substituting the Hamilton's equations into this expression reveals important behaviors and conservation within the system.
First integrals provide insight into the conservation properties of systems, such as conservation of energy, linear momentum, or angular momentum. Recognizing a first integral is crucial for simplifying the analysis and solution of large mechanical systems.
Generalized Coordinates
Generalized Coordinates are an extension of usual coordinates used in describing the state of a system. While traditional coordinates might focus strictly on spatial dimensions, generalized coordinates allow for the representation of additional dimensions necessary to fully describe the system's configuration.
In systems with constraints, the use of generalized coordinates can simplify their dynamics significantly, maintaining the correctness of the description by reducing the number of variables involved. Each coordinate \( q_i \) corresponds directly with a specific degree of freedom within the system.
The advantage here is adaptability. By defining \( q_i \) as generalized coordinates, even very complex mechanical systems become manageable, and these coordinates simplify the application of Hamilton's equations. They are not limited to traditional space coordinates but could encompass angles, phases, or any other descriptive quantities necessary for a thorough dynamic analysis.
In systems with constraints, the use of generalized coordinates can simplify their dynamics significantly, maintaining the correctness of the description by reducing the number of variables involved. Each coordinate \( q_i \) corresponds directly with a specific degree of freedom within the system.
The advantage here is adaptability. By defining \( q_i \) as generalized coordinates, even very complex mechanical systems become manageable, and these coordinates simplify the application of Hamilton's equations. They are not limited to traditional space coordinates but could encompass angles, phases, or any other descriptive quantities necessary for a thorough dynamic analysis.
Time Derivative
The Time Derivative is a fundamental concept in dynamics, representing how a quantity changes with respect to time. Within Hamiltonian mechanics, time derivatives play a critical role in describing how both generalized coordinates and momenta evolve over time.
For example, in Hamilton's equations, the time derivatives \( \dot{q}_i \) and \( \dot{p}_i \) indicate:
In terms of first integrals, a zero time derivative \( \frac{d F}{d t} = 0 \) indicates conservation, aligning with the laws of physics that tell us energy or other quantities might remain constant under certain conditions. Recognizing when these derivatives equate to zero can simplify analyzing and solving real-world problems.
For example, in Hamilton's equations, the time derivatives \( \dot{q}_i \) and \( \dot{p}_i \) indicate:
- \( \dot{q}_i = \frac{\partial H}{\partial p_i} \), the rate of change of generalized coordinates, and
- \( \dot{p}_i = -\frac{\partial H}{\partial q_i} \), the rate of change of momenta.
In terms of first integrals, a zero time derivative \( \frac{d F}{d t} = 0 \) indicates conservation, aligning with the laws of physics that tell us energy or other quantities might remain constant under certain conditions. Recognizing when these derivatives equate to zero can simplify analyzing and solving real-world problems.