Question

Verify that if \(U=\frac{1}{2} \sum_{j} \sum_{k} K_{j k} q_{j} q_{k},\) where the coefficients \(K_{j k}\) are all constant and satisfy \(K_{i j}=K_{j i}\) then \(\partial U / \partial q_{i}=\sum_{j} K_{i j} q_{j},\) as claimed in Equation (11.58).

Short answer

The partial derivative is \(\sum_{j} K_{ij} q_j\), as required.

Step-by-step solution

Understand the Problem

We are given a potential energy function \(U\) expressed as a double summation involving constants \(K_{jk}\) and variables \(q_j\), and are asked to verify a partial derivative involving these terms. The constants \(K_{jk}\) are symmetric, meaning \(K_{ij} = K_{ji}\).

Express the Double Summation

The expression for \(U\) is given by: \[ U = \frac{1}{2} \sum_{j} \sum_{k} K_{jk} q_j q_k. \] This means that each term in the summation involves a combination of two variables \(q_j\) and \(q_k\), multiplied by the constant \(K_{jk}\).

Differentiate \(U\) with Respect to \(q_i\)

We need to compute the partial derivative \(\frac{\partial U}{\partial q_i}\). Since \(K_{jk}\) are constants and we are differentiating with respect to \(q_i\), terms where neither index is \(i\) will have a derivative of 0. Consider terms with \(j=i\) and \(k=i\), and use symmetry.

Differentiate Terms with j=i and k=i

Differentiate the expression with respect to \(q_i\) focusing on both \(j = i\) and \(k = i\):- For terms with \(j = i\): \(\frac{1}{2} \sum_{k} K_{ik} q_k q_i = \frac{1}{2} q_i \sum_{k} K_{ik} q_k\), differentiate to get \(\sum_{k} K_{ik} q_k\).- For terms with \(k = i\): \(\frac{1}{2} \sum_{j} K_{ji} q_j q_i = \frac{1}{2} q_i \sum_{j} K_{ji} q_j\), differentiate to get \(\sum_{j} K_{ji} q_j\).

Use Symmetry of Coefficients

By using the fact \(K_{ij} = K_{ji}\), both differentiated parts \(\sum_{k} K_{ik} q_k\) and \(\sum_{j} K_{ji} q_j\) collapse to the same expression: \(\sum_{j} K_{ij} q_j\), confirming the equality \(\frac{\partial U}{\partial q_i} = \sum_{j} K_{ij} q_j\).

Key concepts

Potential Energy Function

A potential energy function in physics is a way to describe energy stored in a system due to its position or configuration. In this exercise, the potential energy function is expressed as:\[ U = \frac{1}{2} \sum_{j} \sum_{k} K_{jk} q_j q_k. \]Here, the terms \(K_{jk}\) represent constant symmetric coefficients and \(q_j\) are variables indicating specific system configurations.

• The factor \(\frac{1}{2}\) is used to avoid double counting, as each pair of interactions \(q_j q_k\) is considered only once.
• The double summation involves iterating over each pair \((j, k)\), capturing all possible interactions within the system.
• This formulation is common in physics when dealing with quadratic energy expressions, reflecting how the state variables interact via the constant coefficients \(K_{jk}\).

Potential energy functions are vital in understanding system dynamics and are often used in fields like mechanics, electromagnetism, and thermodynamics.

Symmetric Coefficients

Symmetric coefficients are a crucial element in the potential energy function in this context. When we say the coefficients \(K_{ij}\) are symmetric, it means:

• For any indices \(i\) and \(j\), \(K_{ij} = K_{ji}\).
• This symmetry property allows us to simplify the calculation when differentiating.

The symmetry of the coefficients implies that the interaction between variables is direction-independent. This can make physical models more realistic because, in many real-world systems, the interaction strength does not depend on the order of the elements. When working with these equations, recognizing symmetry helps streamline computations and can reduce potential errors.

Differentiation

Differentiation in this context involves taking the partial derivative of the potential energy function \(U\) with respect to one of the variables, say \(q_i\). Partial differentiation is used because \(U\) is a multivariable function.

• Only terms involving \(q_i\) need to be considered during differentiation, reducing the complexity of the operation.
• Differentiating \(U\) with respect to \(q_i\) isolates the terms where \(j\) or \(k\) equals \(i\), due to the symmetric property \(K_{ij} = K_{ji}\).
• As a result, the expression simplifies, and you end up with \(\frac{\partial U}{\partial q_i} = \sum_{j} K_{ij} q_j\).

Differentiation is a fundamental tool in calculus for analyzing changes in quantities. In physics, it helps in understanding how one aspect of a system can affect another, making it indispensable for solving problems in dynamics and thermodynamics.