Question

In Example \(3.8\), find \(\tilde{Q}_{1}\) and \(\tilde{Q}_{2}\) in terms of the components of \(\boldsymbol{F}\) by considering the work done under suitable small displacements. Check that the same expressions for the \(\tilde{Q}_{a}\) s follow from the chain rule in the case that \(\boldsymbol{F}=-\boldsymbol{\nabla} U\).

Short answer

In this exercise, we first derived the expressions for the generalized forces, \(\tilde{Q}_{1}\) and \(\tilde{Q}_{2}\), using the work-energy principle. We related the small displacements in Cartesian and generalized coordinates through the Jacobian matrix. Then, we verified the expressions for the generalized forces using the chain rule when the force vector was derived from the potential energy. The consistency of these expressions shows that the work-energy principle can be effectively used to find generalized forces in terms of the components of the force vector in a given coordinate system.

Step-by-step solution

Derive the expression for generalized force using work-energy principle

To derive the generalized forces \(\tilde{Q}_{1}\) and \(\tilde{Q}_{2}\), we will use the work-energy principle, which states that the work done by a force during a displacement is equal to the change in the system's energy. The work done under small displacements is given by \(\delta W = \boldsymbol{F} \cdot \delta\boldsymbol{r}\), where \(\delta\boldsymbol{r}\) is the small displacement vector. Let the generalized coordinates be \(\tilde{q}_{1}\) and \(\tilde{q}_{2}\). The small displacements in these coordinates are given by \(\delta\tilde{q}_{1}\) and \(\delta\tilde{q}_{2}\), respectively. The relationship between the small displacements in Cartesian and generalized coordinates is given by the Jacobian matrix: \(\delta\boldsymbol{r} = J\delta\boldsymbol{\tilde{q}}\) where \(J\) is the Jacobian matrix and \(\delta\boldsymbol{\tilde{q}}\) is the vector of small displacements in generalized coordinates. The work done during these small displacements can be expressed as: \(\delta W = \boldsymbol{F} \cdot J\delta\boldsymbol{\tilde{q}}\) The generalized forces are the coefficients of the infinitesimal displacements in generalized coordinates: \(\tilde{Q}_{1} = \boldsymbol{F} \cdot J_{1\cdot}\), \(\tilde{Q}_{2} = \boldsymbol{F} \cdot J_{2\cdot}\) where \(J_{1\cdot}\) and \(J_{2\cdot}\) are the first and second rows of the Jacobian matrix, respectively.

Verify the expressions using the chain rule

Now, we need to verify the expressions derived in Step 1 using the chain rule in the case that \(\boldsymbol{F} = -\boldsymbol{\nabla}U\). The chain rule states that: \(\frac{\partial U}{\partial\tilde{q}_{a}} = \frac{\partial U}{\partial x_{i}}\frac{\partial x_{i}}{\partial\tilde{q}_{a}}\) where \(\frac{\partial U}{\partial x_{i}} = -F_{i}\), and \(\frac{\partial x_{i}}{\partial\tilde{q}_{a}}\) are the elements of the Jacobian matrix \(J^{T}\) (transpose of the Jacobian matrix). Therefore, the above equation can be written as: \(-\tilde{Q}_{a} = -F_{i}\frac{\partial x_{i}}{\partial\tilde{q}_{a}} = F_{i}J_{a\cdot}^{T}\). Now, comparing this equation with the expressions derived in Step 1, we find that the two expressions for generalized forces in terms of the components of force vector \(\boldsymbol{F}\) are consistent: \(\tilde{Q}_{a} = F_{i}J_{a\cdot}\) Hence, we have verified the expressions for the generalized forces using the chain rule.

Key concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics, relating the work done by forces on a system to the change in energy of that system. Simply put, when a force causes a displacement, it does work, and this work changes the energy of the object or system involved.

This principle is often expressed as \( \delta W = \boldsymbol{F} \cdot \delta \boldsymbol{r} \), where \( \delta W \) is the work done, \( \boldsymbol{F} \) is the force applied, and \( \delta \boldsymbol{r} \) is the displacement.

• Work Done by a Force: This is the dot product of the force vector and the displacement vector. It's positive when the force and displacement are in the same direction.
• Generalized Forces: These emerge when considering displacements in generalized coordinates (like angles or distances not directly measured in traditional coordinates).

Applying this principle helps us derive expressions for generalized forces \( \tilde{Q}_{1} \) and \( \tilde{Q}_{2} \) by considering the small displacements and the work done. Understanding how energy changes in response to force and movement is crucial in analyzing mechanical systems.

Jacobian Matrix

The Jacobian matrix is a mathematical tool that relates small changes in one coordinate system to changes in another. It's particularly useful in dealing with problems where we switch from Cartesian coordinates (like \( x, y, z \)) to generalized coordinates (like \( \tilde{q}_{1}, \tilde{q}_{2} \)).

Consider the relationship \( \delta \boldsymbol{r} = J \delta \boldsymbol{\tilde{q}} \), where \( J \) is the Jacobian matrix.

• Role of the Jacobian: This matrix contains partial derivatives that describe how a small change in a generalized coordinate affects the Cartesian coordinates.
• Using the Jacobian: The forces \( \boldsymbol{F} \) and the displacements are connected via this matrix, hence \( \delta W = \boldsymbol{F} \cdot J \delta \boldsymbol{\tilde{q}} \).

By breaking down the Jacobian matrix into its components, each row can guide us to find the generalized forces as \( \tilde{Q}_{1} = \boldsymbol{F} \cdot J_{1\cdot} \) and \( \tilde{Q}_{2} = \boldsymbol{F} \cdot J_{2\cdot} \). This gives us a clearer path to express forces in terms of different coordinates.

Chain Rule

The chain rule is a powerful calculus tool that helps us differentiate functions that are composed of other functions. In this context, it allows us to express derivatives of a potential function \( U \) in terms of generalized coordinates.

When \( \boldsymbol{F} = -\boldsymbol{abla} U \), the chain rule gives us:\[\frac{\partial U}{\partial \tilde{q}_{a}} = \frac{\partial U}{\partial x_{i}} \frac{\partial x_{i}}{\partial \tilde{q}_{a}} \]Here:

• Gradient \( abla U \): Represents how \( U \) changes in space and is linked to the force \( \boldsymbol{F} \).
• Jacobian Connection: \( \frac{\partial x_{i}}{\partial \tilde{q}_{a}} \) are elements from the Jacobian matrix, which help in translating the effects of changes in coordinates.

Using the chain rule, we confirm that the expressions previously derived for generalized forces \( \tilde{Q}_{a} = F_{i} J_{a\cdot} \) are indeed consistent with this alternative approach. It beautifully connects derivatives and coordinate transformations, enriching our understanding of dynamics.