Chapter 7: Problem 48
Let \(F=F\left(q_{1}, \cdots, q_{n}\right)\) be any function of the generalized coordinates \(\left(q_{1}, \cdots, q_{n}\right)\) of a system with Lagrangian \(\mathcal{L}\left(q_{1}, \cdots, q_{n}, \dot{q}_{1}, \cdots, \dot{q}_{n}, t\right) .\) Prove that the two Lagrangians \(\mathcal{L}\) and \(\mathcal{L}^{\prime}=\mathcal{L}+d F / d t\) give exactly the same equations of motion.
Short Answer
Step by step solution
Understand the problem
Write the Euler-Lagrange equations
Differentiate the new Lagrangian
Simplify total derivative effect
Ensure both constraints are satisfied
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Generalized Coordinates
Generalized coordinates can be any parameters that define a system's position, such as angles for rotational systems or distances for translational ones. The advantage of using them is that they provide a tailored approach:
- They reduce the number of variables needed, focusing only on the ones necessary for describing the specific motion.
- They can be chosen to simplify the equations of motion, making solving them more straightforward.
Euler-Lagrange Equations
Mathematically, the Euler-Lagrange equation for each generalized coordinate \( q_i \) is given by:\[\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0.\]Here, \( \mathcal{L} \) represents the Lagrangian, a function that encapsulates the dynamics of the system. This equation essentially states:
- The change in the derivative of the Lagrangian concerning the generalized velocities and time equals the derivative of the Lagrangian concerning the coordinates.
- It involves both temporal and spatial derivatives that describe how the system's state evolves over time.
- By solving these equations, one can find the trajectory or path of the system under given conditions.
Total Time Derivative
In Lagrangian mechanics, the total time derivative of a function, such as \( F(q_1, q_2, \ldots, q_n, t) \), is expressed as:\[\frac{dF}{dt} = \sum_{i} \left( \frac{\partial F}{\partial q_i} \dot{q}_i + \frac{\partial F}{\partial \dot{q}_i} \ddot{q}_i \right) + \frac{\partial F}{\partial t}.\]Here’s why it’s noteworthy:
- It incorporates the rates of all variables — both generalized coordinates and time — capturing how a function’s value evolves.
- This derivative plays a crucial role in mechanics because it can modify the form of the Lagrangian without changing the equations of motion.
- Specifically, if a Lagrangian \( \mathcal{L}' = \mathcal{L} + \frac{dF}{dt} \) only differs by a total derivative \( \frac{dF}{dt} \) of a function of coordinates, it results in the same equations of motion as \( \mathcal{L} \). This universality is critical as it allows for transformations of the problem that simplify the description without altering the physical content.