Chapter 7: Problem 48
Let
Short Answer
Expert verified
Both Lagrangians have the same equations of motion because the total derivative doesn't affect them.
Step by step solution
01
Understand the problem
We're given a Lagrangian and a second Lagrangian , where is a function of generalized coordinates. The task is to show that these two Lagrangians produce the same equations of motion.
02
Write the Euler-Lagrange equations
The Euler-Lagrange equations for a system are given by: for each coordinate . We need to apply this to both and .
03
Differentiate the new Lagrangian
For the new Lagrangian , we find: Notice that the term involves differentiating a total derivative, which only depends on and not on , thus it does not contribute.
04
Simplify total derivative effect
For , since it is a total time derivative, differentiate with respect to leads to zero because . Time derivative of affects only but doesn't affect , thus .
05
Ensure both constraints are satisfied
After computing and , so the Euler-Lagrange equation for will be: which is the same as before for .
06
Conclusion
The total time derivative does not affect the Euler-Lagrange equations. Both Lagrangians and result in the same equations of motion for the system.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Generalized Coordinates
In Lagrangian mechanics, the concept of generalized coordinates is central to describing motion in dynamic systems. Instead of using traditional coordinates like Cartesian coordinates, generalized coordinates take the complexity of a system and simplify it by representing all the possible ways a system can move. These coordinates, denoted usually by , are a set of variables that uniquely describe the configuration of the system.
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:
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
The Euler-Lagrange equations are pivotal in deriving the equations of motion for a system using Lagrangian mechanics. Derived from a principle of stationary action, these equations help find paths which make the action integral stationary, leading to the natural evolution of the system.
Mathematically, the Euler-Lagrange equation for each generalized coordinate is given by: Here, represents the Lagrangian, a function that encapsulates the dynamics of the system. This equation essentially states:
Mathematically, the Euler-Lagrange equation for each generalized coordinate
- 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
A total time derivative in dynamics involves understanding how a function changes when all its variables are functions of time. In this context, it is essentially a rate of change of a function as seen from a moving reference point.
In Lagrangian mechanics, the total time derivative of a function, such as , is expressed as: Here’s why it’s noteworthy:
In Lagrangian mechanics, the total time derivative of a function, such as
- 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
only differs by a total derivative of a function of coordinates, it results in the same equations of motion as . This universality is critical as it allows for transformations of the problem that simplify the description without altering the physical content.