15. Introduction to Lagrange With Examples

The script begins with an introduction to the Lagrange method, emphasizing its importance in physics and engineering. It mentions the Creative Commons license under which the content is provided and encourages donations to MIT OpenCourseWare. The professor outlines the topics for the day, including generalized coordinates and forces, and recommends a set of notes on Stellar for further reading. The lecture also covers the structure of the upcoming quiz and introduces the concept of the Lagrangian, which is the difference between the kinetic and potential energy of a system. The importance of understanding generalized coordinates and forces is highlighted, and the lecture notes their relevance in formulating the Lagrange equations.

This paragraph delves deeper into the practical application of Lagrange equations. It explains the simplification of the equations for mechanical systems where potential energy is not a function of time or velocity. The professor emphasizes the importance of bookkeeping in Lagrange's method and introduces the concept of generalized coordinates being independent and complete for a holonomic system. The explanation includes a practical approach to writing down the equations and the process of identifying the correct coordinates for a system, such as a double pendulum, to ensure they meet the criteria of independence and completeness.

The script continues with a detailed discussion on the selection of generalized coordinates and the concept of holonomic systems. It uses the example of a double pendulum to illustrate the independence and completeness of coordinates and contrasts it with an incorrect choice of coordinates using Cartesian coordinates. The importance of the system being holonomic is highlighted, meaning the number of degrees of freedom equals the number of coordinates needed to describe the system's motion. A counterexample of a non-holonomic system involving a rolling ball with constraints is provided to demonstrate the difference.

The professor moves on to the practical application of Lagrange's equations, breaking down the process into systematic steps. These steps include determining the number of degrees of freedom, choosing appropriate coordinates, verifying their completeness, independence, and holonomicity, and computing the kinetic and potential energies for each coordinate. The paragraph also discusses the calculation of generalized forces through virtual work, using a simple mass-spring-damper system as an example to demonstrate the process.

This section introduces a more complex problem involving multiple degrees of freedom, specifically a steel pipe system with a pendulum and a sliding sleeve. The challenge involves determining the appropriate coordinates to describe the system's motion, which is identified as two: an angle and a deflection. The paragraph outlines the need to calculate potential and kinetic energies for this system, considering the constraints of planar motion and the absence of certain degrees of freedom.

The script presents the process of deriving the equations of motion for the complex system introduced earlier. It involves calculating the potential and kinetic energies, taking into account the system's geometry and the forces acting upon it. The potential energy calculation is detailed, considering the unstretched length of the spring and the gravitational potential energy for both the rod and the sleeve. The kinetic energy is broken down into rotational and translational components, with specific attention given to the moving parts of the system.

This paragraph continues the application of Lagrange's method to the coupled system, focusing on the derivation of the equations of motion. It discusses the need to apply the method to each coordinate, resulting in two equations for the system. The first equation is developed for the translational motion of the sliding sleeve, and the paragraph outlines the expected acceleration terms, such as the centripetal acceleration and the effects of the spring force and gravity.

The script emphasizes the importance of validating the derived equations of motion against Newtonian physics principles. It suggests checking for the presence of expected acceleration terms and ensuring the equations make sense physically. The paragraph also introduces the concept of static equilibrium as a method to verify the equations, showing that at rest, the forces and torques balance out according to the system's geometry and the forces acting upon it.

The final part of the script focuses on deriving the second equation of motion for the swinging motion of the system. It prompts the audience to predict the types of acceleration terms that would appear in this equation, such as the Eulerian acceleration term due to the swinging motion and the Coriolis term resulting from the interaction between the sliding motion and the swinging motion. The paragraph outlines the process of finding these terms by considering the system's kinetic and potential energies and the work done by external forces.