Hamilton's Method with Lagrange Equation and Constraint

In summary, the individual is comfortable with using Hamilton's Principle and the Euler Lagrange Equations, but struggles with advanced mathematical theory and has minimal understanding of Lagrange multipliers. They are seeking a simple example of applying a constraint to Hamilton's Principle, specifically in the case of a particle moving in the vertical plane with gravity and a linear constraint. As a retiring mechanical engineer with a limited mathematical background, they are interested in learning new equations and concepts.
  • #1
Trying2Learn
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TL;DR Summary
How do you apply constraint to calculus of variations
Good Morning

I am "comfortable" with formulating Hamilton's Principle with a Lagrangian (KE - PE), conducting the calculus of variations and obtaining the Euler Lagrange Equations. Advanced mathematical theory, is beyond me.

I also have a minimal understanding of using Lagrange multipliers.

I would like to combine both, say, for a closed loop on a robot linkage system

My issue is that I would like to see a SIMPLE example of how to apply a constraint (any constraint but formulated as an equality) to Hamilton's Principle

Could someone point me to a simple example?

I am a retiring mechanical engineer with a minimal mathematical skill set. My recent equations are now curiosity -- things I never learned.
 
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  • #2
Perhaps the simplest example is to write formulas for the case when a particle moves in the vertical plane (x,y) in the standard gravity g and the constraint is ax+by=0
 

1. What is Hamilton's method with Lagrange equation and constraint?

Hamilton's method with Lagrange equation and constraint is a mathematical approach used to solve problems in classical mechanics. It combines the Lagrange equation, which describes the motion of a system based on its potential and kinetic energy, with the Hamiltonian, which represents the total energy of the system. This method is used to find the equations of motion for a system with constraints, such as a particle moving on a curved surface.

2. How does Hamilton's method differ from other methods in classical mechanics?

Hamilton's method differs from other methods in classical mechanics, such as Newton's laws or the Lagrange equations, in that it takes into account both the position and momentum of a system. This allows for a more complete description of the system's motion and can be particularly useful when dealing with systems with constraints.

3. What is the role of the Lagrange multiplier in Hamilton's method?

The Lagrange multiplier is a term that is added to the Lagrange equation when dealing with constrained systems. It serves as a way to incorporate the constraints into the equations of motion and is essential in finding the correct solutions for the system.

4. Can Hamilton's method be applied to any system?

Yes, Hamilton's method can be applied to any system in classical mechanics, including systems with multiple particles, constraints, and external forces. However, it is most commonly used for systems with constraints, as it provides a more efficient and elegant approach compared to other methods.

5. What are the advantages of using Hamilton's method over other methods in classical mechanics?

One of the main advantages of using Hamilton's method is its ability to incorporate constraints into the equations of motion, making it particularly useful for systems with constraints. Additionally, it provides a more complete description of the system's motion by taking into account both position and momentum. This method also has applications in other areas of physics, such as quantum mechanics and statistical mechanics.

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