Lagrange equations of the first kind

Thread starter saRisky
Start date May 31, 2020
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In summary, Lagrange equations of the first kind, also known as the constrained equations of motion, are a set of equations used to describe the motion of a system with constraints. They are useful because they simplify the process of determining the equations of motion and allow for a more elegant and unified approach to solving problems in mechanics. These equations are derived using the principle of virtual work and take into account constraints in the system. They differ from Lagrange equations of the second kind, which do not consider constraints. Some applications of Lagrange equations of the first kind include classical mechanics, celestial mechanics, structural mechanics, robotics, control theory, and optimization.

May 31, 2020

saRisky

Homework Statement: A material point of mass m is moving in a constant gravitational field along the intersection of a sphere and a moving plane (z=Rsin(omega*t)). Find Lagrange's equations (first kind), the point's movement, and the constraint forces. Ignore friction and assume that the gravitational field is parallel to the "z" axis.

Relevant Equations: Lagrange equations of first kind

We cannot make it anyhow

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Jun 7, 2020

anuttarasammyak

Gold Member

If I take the setting correctly, gravity force (z direction ) is always perpendicular to free degree motion of the body ( on xy plane ) so it makes no effect to the motion. Only the inflating-shrinking circle boundary of the intersection would matter.

Last edited: Jun 7, 2020

1. What are Lagrange equations of the first kind?

The Lagrange equations of the first kind are a set of equations used in classical mechanics to describe the motion of a system of particles subject to constraints. They were developed by Italian mathematician and astronomer Joseph-Louis Lagrange in the late 18th century.

2. How are Lagrange equations of the first kind derived?

The Lagrange equations of the first kind are derived using the principle of virtual work. This principle states that the work done by the applied forces on a system is equal to the change in the kinetic energy of the system.

3. What is the significance of Lagrange equations of the first kind in classical mechanics?

The Lagrange equations of the first kind are significant because they provide a systematic and efficient way to analyze the motion of a system of particles. They allow for the incorporation of constraints into the equations of motion, making it easier to solve complex problems.

4. Can Lagrange equations of the first kind be used in any system?

Yes, Lagrange equations of the first kind can be used in any system as long as the constraints are holonomic (dependent only on the position of the particles) and the forces are conservative.

5. Are there any limitations to using Lagrange equations of the first kind?

One limitation of Lagrange equations of the first kind is that they cannot be used to analyze systems with non-conservative forces, such as friction. Additionally, they may become more complex and difficult to solve for systems with a large number of particles or complex constraints.