The Lagrange–d'Alembert principle for rigid bodies

In summary, the conversation discusses the use of the Lagrange-d'Alembert principle in nonholonomic mechanics. It presents an example problem of a ball moving in a chute formed by two planes, and the acceleration of the ball is found using the principle. The conversation also highlights the benefits of using this principle, such as not needing to consider reaction forces at certain points. The suggestion is made to turn this into a PF insight article.
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wrobel
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This post is also an invitation to compose problems of the presented below type.
The Lagrange–d'Alembert principle by itself is used very seldom. It is usually used to derive the Lagrange equations and that is all. But actually it is a powerful tool in nonholonomic mechanics.
As an example consider a problem which is interesting by itself.
A chute is formed by two planes. The angle between the planes equals ##\alpha\in(0,\pi)## and it does not depend on time. The planes intersect by a horizontal line ##\ell##. The line ##\ell## does not change its position in the space during the time.
One of the planes is fixed while other plane moves in horizontal direction along the line ##\ell##. The velocity of this plane is a given function ##\boldsymbol u=\boldsymbol u(t)##.
Then one puts a homogeneous ball inside the chute. The ball does not slip by the both planes. Find the acceleration of the center of the ball.
pic.jpg

The answer is
$$\frac{\dot u}{\frac{mr^2}{J}\Big(1-\cos\alpha\Big)+2},\quad J=2mr^2/5.$$
Here ##r## is a radius of the ball; ##m## is its mass.
Let us now sketch the solution by means of the Lagrange–d'Alembert principle.
Let ##S## stand for the center of the ball; ##A## stand for the contact point of the ball and the moving plane and let ##B## be the contact point of the ball and the fixed plane.
The conditions of non-slipness are
$$\boldsymbol v_S+\boldsymbol \omega\times \boldsymbol {SA}=\boldsymbol u;\quad\boldsymbol v_S+\boldsymbol \omega\times \boldsymbol {SB}=0.$$
Here ##\boldsymbol \omega## is the angular velocity of the ball.
The Lagrange–d'Alembert principle is
$$(J\boldsymbol{\dot\omega},\boldsymbol\xi)+(m\boldsymbol{\dot v}_S,\boldsymbol\gamma)=0.$$
This equation must be satisfied for all virtual displacements ##\boldsymbol\xi,\boldsymbol\gamma## such that
$$\boldsymbol \gamma+\boldsymbol \xi\times \boldsymbol {SA}=0;\quad\boldsymbol \gamma+\boldsymbol \xi\times \boldsymbol {SB}=0.$$These are the equations of motion. They allow to accomplish the solution of the problem.

The gain of using the Lagrange–d'Alembert principle is as follows. The equations of motion do not contain reaction forces at points ##A,B##. Moreover these reactions can not be defined uniquely.

Note also that if ##\boldsymbol u=0## then the problem is holonomic.
 
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Nice! Why not make this a PF insight article?
 

Related to The Lagrange–d'Alembert principle for rigid bodies

1. What is the Lagrange-d'Alembert principle for rigid bodies?

The Lagrange-d'Alembert principle for rigid bodies is a fundamental principle in classical mechanics that is used to describe the motion of rigid bodies in a system. It states that the equations of motion for a rigid body can be derived from the principle of virtual work, which states that the total work done by all forces acting on a system is equal to the change in the kinetic energy of the system.

2. How is the Lagrange-d'Alembert principle different from Newton's laws of motion?

The Lagrange-d'Alembert principle is a more general and powerful approach to describing the motion of rigid bodies compared to Newton's laws of motion. While Newton's laws are based on the concept of force, the Lagrange-d'Alembert principle is based on the concept of virtual work, which takes into account both external and internal forces acting on a system.

3. Can the Lagrange-d'Alembert principle be applied to non-rigid bodies?

No, the Lagrange-d'Alembert principle is specifically designed for rigid bodies, which are defined as bodies that maintain their shape and size regardless of external forces acting on them. For non-rigid bodies, other principles and laws, such as the principle of virtual displacements or the Euler-Lagrange equations, must be used.

4. What are some practical applications of the Lagrange-d'Alembert principle for rigid bodies?

The Lagrange-d'Alembert principle has wide-ranging applications in engineering and physics, particularly in the fields of mechanics and dynamics. It is used to analyze the motion of complex systems, such as satellites, vehicles, and robots, and is also used in the design and control of mechanical systems, such as engines and machines.

5. Is the Lagrange-d'Alembert principle for rigid bodies still relevant in modern science?

Yes, the Lagrange-d'Alembert principle is still a fundamental principle in classical mechanics and is widely used in modern science and engineering. It provides a powerful and elegant approach to describing the motion of rigid bodies and is essential in understanding and predicting the behavior of complex systems in various fields of study.

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