Variation principle - holonomic constraints

In summary, to extend Hamilton's principle to include holonomic constraints, the lecturer added the constraint equations multiplied by arbitrary constants to the Lagrangian. This allows us to treat these constants as independent coordinates and find new equations that take into account the constraints. This process is similar to using Lagrange multipliers in calculus, where the gradients of the constraint and objective function must be collinear. By treating the constraints as independent coordinates, we can also treat the other coordinates as independent and vary them without considering the constraints. This approach removes degrees of freedom from the problem and transforms it into an optimization problem again.
  • #1
QuasarBoy543298
32
2
in order to extend hamilton's principle to include holonomic constraints, out lecturer did the following :
when we are under constraints, we cannot consider the variations of the coordinates as independent of each other.
we know that the constraint equations are fa = 0.
we can multiply each constraint by an arbitrary constant λa.
∑fa*λa = 0, so we can just add it to the Lagrangian, as we treat the λa
as independent coordinates.

from there we just followed the same thing we did in order to find the original EL equations.my question is - why can we treat the λa as independent coordinates.
also, why doing so gives us the option to treat the other coordinates as independent from each other (vary them by δqi without considering the constraints).

besides, if someone can explain a little the background of this process because to me it feels like we just added something that relates to the constraints into the Lagrangian and got new equations plus the original constraints.thanks in advance !
 
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  • #2
Did you already study Lagrange multipliers in calculus ?
It basically has to do with the two gradients (of the constraint(s) and of the objective function to be minimized): they have to be collinear.
You remove (number of constraints) degrees of freedom from an n##\times##n problem, so you add n ##\lambda##'s to come back to an n##\times##n optimization problem again.
 

1. What is the variation principle?

The variation principle, also known as the principle of least action, is a fundamental concept in physics and mathematics. It states that a system will follow a path that minimizes the action, which is a measure of the system's energy over time.

2. What are holonomic constraints?

Holonomic constraints are constraints on the motion of a system that can be described by a set of equations relating the position, velocity, and acceleration of the system. This means that the constraints do not depend on the path taken by the system, but only on its current state.

3. How do holonomic constraints affect the variation principle?

Holonomic constraints can be incorporated into the variation principle by adding them as additional equations to the system's Lagrangian, which is used to calculate the action. This ensures that the system's motion follows the constraints while still minimizing the action.

4. What is the significance of the variation principle in physics?

The variation principle is significant because it provides a framework for understanding the behavior of physical systems and predicting their motion. It is a fundamental principle in many areas of physics, including classical mechanics, quantum mechanics, and electromagnetism.

5. Can the variation principle be applied to non-physical systems?

Yes, the variation principle can be applied to non-physical systems, such as economic or social systems. In these cases, the action is not necessarily a measure of energy, but can represent other quantities such as utility or cost. The principle can still be used to predict the behavior of these systems by finding the path that minimizes the action.

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