D'alembert's Principle: Doubts Explained

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In summary, the principle states that the sum of all the terms in a sum must vanish, but this doesn't mean that each term must vanish separately.
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/I'm having some doubt with D'alembert's Principle. The principle states that [itex] \sum_{i}(\vec {F}_i - \dot{\vec{p}}_i)\delta\vec{r}_i=0[/itex] but does that mean that each term of the summation must vanish too, or just the sum does? I know that mathematically there's no need that each term shall vanish, but does physical considerations requires them to vanish separately?
 
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The point of d'Alembert's principle is to use it for constraint systems. If the constraints are all holonomic, you can introduce independent coordinates [itex]q_k[/itex] writing [itex]\vec{r}_i=\vec{r}_i(q_1,\ldots,q_f)[/itex], where [itex]f[/itex] is the number of independent degrees of freedom. Then you can vary the [itex]q_k[/itex] independently from each other, and this gives you equations of motion in terms of these variables.

Another method is to keep the Cartesian coordinates [itex]\vec{r}_i[/itex] and implement the constraints with help of Lagrange multipliers. Then you can vary the [itex]\delta \vec{r}_i[/itex] independently. This leads to equations of motion, where additional forces from the constraints are taken into account.
 
  • #3
Let's take this in two steps:

The Principle of Virtual Work tells us that constraints do no work.

D'Alembert's Principle permits the impressed forces of dynamical systems to be handled in the same way as constraints - the application of this is to find the unknown trajectories caused by the impressed forces via the calculus of variations - thanks to Euler and Lagrange.

For a system in equilibrium the independence of the coordinates tells us that each of the terms must be zero; for dynamics it is only the sum which must be zero.
 
  • #4
But why in dynamic situations only the sum must be zero, and not each member of the sum?
 
  • #5
In the case of equilibrium each variation has only a single coefficient: the value for the force of that constraint. In linear algebra it is shown that if the variables (coordinates) are linearly independent then the sum of a set of coefficients times the independent variables can only sum to zero when all of the coefficients are identically zero.

In the dynamical case you have a difference which must be zero ... but now the expression depends upon the trajectories via Newton's second law of motion: so yes, each term must go to zero, and this is the variational principle which is used to find those trajectories.

Sorry if my earlier statement was misleading and unclear.
 

1. What is D'alembert's Principle?

D'alembert's Principle is a fundamental principle in physics that states that the net force acting on a body is equal to its mass times its acceleration. It is used to analyze the motion of objects and is a key concept in classical mechanics.

2. How is D'alembert's Principle used in physics?

D'alembert's Principle is used to simplify the analysis of complex systems in physics. It allows us to treat a system as if it were in equilibrium, even if it is actually accelerating. This simplifies calculations and makes it easier to understand the overall motion of a system.

3. Is D'alembert's Principle applicable to all types of systems?

D'alembert's Principle is applicable to any system that can be described using classical mechanics. This includes systems with rigid bodies, fluids, and particles. However, it may not be applicable to systems that involve quantum mechanics or relativity.

4. What is the difference between D'alembert's Principle and Newton's Second Law?

D'alembert's Principle is based on the concept of virtual work, while Newton's Second Law is based on the concept of actual forces. This means that D'alembert's Principle can be applied to a wider range of systems, as it takes into account both external and internal forces.

5. Can D'alembert's Principle be used to solve real-world problems?

Yes, D'alembert's Principle is commonly used in engineering and physics to solve real-world problems. It can be used to analyze the motion of structures, vehicles, and other mechanical systems. However, it is important to note that D'alembert's Principle is a simplification and may not accurately represent all aspects of a system's motion.

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