Is Calculus of Variations the Next Step After Traditional Calculus?

In summary, The subject of Calculus of Variations is often mentioned in relation to advanced topics such as computational style calculus and Classical Mechanics. It is the basis for the Lagrangian formulation of Classical Mechanics and is also used in modern numerical methods and Finite Element techniques. This topic generalizes the methods of finding extremal points in functions, but the theory can be quite advanced for those without a strong background in mathematics. While there are some examples in Classical Mechanics that may be more accessible, it is important to choose a suitable book on the subject, such as Lanczo's "Variational principles of Mechanics". Additionally, VC is not only important for Classical Mechanics, but also for other major theories in Physics such as Electrodynamics and General Rel
  • #1
Hi, I've seen the words "Calculus of Variations" mentioned quite a bit but never thought too much about them since it seemed too advanced.

Well, I am nearly finished the computational style calculus and am awaiting my Apostol text to get more into the theory but I also picked up a text called Calculus of Variations for like $5 (Second hand Dover book!) and I wonder will this be the kind of subject you can just jump into after calculus?

It seems to me as this subject is the basis of the Lagrangian formulation of Classical Mechanics, is this correct?
 
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  • #2
It is also the underlying maths for some modern numerical methods and some Finite Element techniques.
 
  • #3
Variational Calculus can be seen as a (vast) generalization of the methods for finding extremal points in functions, that you should have covered in Calculus; here, the "points" are functions, that belong to suitable function spaces, instead of the more common "points" that you may be used to, belonging to an euclidian space, for example.

Given this, I can tell you that the theory, as developed by mathematicians, is far too advanced for your background. As for the Physics' applications, it depends; there are some examples in Classical Mechanics that, I think, are within your reach.

As far as books are concerned, Dover re-issues are not a good bet for this topic; odds are that you spent 5$ on a book that will scare you away from VC at the first pages. There is one exception: Lanczo's "Variational principles of Mechanics"; this book is more conceptual than rigorous, but it could give you a fair view of VC, without the more spiky technicalities.

One more thing: VC is not only the foundation of lagrangian Classical Mechanics; it's also the foundation of hamiltonian Classical Mechanics (the two are related by something called a Legendre transform) and practically every major Physics theory (Electrodynamics, classical and quantum, General Relativity, etc.) admits a variational formulation.
 

1. What is the calculus of variations?

The calculus of variations is a mathematical field that deals with finding the optimal path or curve between two points. It involves optimizing a functional, which is a function that takes in other functions as its inputs.

2. How is calculus of variations used in real-world applications?

The calculus of variations is used in a variety of fields, including physics, engineering, and economics. It is used to find the path of least resistance or maximum efficiency in systems such as fluid flow, electrical circuits, and structural design. It is also used in optimization problems in economics and finance, such as maximizing profits or minimizing costs.

3. What is the main difference between calculus of variations and traditional calculus?

The main difference between calculus of variations and traditional calculus is that traditional calculus deals with optimizing functions with fixed endpoints, while calculus of variations deals with optimizing functions with variable endpoints. In other words, traditional calculus finds the maximum or minimum value of a function at a specific point, while calculus of variations finds the function that produces the maximum or minimum value over a range of inputs.

4. What are some common techniques used in calculus of variations?

Some common techniques used in calculus of variations include the Euler-Lagrange equation, which is used to find the optimal function by setting the derivative of the functional to zero. Other techniques include the method of variation of parameters, the Weierstrass-Erdmann corner condition, and the Hamilton-Jacobi-Bellman equation.

5. How does calculus of variations relate to other areas of mathematics?

Calculus of variations is closely related to other areas of mathematics, such as differential equations, optimization, and functional analysis. It is also used in conjunction with other mathematical techniques, such as partial differential equations and numerical methods, to solve complex problems in physics, engineering, and economics.

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