Derivation in classical mechanics

In summary, the conversation is about the relationships between the action, Hamiltonian, Lagrangian, kinetic energy, and potential energy in classical mechanics. The question arises about whether there are any other relationships or identities that exist, and if so, what they are. However, the responders suggest that there is no point in writing down these additional quantities as they can already solve equations of motion without them. They also suggest that the individual asking the question should do more research on their own rather than expecting others to provide examples and write a textbook for them.
  • #1
Jhenrique
685
4
I'm studying classical mechanics and I'm stumbling in the quantity of differential identities.

Being S the action, H the hamiltonian, L the lagrangian, T the kinetic energy and V the potential energy, following the relationships:

attachment.php?attachmentid=70623&stc=1&d=1402838216.png


But, the big question is: that's all? Or has exist more?

Seems be missing
$$\frac{\partial S}{\partial q'} \;\;\; \frac{\partial S}{\partial p} \;\;\; \frac{\partial S}{\partial p'} \;\;\; \frac{\partial S}{\partial q'} \;\;\; \frac{\partial L}{\partial p} \;\;\; \frac{\partial L}{\partial p'} \;\;\; \frac{\partial H}{\partial p'} \;\;\; \frac{\partial H}{\partial q'} \;\;\; \frac{\partial V}{\partial q'} \;\;\; \frac{\partial V}{\partial p} \;\;\; \frac{\partial V}{\partial p'} \;\;\; \frac{\partial T}{\partial q} \;\;\; \frac{\partial T}{\partial p} \;\;\; \frac{\partial T}{\partial p'}$$
These relation exist? Make sense? If yes, how will be the identities?
 

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  • #2
Once you write down the Hamiltonian or the Lagrangian then you certainly can write down all of the rest of those quantities, but there is no point it doing so. You can already solve the equations of motion without them.
 
  • #3
DaleSpam said:
Once you write down the Hamiltonian or the Lagrangian then you certainly can write down all of the rest of those quantities

How? Give me examples...
 
  • #4
First, it's please give me examples. We are not your servants, to be ordered around.

Second, you're essentially asking us to write down a textbook for you. I'm afraid that's beyond what one can reasonably expect PF to do. You are going to have to do some work on your own.

This looks like a good time to close this thread.
 
  • #5


Thank you for your question. In classical mechanics, the fundamental principle is the principle of least action, which states that the path of a system between two points in time is the one that minimizes the action, S, defined as the integral of the Lagrangian, L, over time. From this principle, we can derive the equations of motion for a system using the Euler-Lagrange equations.

The relationships you have listed are all valid and can be derived using the Euler-Lagrange equations. However, there are also other identities that can be derived in classical mechanics, such as the Legendre transformations, which relate the Hamiltonian, H, to the Lagrangian, L, and the equations of motion.

In addition, there are also other identities that arise from specific systems or scenarios, such as the Hamilton-Jacobi equation and the Hamiltonian formulation of classical mechanics.

It is important to note that these identities are not just mathematical equations, but they have physical significance and can be used to understand and analyze the behavior of systems in classical mechanics. So, while there may be more identities than the ones you have listed, the ones you have mentioned are important and fundamental in classical mechanics.
 

1. What is derivation in classical mechanics?

Derivation in classical mechanics is the process of mathematically deriving equations and principles from first principles, such as Newton's laws of motion and the conservation laws of energy and momentum. This involves using mathematical techniques, such as calculus, to model and analyze the motion of objects in the physical world.

2. Why is derivation important in classical mechanics?

Derivation is important in classical mechanics because it allows us to understand and predict the behavior of physical systems. By deriving equations and principles, we can make accurate predictions about the motion of objects and systems in various situations, which is crucial for many practical applications in engineering and physics.

3. What are some common techniques used in derivation in classical mechanics?

Some common techniques used in derivation in classical mechanics include Newton's laws of motion, the principles of conservation of energy and momentum, the equations of motion, and the use of calculus to solve differential equations. Other techniques may include the use of vector calculus, Lagrangian mechanics, and Hamiltonian mechanics.

4. How is derivation different from integration in classical mechanics?

Derivation and integration are closely related processes, but they have different purposes in classical mechanics. Derivation is used to find equations and principles from first principles, while integration is used to solve equations and find the values of physical quantities. In other words, derivation is the process of going from general principles to specific equations, while integration is the process of going from specific equations to specific values.

5. Can anyone learn how to do derivation in classical mechanics?

Yes, anyone with a basic understanding of mathematics and physics can learn how to do derivation in classical mechanics. It may require some practice and effort to become proficient, but with proper instruction and practice, anyone can learn how to use the techniques of classical mechanics to derive equations and principles.

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