# How to integrate velocity squared to solve action integral?

• nabeel17
In summary, the conversation discusses the calculation of the action for a force-free mass point in one-dimensional space with specific boundary conditions. The Lagrangian is determined to be equal to L=1/2mv^2 and the action integral is given by S=∫1/2mv^2dt. The integral cannot be solved without further information about the velocity, either v(t) or v(x), and the conversation concludes with a clarification on this point.
nabeel17
Consider a forcefree mass point in one-dimensional space.
(a) Calculate the action S for the actual path of the mass point in the time interval
[0, T] and for the boundary conditions x(0) = 0 and x(T) = d.

I said the Lagrangian was just equal to L=1/2mv^2. I'm not sure if my reasoning for this is correct (I may have a conceptual error) but since there is no force acting on it, the potential is 0 (or a constant but it can be set to 0?)

so the action integral is S=∫1/2mv^2dt where the limits are 0-T

I'm not sure how to integrate v^2 with respect to t. Even if my approach is wrong, I would still like to know how that integral is done

oops, velocity would be constant...so that solves that. However, if it wasn't constant, how would I go about solving that integral?

You cannot solve it without some information about how velocity varies with time.

You need to either know v(t) or v(x). If you know the second then you can use the fact that v2dt=vdx.

ahh, ok. Thank you

## 1. What is the action integral?

The action integral is a mathematical concept used in classical mechanics to calculate the motion of a physical system. It represents the total energy of a system over a particular period of time.

## 2. Why is velocity squared integrated in the action integral?

Velocity squared is integrated in the action integral because it represents the kinetic energy of a system. By integrating velocity squared over time, we can calculate the total kinetic energy of the system and use it to determine the motion of the system.

## 3. How is the action integral used in classical mechanics?

In classical mechanics, the action integral is used to determine the path a system will take by minimizing the action, which is the integral of the Lagrangian (a measure of the system's energy) over time. This results in the principle of least action, which states that a system will follow the path that minimizes its total energy.

## 4. What is the relationship between the action integral and the Hamiltonian?

The Hamiltonian is a mathematical function used in classical mechanics to calculate the total energy of a system. The action integral is related to the Hamiltonian through Hamilton's principle, which states that the action integral is equal to the integral of the Lagrangian over time. This allows us to use the Hamiltonian to calculate the motion of a system.

## 5. Is the action integral applicable to all physical systems?

Yes, the action integral is applicable to all physical systems that can be described using classical mechanics. It is a fundamental concept that is used to analyze and predict the behavior of a wide range of systems, from a simple pendulum to complex celestial bodies.

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