Does the value of the action S have classical significance?

In summary, the Euler-Lagrange equations are used to obtain equations of motion, with the action ##S## being an extremum along this path. Typically, we are interested in the path ##x(t)## connecting ##x(0)=x_0## and ##x(T)=x_1## in one dimension. However, the numerical value of the resulting action may not have any significance. Its units are energy-time. In the simple case of a particle with mass ##m## under a constant force ##F##, the Euler-Lagrange equation gives ##m \ddot x = F##, but the action itself has a complicated form. It is given by $$S = \frac {(x_1+x_
  • #1
Kostik
58
6
We obtain equations of motion by solving the Euler-Lagrange equations; along this path the action ##S## is an extremum. We are usually interested in the path ##x(t)## (in one dimension) connecting ##x(0)=x_0## and ##x(T)=x_1##. But does the numerical (extremal) value of the resulting action have any significance? Its units are plainly energy-time.

In the extremely simple case of a particle with mass ##m## under a constant force ##F##, the Euler-Lagrange equation gives the obvious ##m \ddot x = F##, but the action itself has the complicated form:
$$S = \frac {(x_1+x_0)FT} {2} + \frac {(x_1-x_0)^2 m} {2T} - \frac {F^2 T^3} {24m}.$$

This strikes me as a peculiarly complicated quantity to have some significance to a particle moving under a constant force.
 
  • Like
Likes nomadreid
Physics news on Phys.org
  • #2
Can you show your steps at getting this equation?
Other than that, action is the same units as angular momentum, but is not directly related to it, like energy and torque.
 

1. What is meant by classical significance in relation to the value of action S?

In classical mechanics, the value of action S refers to the integral of a system's Lagrangian over a certain period of time. This value has significance in determining the path that a system will take, as it follows the principle of least action. This means that the system will follow the path that minimizes this value.

2. How is the value of action S calculated?

The value of action S is calculated by taking the integral of the system's Lagrangian over a specific time interval. This integral is usually written as S = ∫Ldt, where L represents the Lagrangian and t represents time.

3. What is the principle of least action?

The principle of least action states that a physical system will follow the path that minimizes the value of its action. This principle is derived from the fundamental laws of classical mechanics and applies to a wide range of systems.

4. What is the significance of the value of action S in quantum mechanics?

In quantum mechanics, the value of action S has a different meaning. It represents the phase of the wave function of a quantum system. This phase is important in determining the probability of a particle's position and momentum, and plays a crucial role in quantum phenomena such as interference and tunneling.

5. How does the value of action S relate to the concept of energy?

The value of action S is closely related to the concept of energy. In classical mechanics, the Hamiltonian of a system is equal to the sum of its kinetic and potential energies, which can be expressed in terms of the system's Lagrangian. In quantum mechanics, the energy of a system is proportional to the frequency of its wave function, which is related to the phase of the wave function through the value of action S.

Similar threads

  • Classical Physics
Replies
11
Views
2K
Replies
19
Views
1K
Replies
22
Views
311
  • Classical Physics
Replies
1
Views
1K
  • Classical Physics
2
Replies
41
Views
2K
Replies
2
Views
718
Replies
1
Views
503
  • Introductory Physics Homework Help
Replies
2
Views
384
  • Classical Physics
Replies
17
Views
1K
Replies
14
Views
1K
Back
Top