Action Definition: Integral of Lagrangian Over Time

  • Thread starter Thread starter o_neg
  • Start date Start date
  • Tags Tags
    Definition
AI Thread Summary
The action is defined as the integral of the Lagrangian over time because setting its variation to zero yields the equations of motion. This integral represents the measurement of energy used, specifically the difference between kinetic and potential energy. While the interpretation of the integral can be complex, it can be visualized as the path a particle takes that minimizes energy usage. A reference to the Feynman Lectures on Physics provides further clarity on this concept. Understanding this framework is essential for grasping the principles of classical mechanics.
o_neg
Messages
4
Reaction score
0
Hi,

why is the action defined as the integral of the Lagrangian over time?
i don't see the meaning of integral on the (kinetic - potential) energy

thanks,
ori.
 
Physics news on Phys.org
The integral has no meaning by itself, but setting its variation equal to zero leads to the equations of motion.
 
You can see the integral as a way of measuring how much of something is used. So in this case, the integral measures how much (K-U) energy is being used up, and the path that a particle takes minimizes this (this isn't technically correct... it takes a 'stationary' path, but looking at it this way could be beneficial to the student visually).
 
thanks for the replay.

I found a detailed explanation in the Feynman lectures on physics vol 2
 
Thread ''splain this hydrostatic paradox in tiny words'

Thread ''splain this hydrostatic paradox in tiny words'

This is (ostensibly) not a trick shot or video*. The scale was balanced before any blue water was added. 550mL of blue water was added to the left side. only 60mL of water needed to be added to the right side to re-balance the scale. Apparently, the scale will balance when the height of the two columns is equal. The left side of the scale only feels the weight of the column above the lower "tail" of the funnel (i.e. 60mL). So where does the weight of the remaining (550-60=) 490mL go...
View full post »

Thread 'The Effect of Linear and Rotational Motion on Measured Weight'

Consider an extremely long and perfectly calibrated scale. A car with a mass of 1000 kg is placed on it, and the scale registers this weight accurately. Now, suppose the car begins to move, reaching very high speeds. Neglecting air resistance and rolling friction, if the car attains, for example, a velocity of 500 km/h, will the scale still indicate a weight corresponding to 1000 kg, or will the measured value decrease as a result of the motion? In a second scenario, imagine a person with a...
View full post »

Thread 'Coulomb gauge implies instantaneous radial electric field?'

Scalar and vector potentials in Coulomb gauge Assume Coulomb gauge so that $$\nabla \cdot \mathbf{A}=0.\tag{1}$$ The scalar potential ##\phi## is described by Poisson's equation $$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}\tag{2}$$ which has the instantaneous general solution given by $$\phi(\mathbf{r},t)=\frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf{r}',t)}{|\mathbf{r}-\mathbf{r}'|}d^3r'.\tag{3}$$ In Coulomb gauge the vector potential ##\mathbf{A}## is given by...
View full post »

Similar threads

I Action in Lagrangian Mechanics
Replies
5
Views
2K
I Principle of Stationary Action - Intuition
Replies
13
Views
3K
I Why the Lagrangian is the difference of energies?
Replies
5
Views
3K
A Stationary Point of Variation of Action
Replies
16
Views
2K
I Momentum and Action: Understanding Lagrangian Mechanics
Replies
5
Views
2K
I Proving that the Lagrangian of a free particle is independent of q
Replies
1
Views
1K
I Lacking intuition with partial derivatives
Replies
18
Views
2K
I Principle of minimum action and application to real problems
Replies
10
Views
1K
A Lagrange Densities: Intuitive Understanding
Replies
1
Views
1K
I Lagrangian approach for the inclined plane
Replies
21
Views
3K

Hot Threads

Recent Insights

Back
Top