Trajectory from differentiating action with respect to energy?

In summary, the conversation discusses the problem of a particle in a Coulomb field in three dimensions and the use of time-translation invariance and spherical symmetry to write the action as A = -εt + L ∅ + f(r). The form of f(r) can be obtained by substituting in Hamilton-Jacobi equations. The last step involves solving the Hamilton-Jacobi equation by differentiating the action with respect to energy and setting it equal to a constant, resulting in a trajectory solution. This method is known as the Hamilton-Jacobi method of solution and can be found in textbooks on classical or analytical mechanics, such as "Analytical Mechanics" by Carl E. Mungan.
  • #1
Sphysicist
3
0
Hi...

I was looking at the problem of a particle in a Coulomb field in three dimensions. Time-translation invariance and spherical symmetry ensures that the energy and angular momentum are conserved, allowing us to write the action as

A = -εt + L ∅ +f(r)

The form of f(r) in terms of ε and L and r can be obtained by substituting in Hamlton-Jacobi equations.

After having obtained the expression for the action, the author differentiates it with respect to ε and equating it to a constant t0 to obtain the trajectory r(θ).

I do not understand this last step. Can anyone explain or give me a reference?
 
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  • #2
The last step is a way of solving the Hamilton-Jacobi equation. It is based on the idea that, since the action is a constant along the path of the particle, if you differentiate it with respect to the energy and set it equal to a constant, then the resulting trajectory will be a solution to the equation. This method is known as the Hamilton-Jacobi method of solution. You can find more details about this method in any textbook on classical mechanics or analytical mechanics. Here is an example of a book which discusses this method: Mungan, Carl E. Analytical Mechanics. Cambridge University Press, 2007.
 

1. What is the definition of trajectory in relation to differentiating action with respect to energy?

Trajectory is the path that an object follows through space, as described by its position, velocity, and acceleration. Differentiating action with respect to energy means finding the rate of change of action (a measure of the energy required for an object to move from one position to another) with respect to the energy of the system.

2. How is the trajectory of an object affected by changes in energy?

The trajectory of an object is affected by changes in energy because energy determines the speed and direction of the object's motion. A higher energy level will result in a faster and more direct trajectory, while a lower energy level will result in a slower and more curved trajectory.

3. What is the relationship between trajectory and potential energy?

The relationship between trajectory and potential energy is that potential energy is a measure of an object's stored energy due to its position, and this energy is then converted into kinetic energy as the object follows its trajectory. In other words, the trajectory of an object is influenced by the potential energy present in the system.

4. How do you calculate the trajectory of an object using differentiating action with respect to energy?

To calculate the trajectory of an object using differentiating action with respect to energy, you will need to know the initial energy of the system and the forces acting on the object. By differentiating the action (or energy) with respect to time, you can determine the object's velocity and acceleration, which can then be used to calculate its trajectory.

5. What are some real-life applications of using trajectory from differentiating action with respect to energy?

Some real-life applications of using trajectory from differentiating action with respect to energy include predicting the path of a projectile, understanding the motion of particles in a chemical reaction, and designing efficient spacecraft trajectories. It can also be used in sports science to analyze the trajectory of a ball or in engineering to optimize the path of a moving object.

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