Find a trajectory such that the action is a minimum
A particle is subjected to the potential V (x) = −F x, where F is a constant. The particle travels from x = 0 to x = a in a time interval t0 . Assume the motion of the particle can be expressed in the form
. Find the values of A, B, and C such that the action is a minimum.
I was thinking it can solved using Lagrangian rather than Hamilton. There's no frictional force.
Differentiate
For finding B I was thinking to integrate
initial position is 0 so, not writing constant.
Differentiate
Again, going to integrate
initial velocity and initial position is 0.
According to my, I think that C is the minimum (I think B is cause, B is negative; negative is less than positive). And, A is maximum.
A person were saying that It asked you to minimise the action; it told you the particle moved from $0$ to $a$ in time $t_0$; it gave you the equation of the trajectory.
In my work where should I put the interval?
1 answer
The Euler-lagrangian equation gives the equations of motion that once solved give you a family of solutions that minimize the action. A unique solution is given by specifying boundary conditions. It is just a case of inputing those boundary conditions.
Wlog let
and . Integrating gives the general solution , fixing C. Subbing in gives and subbing gives as . ~ https://physics.stackexchange.com/a/664865/313317
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