Classical Path using Lagrangian and EOM

In summary, the classical path satisfying the given conditions and time duration can be represented by the equation ##\bar{x}(t) = x_b\frac{\sin\omega (t-t_a)}{\sin\omega T} + x_a\frac{\sin\omega (t_b-t)}{\sin\omega T}##. The process of solving this differential equation may involve using trigonometric identities to rearrange the solution and finding the constants. As long as ##\omega T \neq \pi n##, the linear combination of ##\sin(\omega t)## and ##\cos(\omega t)## will also be a general solution. Further explanation and examples can be found online.
  • #1
spacetimedude
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Homework Statement


Show that the classical path satisfying ##\bar{x}(t_a) = x_a##, ##\bar{x}(t_b) = x_b## and ##T = t_b-t_a## is
$$\bar{x}(t) = x_b\frac{\sin\omega (t-t_a)}{\sin\omega T} + x_a\frac{\sin\omega (t_b-t)}{\sin\omega T}$$

Homework Equations


The Lagrangian: ##L = \frac{1}{2}m(\dot{x}^2-\omega^2x^2)##
The EOM: ##\ddot{\bar{x}}+\omega^2\bar{x}=0##

The Attempt at a Solution


The initial step to this problem is confusing me. I have only been exposed to SHM problems in which the solution to its differential equation is ##x(t) = A\cos(\omega t) + B\sin(\omega t)##. But in this particular question, the solution starts with
##\bar{x}(t) = A\sin\omega (t-t_a) + B\sin\omega (t-t_b)##.

My original thought was that it had something to do with the trig identities and rearrangements, but I could not get to the solution.
It would be great if someone can lead me to the process of solving this differential equation or linking me to a site with an explanation. I have only taken a course in differential equation that was integrated into my physics course and have not taken a standalone maths course.
 
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  • #2
You can start with your approach too. However, once you have solved for your constants, you should be able to rearrange it to the same form using trigonometric identities. The thing is that both ##\sin(\omega(t-t_a))## and ##\sin(\omega(t-t_b))## are linear combinations of ##\sin(\omega t)## and ##\cos(\omega t)## and so a linear combination of them is a linear combination of ##\sin(\omega t)## and ##\cos(\omega t)## as well. As long as ##\omega T \neq \pi n##, the linear combination is also general.
 
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Related to Classical Path using Lagrangian and EOM

What is the concept of Lagrangian in classical mechanics?

The Lagrangian is a mathematical function that describes the dynamics of a system in classical mechanics. It is defined as the difference between the kinetic energy and potential energy of the system. It is used to derive the equations of motion for a system in terms of generalized coordinates and their derivatives.

What is the advantage of using Lagrangian in classical mechanics?

The use of Lagrangian in classical mechanics has many advantages. It simplifies the equations of motion by reducing the number of variables and making them independent of the chosen coordinate system. It also provides a more elegant and general approach to solving problems in classical mechanics.

What are the equations of motion derived from Lagrangian?

The equations of motion derived from Lagrangian are known as the Euler-Lagrange equations. They describe the time evolution of a system in terms of the generalized coordinates, their derivatives, and the Lagrangian function. They are given by d/dt (∂L/∂q̇i) - ∂L/∂qi = 0, where qi are the generalized coordinates and L is the Lagrangian function.

What is the principle of least action in classical mechanics?

The principle of least action, also known as Hamilton's principle, states that the true path of a system is the one that minimizes the action integral, which is the integral of the Lagrangian over time. This principle is a fundamental concept in classical mechanics and is used to derive the equations of motion for a system.

What is the relation between Lagrangian and Hamiltonian in classical mechanics?

The Hamiltonian is another mathematical function that describes the dynamics of a system in classical mechanics. It is defined as the sum of the kinetic energy and potential energy of the system. The Hamiltonian is related to the Lagrangian through a mathematical transformation, known as the Legendre transformation. The Hamiltonian is often used in problems involving energy conservation and in quantum mechanics.

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