Lagrangian and Hamiltonian equations of motion

In summary, the conversation discusses using three different methods (Lagrangian, Newton, and Hamilton) to find the equations of motion for a particle acted on by a constant force over time. The equations involve kinetic and potential energy, and can be solved using differential equations.
  • #1
bnz23
3
0

Homework Statement


To try and relate the three ways of calculating motion, let's say you have a particle of some mass, completely at rest, then is acted on by some force, where F equals a constant, C, times time. (C*t).
I want to find the equations of motion using Lagrangrian, but also Newton and Hamilton

Homework Equations


I know I need L= T - V
T = 1/2mv^2, where v is x dot
This needs to be altered for the Force equation
I feel PE (V) is just mvx

The Attempt at a Solution


Then if these T and V are correct, I need to solve the DE for Lagrange. That is easy once I know my L equation is correct.
Next, Hamilton.

How would this be done with Newton's equations of motion? Simple I'm sure.

Thanks!
 
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  • #2
If you're working in one dimension, Newton's second law is just m a = C t, where a is the acceleration. Since F = -dU/dx, we can choose U = - C t x. Then L = 1/2 m v^2 + C t x, and H = 1/2 m v^2 - C t x.
 
  • #3
Thank you! That did it. All solved.
 

Related to Lagrangian and Hamiltonian equations of motion

1. What is the difference between Lagrangian and Hamiltonian equations of motion?

The Lagrangian and Hamiltonian equations of motion are two different mathematical formulations used to describe the dynamics of a physical system. The key difference between them lies in the variables that are used to describe the system. The Lagrangian formulation uses generalized coordinates, while the Hamiltonian formulation uses generalized coordinates and momenta.

2. How are the Lagrangian and Hamiltonian equations of motion related?

The Lagrangian and Hamiltonian equations of motion are related through the principle of least action. This principle states that the actual path taken by a system between two points in time is the one that minimizes the action, which is a mathematical quantity calculated from the Lagrangian. The Hamiltonian is derived from the Lagrangian and is used to describe the evolution of the system over time.

3. What physical systems can be described using Lagrangian and Hamiltonian equations of motion?

Lagrangian and Hamiltonian equations of motion can be used to describe a wide range of physical systems, including classical mechanics, quantum mechanics, and electromechanical systems. They are also commonly used in fields such as fluid dynamics, astrophysics, and control theory.

4. How are Lagrangian and Hamiltonian equations of motion used in practice?

In practice, Lagrangian and Hamiltonian equations of motion are used to solve problems in physics and engineering. They are used to study the behavior of systems and predict their future state based on initial conditions. They are also used to design control systems and optimize the performance of complex systems.

5. What are some limitations of Lagrangian and Hamiltonian equations of motion?

While Lagrangian and Hamiltonian equations of motion are powerful tools for solving problems in physics and engineering, they do have some limitations. They may not accurately describe systems with strong nonlinearities or chaotic behavior. They also do not take into account the effects of external forces such as friction and air resistance, which can sometimes be significant in real-world systems.

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