What are the equations of motion for a time dependent Lagrangian?

However, when the action is varied, the time part does not get lost, but rather it is taken into account through the integration by parts step. This allows for the factorization of the dx term and ultimately leads to the integrand being zero. Therefore, the time part is not lost, but rather it is accounted for in the derivation of the equations of motion.
  • #1
cheeseits
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Let's say that L=((1/2)m*v^2-V(x))*f(t), or something similar. What are the equations of motion? For time independent it should be: (d/dt) (dL/dx_dot)=dL/dx .
Using this I get m[tex]\ddot{x}[/tex]+m f_dot/f x_dot+dV/dx=0.
Is this right? I keep thinking about the derivation of the equations and it seems like there should be a time varying term. When the action is varied, there is a term from x, x_dot, and t, right? Then integrate by parts, and factor out the dx term, argue that the integrand must be zero. Where does the time part get lost or where does it show up?

Thanks.
 
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  • #2
Yes, your equation of motion is correct. The time varying term you are referring to is the f(t) term, which is included in the equation. The time varying part comes from the fact that f(t) is itself a function of time, and thus its derivative with respect to time (f_dot/f) is non-zero. This is what gives rise to the time-varying term in the equation of motion.
 

1. What is a time dependent Lagrangian?

A time dependent Lagrangian is a mathematical function that describes the dynamics of a system over time. It takes into account the positions, velocities, and accelerations of all the particles in the system, as well as any external forces acting on them.

2. What are the equations of motion for a time dependent Lagrangian?

The equations of motion for a time dependent Lagrangian are called the Euler-Lagrange equations. They are a set of differential equations that describe how the positions and velocities of the particles in a system change over time.

3. How do the equations of motion change if the Lagrangian is time dependent?

When the Lagrangian is time dependent, the Euler-Lagrange equations become partial differential equations. This means that in addition to the positions and velocities, they also take into account the time derivative of the Lagrangian.

4. Can the equations of motion be solved analytically for a time dependent Lagrangian?

In most cases, the equations of motion for a time dependent Lagrangian cannot be solved analytically. However, they can be solved numerically using computational methods.

5. What are some real-world applications of time dependent Lagrangians?

Time dependent Lagrangians are used in many areas of physics and engineering, including classical mechanics, quantum mechanics, and control systems. They are essential for understanding the dynamics of complex systems, such as celestial bodies, particles in accelerators, and robots.

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