Time dependent lagrangian

In summary, the equations of motion that derive from the Lagrangian M will correctly describe the motion of the constrained system.
  • #1
Petr Mugver
279
0
Suppose I have a mechanical system with l + m degrees of freedom and an associated lagrangian

[itex]L(\alpha,\beta,\dot{\alpha},\dot{\beta},t)[/itex]

where [itex]\alpha\in\mathbb{R}^l[/itex] and [itex]\beta\in\mathbb{R}^m[/itex].
Now suppose I have a known [itex]\mathbb{R}^l[/itex]-valued function f(t) and define a new lagrangian

[itex]M(\beta,\dot{\beta},t)=L(f(t),\beta,\dot{f}(t), \dot{\beta},t)[/itex]

Do the equations that derive from M correctly describe the motion of the initial mechanical system, where the first l degrees of freedom are constrained to the motion f(t) (by means of an external force)?
 
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  • #2
[tex]\frac{\partial L}{\partial q} = \frac{d}{dt} \frac{\partial L}{\partial \dot{q} } [/tex]
where q is anyone of the coordinates of [itex]\alpha[/itex] or [itex]\beta[/itex] (i.e. there are l+m separate equations).
From this, I would assume that [itex]M(\beta , \dot{\beta} , t) [/itex] would be correct, because this simply gives the m equations, which correspond to [itex]\beta[/itex].
(I know I've not done a rigorous proof or anything, but this seems to make sense to me).
 
  • #3
BruceW said:
[tex]\frac{\partial L}{\partial q} = \frac{d}{dt} \frac{\partial L}{\partial \dot{q} } [/tex]
where q is anyone of the coordinates of [itex]\alpha[/itex] or [itex]\beta[/itex] (i.e. there are l+m separate equations).
From this, I would assume that [itex]M(\beta , \dot{\beta} , t) [/itex] would be correct, because this simply gives the m equations, which correspond to [itex]\beta[/itex].
(I know I've not done a rigorous proof or anything, but this seems to make sense to me).

Uhm, please don't let me write the formula, but when you take the derivative with respect to t of the momentum dM/dv, don't you get extra terms due to f(t) and df(t)/dt?
 
  • #4
Do the equations that derive from M correctly describe the motion of the initial mechanical system, where the first l degrees of freedom are constrained to the motion f(t) (by means of an external force)?
Yes, this is correct. In fact it's done all the time: write down T and V as if the particles were entirely free, and then impose the constraints on them. The Lagrangian M you get from this will describe the motion of the constrained system.

What you cannot do is the other way around: trying to substitute f(t) into the equations of motion you originally derived from L.
 
  • #5
Bill_K said:
Yes, this is correct. In fact it's done all the time: write down T and V as if the particles were entirely free, and then impose the constraints on them. The Lagrangian M you get from this will describe the motion of the constrained system.

What you cannot do is the other way around: trying to substitute f(t) into the equations of motion you originally derived from L.

Yes, it sounds so obvious. I feel stupid now... :)
 

1. What is a time dependent Lagrangian?

A time dependent Lagrangian is a mathematical function used in classical mechanics to describe the motion of a system over time. It takes into account the kinetic and potential energies of the system, as well as any external forces acting on it.

2. How is a time dependent Lagrangian different from a time independent Lagrangian?

A time dependent Lagrangian takes into account the changing nature of a system over time, while a time independent Lagrangian assumes that the system's properties remain constant. This makes the time dependent Lagrangian more suitable for describing dynamic systems.

3. What is the significance of the time dependent Lagrangian in physics?

The time dependent Lagrangian is a fundamental concept in classical mechanics and is used to derive the equations of motion for a system. It allows us to analyze the behavior of a system over time and make predictions about its future motion.

4. How is the time dependent Lagrangian used in practical applications?

The time dependent Lagrangian is used in a variety of practical applications, such as in engineering and physics simulations. It is also used in the study of celestial mechanics and in the development of control systems for machines and robots.

5. Are there any limitations to using a time dependent Lagrangian?

One limitation of using a time dependent Lagrangian is that it assumes the system is conservative, meaning that the total energy of the system remains constant. This may not be applicable to all systems, especially those that experience non-conservative forces, such as friction.

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