Space-time symmetry (Langrangian Mechanics)

In summary, when deriving the conserved quantity in the case of space-time symmetry, a line in the notes goes from \int{dt.(1+\epsilon\dot{\xi}).L[q(t+\epsilon\xi)+{\delta}q(t+\epsilon\xi)]} - \int{dt.L[q(t)+{\delta}q(t)]} to \int{\dot{\xi}L+\xi\frac{dL}{dt}+O(\epsilon^{2})} due to a Taylor expansion in \epsilon. The missing factor in the last expression should be \epsilon. The derivation involves expanding the Lagrangian L[q(t+\epsilon\xi(t))] and simplifying terms to reach the final result.
  • #1
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When deriving the conserved quantity in the case of space-time symmetry, a line in my notes goes from:

[tex]\int{dt.(1+\epsilon\dot{\xi}).L[q(t+\epsilon\xi)+{\delta}q(t+\epsilon\xi)]} - \int{dt.L[q(t)+{\delta}q(t)]}[/tex]

where L is the Lagrangian and [tex]\xi[/tex] is a function of time and both integrals are over the same time interval, to:

[tex]\int{\dot{\xi}L+\xi\frac{dL}{dt}+O(\epsilon^{2})}[/tex]

I can't see how these two lines equal one another.

How does the [tex]O(\epsilon^{2})[/tex] come about?

Thanks.
 
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  • #2
That would be the result of a Taylor expansion in [itex]\epsilon[/itex]. Also I think you're missing a factor in the last expression, and that it should be
[tex]\int \mathrm{d}t\,\epsilon\biggl(\dot{\xi}L + \xi\frac{\mathrm{d}L}{\mathrm{d}t}\biggr) + O(\epsilon^2)[/tex]
 
  • #3
thanks diazona, much appreciated:

I tried Taylor expansion as you suggested (and I did get the right answer) but I am not sure whether I have done so mathematically correctly or not. Can you check whether or not I am on the right lines...

We have [tex]\int{dt.(1+\epsilon\dot{\xi(t)}).L[q(t+\epsilon\xi(t))+{\delta}q(t+\epsilon\xi(t))]} - \int{dt.L[q(t)+{\delta}q(t)]}[/tex]

start with [tex]L[q(t+\epsilon\xi(t))][/tex]

if we first Taylor expand the bit in parenthesis (ie. [tex]q(t+\epsilon\xi(t))[/tex]) we get

[tex]q(t+\epsilon\xi(t))=q(t)+\frac{{\partial}q(t)}{{\partial}t}\epsilon\xi(t)+...[/tex]

therefore[tex]L[q(t+\epsilon\xi(t))=L[q(t)+\frac{{\partial}q(t)}{{\partial}t}\epsilon\xi(t)+...][/tex]

This is equal to:

[tex]L(q)+\frac{{\partial}L}{{\partial}q}\frac{{\partial}q(t)}{{\partial}t}\epsilon\xi(t)+...[/tex]

but

[tex]\frac{{\partial}L}{{\partial}q}\frac{{\partial}q(t)}{{\partial}t} = \frac{{\partial}L}{{\partial}t}[/tex] (CANCELLING the 'dq's)

So:

[tex]L[q(t+\epsilon\xi(t))]=L(q)+\frac{{\partial}L}{{\partial}t}\epsilon\xi(t)+smaller terms][/tex]

Thanks.
 

Related to Space-time symmetry (Langrangian Mechanics)

What is space-time symmetry in Lagrangian mechanics?

Space-time symmetry in Lagrangian mechanics refers to the principle that the laws of physics remain the same regardless of the position and orientation of an observer in space and time. This means that the laws of physics are invariant under translations in space and time, rotations in space, and reflections in space.

Why is space-time symmetry important in Lagrangian mechanics?

Space-time symmetry is important in Lagrangian mechanics because it allows us to simplify and generalize the equations of motion for a system. By assuming that the laws of physics are the same for all observers, we can use a single set of equations to describe the behavior of a system, rather than having to use different equations for each possible observer.

How is space-time symmetry related to conservation laws?

Space-time symmetry is closely related to conservation laws. In fact, Noether's theorem states that for every continuous symmetry in space and time, there is a corresponding conservation law. For example, the translation symmetry in space leads to the conservation of momentum, and the translation symmetry in time leads to the conservation of energy.

Can space-time symmetry be violated?

While the laws of physics are generally believed to be symmetric, there are certain cases where space-time symmetry can be violated. For example, in extreme conditions such as near a black hole or in the early universe, the laws of physics may no longer be symmetric. However, these violations are not yet fully understood and are an area of ongoing research.

How is space-time symmetry used in practical applications?

Space-time symmetry is used in many practical applications, particularly in the field of particle physics. The Standard Model, which describes the interactions between particles and the fundamental forces, is based on the principle of space-time symmetry. Additionally, the concept of symmetry is important in the development of new theories and models in physics.

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