Why L=L（v^2) in inertial reference system?

In summary, the equation L = L(v^2) in an inertial reference system illustrates how the length of an object is affected by its velocity relative to the observer. As an object moves faster, its length contracts due to relativistic effects, leading to the relationship where the proper length (L) is modified by the factor of the square of its velocity (v^2). This concept aligns with the principles of special relativity, emphasizing that measurements of length are dependent on the relative motion between the observer and the object.
  • #1
Dr turtle
2
0
TL;DR Summary
Why Landau pointed out that Lagrange function shall only be affected by v square in inertial reference system?
Why he said that beacause space's propertiy is the same in both direction, so L=L(v^2), or do I misunderstand him incorrectly?
btw this conclusion appears in somewhere like page 5 and its about Galilean principle of relativity.
 
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  • #2
The direction should not matter, so only magnitude of velocity |v| should matter.
[tex]f(|v|)=f(\sqrt{v^2})=g(v^2)[/tex]
So we can say only v^2 matters.
 
  • #3
That's really helpful, lots of thanks
 

FAQ: Why L=L（v^2) in inertial reference system?

What does L=L(v^2) represent in an inertial reference system?

L=L(v^2) represents the relationship between the Lagrangian (L) of a system and the square of its velocity (v^2) in an inertial reference frame. This form is often used in classical mechanics to describe systems where the Lagrangian depends only on the kinetic energy, which is a function of the velocity squared.

Why is the Lagrangian often expressed as a function of velocity squared?

The Lagrangian is frequently expressed as a function of velocity squared because, in classical mechanics, the kinetic energy of a particle is proportional to the square of its velocity. This quadratic dependence simplifies the formulation of the equations of motion using the principle of least action.

How does the form L=L(v^2) simplify solving problems in mechanics?

The form L=L(v^2) simplifies solving problems because it directly relates to the kinetic energy, which is a scalar quantity. This allows for straightforward application of the Euler-Lagrange equations to derive the equations of motion, reducing the complexity of the calculations involved.

Can L=L(v^2) be applied to all types of physical systems?

No, L=L(v^2) cannot be applied to all types of physical systems. It is most applicable to systems where the potential energy is either absent or does not depend on velocity. For systems with velocity-dependent potentials or in relativistic contexts, the Lagrangian may take more complex forms.

What are the limitations of using L=L(v^2) in describing physical systems?

The primary limitation of using L=L(v^2) is that it assumes the potential energy is not a function of velocity, which is not true for all systems. Additionally, this form is not suitable for relativistic systems where the relationship between energy and velocity is not quadratic. For such cases, more general forms of the Lagrangian must be used.

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