Relativistic covariance in classical mechanics?

In summary: Your friend.In summary, the conversation discusses a similarity between classical mechanics and relativity in regards to the concept of action and the use of the Hamilton-Jacobi equations. The use of the variable x^0 = ct in the equations is a result of the fundamental principles of relativity. The wave-like behavior of free particles in this context is not a coincidence and is a key principle of relativity. The choice of metric is not a matter of which one is "better" but rather which one is more appropriate for the theory being used.
  • #1
Jano L.
Gold Member
1,333
75
Hi friends,

long time ago I noticed the following interesting similarity between
classical mechanics and relativity. Consider particle moving in an external field and the
action defined as a function of actual time t and position q:

[tex]
S(t,q) = \int_0^t L(q^r,\dot q^r,t´)dt´
[/tex]

The motion [tex]q^r[/tex] is the real motion of the particle which finishes at the
position q at the time t. It is possible to derive Hamilton-Jacobi
equations for S:

[tex]
\frac{\partial S}{\partial t} = -H, \frac{\partial S}{\partial x^i} = p_i.
[/tex]

Here comes the point. Let us define new variable [tex]x^0 = ct[/tex]. We see that the Hamilton Jacobi equations can be written in the compact form

[tex]
\frac{\partial S}{\partial x^\mu} = p_\mu,
[/tex]

which is relativistically covariant equation! Its solution for free particle is

[tex]
S = -Et + \mathbf{p\cdot r}
[/tex]

How can we get the right STR metric (-1,1,1,1)? We did not assume the Lorentz invariance anywhere! In STR, this equation is still valid, although L and therefore S is different. Is there some reason for this (maybe waves with phase -Et+pr are fundamental?), or is it only a random coincidence? Another interesting point is that this wave behaviour of free particle is similar to de Broglie wave hypothesis, which lead Schroedinger to his wave equation in QM.
At least, does CM say us that -1,1,1,1 is better than 1,-1,-1,-1 ? ;-)

Jano L.
 
Last edited:
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  • #2


Hello Jano,

Thank you for sharing your observation with us. It is indeed an interesting similarity between classical mechanics and relativity. The fact that the Hamilton-Jacobi equations can be written in a compact and covariant form when using the variable x^0 = ct is intriguing.

The reason for this coincidence lies in the fundamental principles of relativity. In both classical mechanics and relativity, the concept of action plays a crucial role. In classical mechanics, the action is defined as the integral of the Lagrangian over time. In relativity, the action is defined as the integral of the Lagrangian over space-time. This is why we see a similar form of the Hamilton-Jacobi equations in both cases.

Furthermore, the wave-like behavior of the free particle in your equation is not a coincidence. In fact, this is one of the key principles of relativity – the idea that all particles, including free ones, can exhibit wave-like behavior. This is what led de Broglie to his wave hypothesis and eventually to the development of quantum mechanics.

As for the choice of metric, it is not a matter of which one is "better" but rather which one is more appropriate for the theory being used. In classical mechanics, the metric is not a relevant concept, but in relativity, it is crucial for understanding the behavior of space and time.

I hope this helps to shed some light on your observation. Keep exploring and questioning, as these are important qualities of a scientist. Best of luck in your studies.
 
  • #3


Thank you for sharing your observations. It is indeed interesting to see the similarities between classical mechanics and relativity. The fact that the Hamilton-Jacobi equations can be written in a covariant form when using the variable x^0 = ct suggests that there may be a deeper connection between the two theories.

One possible explanation for this is the principle of least action, which is a fundamental principle in both classical mechanics and relativity. This principle states that the actual path taken by a particle is the one that minimizes the action, which is a measure of the energy of the system.

In classical mechanics, the action is defined as the integral of the Lagrangian over time. In relativity, the action is defined as the integral of the Lagrangian over proper time, which is related to the coordinate time t through the Lorentz transformation. This could explain why the Hamilton-Jacobi equations can be written in a covariant form when using the variable x^0 = ct.

As for the choice of metric, it is a fundamental aspect of the theory of relativity and is derived from the principle of relativity and the constancy of the speed of light. In classical mechanics, we do not have these principles, so there is no reason to assume a specific metric.

It is interesting to note the similarities between the wave behavior of a free particle in classical mechanics and the de Broglie wave hypothesis. This could suggest a deeper connection between the two theories, but further research is needed to fully understand this connection.

Overall, the observation of relativistic covariance in classical mechanics is a fascinating one and could potentially lead to further insights into the relationship between the two theories. Thank you for bringing this to our attention.
 

1. What is relativistic covariance in classical mechanics?

Relativistic covariance is the principle that the laws of classical mechanics should remain unchanged under the transformations of special relativity. This means that the laws describing the behavior of objects in classical mechanics should still hold true when observed from different reference frames moving at constant velocities.

2. Why is relativistic covariance important in classical mechanics?

Relativistic covariance is important because it allows us to describe the behavior of objects in classical mechanics in a way that is consistent with the principles of special relativity. This allows us to make accurate predictions and understand the behavior of objects moving at speeds close to the speed of light.

3. How is relativistic covariance different from Galilean covariance?

Relativistic covariance takes into account the effects of special relativity, such as time dilation and length contraction, while Galilean covariance does not. Relativistic covariance also considers the speed of light to be the same in all reference frames, while Galilean covariance assumes that the speed of light is infinite.

4. Can relativistic covariance be applied to all laws of classical mechanics?

Yes, relativistic covariance can be applied to all laws of classical mechanics. However, it is most commonly applied to Newton's laws of motion and the principle of conservation of energy and momentum.

5. How does relativistic covariance affect the equations of motion in classical mechanics?

Relativistic covariance results in slight modifications to the equations of motion in classical mechanics. For example, the famous equation E=mc^2 is a result of relativistic covariance applied to the conservation of energy and momentum. Additionally, the equations for time, velocity, and acceleration may also be slightly modified to account for the effects of special relativity.

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