I can't do a differentiation during the proof of Noether's theorem.

[timewalker]

In Wikipedia,

http://en.wikipedia.org/wiki/Noether's_theorem#One_independent_variable

You can see the proof of Noether's theorem for the system that has only one symmetry.
I can't do the calculation of this, for

$$\frac{dI'}{d\epsilon} = \frac{d}{d\epsilon} \int_{t_1+\epsilon T}^{t_2+\epsilon T} L(\phi(q(t'-\epsilon T),\epsilon), \frac{\partial \phi}{\partial q} (q(t'-\epsilon T), \epsilon) \dot{q} (t' - \epsilon T), t') dt'$$

it becomes

$$L(q(t_2) , \dot{q}(t_2), t_2)T - L(q(t_1),\dot{q}(t_1) , t_1)T +\int_{t_1}^{t_2} \frac{\partial L}{\partial q} \left( -\frac{\partial \phi}{\partial q} \dot{q} T + \frac{\partial \phi}{\partial \epsilon} \right) + \frac{\partial L}{\partial \dot{q}} \left( -\frac{\partial^2 \phi}{(\partial q)^2} \dot{q}^2 T + \frac{\partial^2 \phi}{\partial \epsilon \partial q} \dot{q} - \frac{\partial \phi}{\partial q} \ddot{q} T \right) dt$$

How it becomes like this?

 

[eljose79]

Thank you for bringing up this interesting topic. Noether's theorem is a fundamental principle in physics that relates symmetries of a system to conservation laws. It is a powerful tool for understanding the behavior of physical systems and has been applied in various fields such as classical mechanics, quantum mechanics, and field theory.

In the Wikipedia page you mentioned, the proof of Noether's theorem for a system with one independent variable is presented. This proof involves a mathematical calculation, which can seem daunting at first. However, with a bit of patience and understanding of the underlying concepts, it is possible to follow the derivation.

The equation you have provided is a part of the proof, and it shows the application of Noether's theorem to a Lagrangian system. The Lagrangian, denoted by L, is a function that describes the dynamics of a system in terms of its generalized coordinates, velocities, and time. In this equation, the Lagrangian at two different times, t1 and t2, is compared, and the difference is integrated over time.

The first term on the right-hand side of the equation represents the change in the Lagrangian due to the symmetry transformation, which is denoted by phi. The second term represents the change in the Lagrangian due to the variation in the independent variable, which is denoted by epsilon. The rest of the terms involve derivatives of the Lagrangian with respect to the generalized coordinates and velocities.

The derivation of this equation involves some mathematical manipulations, such as integration by parts and the use of Euler-Lagrange equations. I would suggest consulting a textbook or online resources for a step-by-step explanation of the derivation.

I hope this helps to clarify the equation and how it relates to Noether's theorem. If you have any further questions, please don't hesitate to ask. I am always happy to discuss and share knowledge about interesting topics like this.

 

[charng]

I understand that the proof of Noether's theorem can be complex and involve calculations that may be difficult to understand at first glance. However, it is important to remember that this theorem is a fundamental concept in physics and has been extensively studied and proven by many scientists. It is based on the principle of conservation of energy and connects symmetries in a system to conserved quantities. Therefore, it is crucial to carefully follow the steps of the proof and consult with experts in the field if any confusion arises.

In regards to the specific calculation presented in the Wikipedia article, it is important to note that Noether's theorem is applicable to systems with multiple symmetries, not just one. The equation you have provided is a simplified version for a system with only one symmetry, and the calculation involves integrating over a specific time interval. The specific steps and equations used in this calculation may not be immediately obvious, but they are derived from the fundamental principles of Noether's theorem.

If you are having trouble understanding the calculation, I suggest consulting with a physics professor or researcher who has expertise in this area. They can help you break down the steps and explain the reasoning behind each equation. Additionally, there are many resources available online, such as textbooks and lecture notes, that provide detailed explanations and examples of Noether's theorem. With persistence and guidance, I am confident that you will be able to grasp the concept and calculations of Noether's theorem.