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# Noethers theoram theoritical approach

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PART II.2 - Modern Physics
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# Noethers theoram theoritical approach

this theory discuss about the symmetry of a system and changes related to this.

this theory discuss about the symmetry of a system and changes related to this.

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### Noethers theoram theoritical approach

1. 1. BIRLA INSTITUTE OF TECHNOLOGY Submission of assignment of the subject Advance Electrodynamics [SAP3007] on the topic Noether’s Theorem and Its Consequences To - Dr Saurabh Lahiri Department of physics 30th September 2021 By – Shivam Parasar IPH/10033/17 Aditya Narayan Singh IPH/10034/17
2. 2. INTRODUCTION Noether’s Theorem If the physical system behaves the same no matter how it is oriented in space, its Lagrangian is symmetric under continuous rotation. According to this symmetry, Noord's theorem shows that the angular momentum of the system is conserved. This is the result of the law. The physical system itself does not have to be symmetric. Zigzag asteroids that fall into space maintain angular momentum even if they are asymmetric. Basically, the laws of motion are symmetric. To give another example, if a physical process shows the same result regardless of location or time, its Lagrangian quantity is symmetric under a continuous transformation of space and time: according to Noether's theorem, these symmetries explain the law of conservation of momentum. The linearity and energy of this system. Noether's theorem is important because it provides information about conservation law and is a practical calculus tool. This allows researchers to determine the conserved quantity (invariant) from the symmetry of the observed physical system. Instead, researchers can use the given invariants to examine the entire virtual Lagrangian category to describe the physical system. Noether's theorem can be expressed as follows. If the system has continuous symmetry, there is a corresponding number of values stored in time. More accurate version: Conserved currents correspond to all differentiable symmetries produced by local interactions. The term "symmetry" in the previous statement more accurately refers to the form of covariance adopted by the laws of physics for a series of one-dimensional Lie group of
3. 3. transformations that meet certain technical standards. The conservation law of physical quantities is generally expressed as a continuity equation. The formal proof of the theorem uses invariants to derive the equation for the current associated with the conserved physical quantity. The conservation law says that in the mathematical description of the evolution of a system, the value of X remains constant throughout its entire motion - it is irreversible. Mathematically, the rate of change of X (time derivative) is zero. 𝑑𝑥 𝑑𝑡 = 𝑥̇ = 0 It is said that this amount is preserved. It is often said that they are an integral part of the movement (it is not necessary to include the movement itself, but only the evolution of time). For example, if the energy of a system is conserved, that energy is always irreversible, which can impose constraints on the behavior of the system and help solve the system. These constant movements not only provide insight into the nature of the system, but are also useful computational tools. For example, you can change the approximate solution by finding the closest position that satisfies the corresponding conservation law. From the late eighteenth to the early nineteenth century, physicists developed more systematic methods for discovering inventions. In 1788, great progress was made in the development of Lagrangian mechanics associated with the principle of least action. With this approach, the state of the system can be described in any number of generalized coordinates q. As is customary in Newtonian mechanics, there is no need to represent the laws of motion in Cartesian coordinate systems. The function is defined as the integral I multiplied by a function known as the Lagrangian L. 𝐼 = ∫ 𝐿(𝑞, 𝑞̇, 𝑡)𝑑𝑡 Here, the coordinates of the point at q represent the rate of change of q. 𝑞̇ = 𝑑𝑞 𝑑𝑡 Hamilton's principle states that a physical path q(t) (the path actually used by the system) is a path in which small changes in this path do not change I, at least until the first order. This principle gives rise to the Euler-Lagrange equation. 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑞̇ ) = 𝜕𝐿 𝜕𝑞 Thus, if one of the coordinates, for example qk, does not appear in the Lagrangian, then the right side of the equation is zero, and the left side needs it. 𝑑 𝑑𝑡 ( 𝜕𝐿 𝜕𝑞̇𝑘 ) = 𝑑𝑝̇𝑘 𝑑𝑡 = 0
4. 4. Momentum defined as 𝑝𝑘 = 𝜕𝐿 𝜕𝑞𝑘 ̇ speed is maintained means momentum is conserved (on the physical path). Therefore, the absence of a negligibly small coordinate qk in the Lagrangian means that the Lagrangian is not affected by changes or transformations of qk. Lagrangian are invariant and are said to have symmetry under such transformations. This is the original idea, generalized by Noether's theorem. We can directly make a statement that Noether’s theorem gives the relation ship between symmetry and conservation laws. It is very essential to know that by which transformation we get what physical quantity conservation. And this theorem can help up. Now we would dig little deep to know about symmetries. Symmetry Physical symmetry is generalized to be invariant or immutable with respect to all kinds of transformations, such as arbitrary coordinate transformations. This concept has become one of the most powerful tools in theoretical physics, and virtually all laws of nature follow from symmetry. Indeed, this role should be ascribed to Nobel laureate P. V. Anderson in his widely read 1972 paper "There Is More Other": "It would not be an exaggeration to say that physics is the study of symmetry." said. Noether's theorem (in a very simplified form states that for all continuous mathematical symmetries there are corresponding conserved quantities such as energy and momentum, and in Noether's native language there are conserved growth currents). Wigner also believes that the symmetry of the laws of physics determines the properties of particles found in nature. Important symmetries in physics include continuous space-time symmetry and discrete symmetry. The internal symmetry of particles, super symmetry of the theory of physics. Continuous symmetry is an intuitive idea that fits with the idea of some symmetry as motion rather than discrete symmetry. Reflection symmetry, which is irreversible from state to state with one type of inversion. However, discrete symmetry can always be rethought as a subset of higher dimensional continuous symmetry. The reflectance of a 2D object in 3D space can be obtained by continuously rotating the object 180 degrees in a non-parallel plane. Discrete symmetry is symmetry that represents discontinuous changes in the system. For example, a square has discrete rotation symmetry because only rotation through an angle that is a multiple of a straight line retains the original square shape. Discrete symmetry can include a kind of "permutation". These swaps are commonly referred to as reflections or swaps. In mathematics and theoretical physics, discrete symmetry is a symmetry about the transformation of discrete groups. A group of topologies with a discrete topology, the elements of which form a finite or countable set. In supersymmetric theory, the equation of force and the equation of matter are the same. In theoretical and mathematical physics, theories with this property are called supersymmetry
5. 5. theory. There are dozens of supersymmetric theories. Supersymmetry is the space-time symmetry between the two main classes of particles. Bosons have integer spins and obey boson-Einstein statistics. Fermions have half-integer spins and obey the Fermi - Dirac statistics. In supersymmetry, every particle of one class has a particle attached to another class, called its super partner, whose spins differ by half of the integers. For example, when an electron is present in supersymmetric theory, there is a particle called an "electron" (super partner electron) that is the bosonic partner of the electron. In the simplest supersymmetry theory, which has a completely "monolithic" supersymmetry, each pair of supersymmetric particles has the same mass and internal quantum number, with the exception of spin. More sophisticated supersymmetry theories can spontaneously break symmetry, and supersymmetric particles have different masses. Supersymmetry includes quantum mechanics, statistical mechanics, quantum field theory, condensed matter physics, nuclear physics, optics, stochastic mechanics, particle physics, astronomical physics, quantum gravity, string theory, cosmology, etc. It is used in different ways in different areas of physics. Supersymmetry is applicable not only to physics, such as finance. Continuous Symmetries and Conservation Law A conservation law states a particular measurable property of an isolated physical system does not change as the system evolves over time. If we talk over exact conservation laws then we have conservation of mass, linear momentum, angular momentum, electric charge, etc. And again, if we talk about approximate conservation then we have conservation law for mass parity, lepton number, baryon number, hypercharge. Also, one other conservation (local conservation) states that the amount of the conserved quantity at a point or within a volume can only be changed by the amount of quantity that flows in or out of the volume. More definition of Noether’s theorem: there is one to one correspondence between each one of them and the differentiable symmetry of nature. Derivation: Let A be invariant under a set of transformation. 𝑞(𝑡) → 𝑞′(𝑡) ⇒ 𝑓(𝑞(𝑡), 𝑞′(𝑡)) { 𝑓(𝑞(𝑡), 𝑞′(𝑡))𝑖𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑞(𝑡) 𝑞(𝑡)𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡, 𝑞 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑚𝑜𝑡𝑖𝑜𝑛 Perform symmetry again and again then it is again symmetry and in general, called symmetric group. So 𝛿𝑠𝑞(𝑡) = 𝑞′(𝑡) − 𝑞(𝑡) → 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑖𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑛𝑑 𝑓𝑖𝑛𝑎𝑙 𝑠𝑦𝑠𝑡𝑒𝑚 𝛿𝑠 → 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 𝑣𝑎𝑟𝑖𝑎𝑖𝑜𝑛 Equation 1
6. 6. 𝛿𝑠𝑞(𝑡) = ∈ ∆(𝑞(𝑡) , 𝑞′(𝑡), 𝑡) → 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑚 We must note that symmetry variations are non-zero at the ends unlike the euler-lagrange equation. using chain rule differential and integration by parts we have, 𝛿𝑠𝐴 = ∫ 𝑑𝑡 [ 𝜕𝐿 𝜕𝑞(𝑡) − 𝜕𝑡 𝜕𝐿 𝜕𝑞′(𝑡) ] 𝛿𝑠𝑞(𝑡) + 𝜕𝐿 𝜕𝑞′(𝑡) 𝛿𝑠𝑞(𝑡) 𝑡𝑏 𝑡𝑎 |𝑡𝑎 𝑡𝑏 For the path q(t) that satisfy Euler-lagrange equation the first term gets vanish as it is equal to zero. So, we are left with 𝛿𝑠𝐴 = ∈ 𝜕𝐿 𝜕𝑞′ ∆(𝑞, 𝑞′ , 𝑡)|𝑡𝑏 𝑡𝑎 Sins according to symmetry group assumption 𝛿𝑠𝐴 can vanish anytime 𝑄(𝑡) = 𝜕𝐿 𝜕𝑞′ ∆(𝑞, 𝑞′ , 𝑡) Q(t) Is independent of time t so we can write it Q(t) = Q so, it is a conserved quantity, also called a constant of motion and 𝜕𝐿 𝜕𝑞′ ∆(𝑞, 𝑞′ , 𝑡) is Noether’s charge. If we generalize equation 5 for which the action is directly not invariant but its symmetry variation is equal to an arbitrary boundary condition. So, 𝛿𝑠𝐴 = ∈∧ (𝑞, 𝑞̇, 𝑡)|𝑡𝑎 𝑡𝑏 Using equation 5 𝑄(𝑡) = 𝜕𝐿 𝜕𝑞′ ∆(𝑞, 𝑞′ , 𝑡) − ∧ (𝑞, 𝑞′ , 𝑡) { 𝑄(𝑡) → 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 𝜕𝐿 𝜕𝑞′ ∆(𝑞, 𝑞′ , 𝑡) → 𝑁𝑜𝑒𝑡ℎ𝑒𝑟′ 𝑠 𝐶ℎ𝑎𝑟𝑔𝑒 Now if we consider the action considering Lagrangian to symmetry variation can be stated as follows, 𝛿𝑠𝐿̇ = 𝐿(𝑞 + 𝛿𝑠𝑞 , 𝑞̇ + 𝛿𝑠𝑞̇) − 𝐿̇(𝑞 − 𝑞̇) 𝛿𝑠𝐿̇ = [ 𝜕𝐿 𝜕𝑞(𝑡) − 𝜕𝑡 𝜕𝐿 𝜕𝑞̇(𝑡) ] 𝛿𝑠𝑞̇(𝑡) + 𝑑 𝑑𝑡 [ 𝜕𝐿 𝜕𝑞(𝑡) 𝛿𝑠𝑞(𝑡)] Equation 2 Equation 3 Equation 4 Equation 5 Equation 6 Equation 7 Equation 8 Equation 9
7. 7. Again, the Euler-lagrange term vanishes. So, the assumption of invariance of action in equation 7 is equivalent to assuming that the symmetry variation of Lagrangian is the total time derivative of some function ∧ (𝑞, 𝑞̇, 𝑡). 𝛿𝑠𝐿(𝑞, 𝑞̇, 𝑡) = ∈ 𝑑 𝑑𝑡 ∧ (𝑞, 𝑞̇, 𝑡) So, using equation five again we can write equation 10 as, ∈ 𝑑 𝑑𝑡 [ 𝑑𝐿 𝑑𝑞̇ ∆(𝑞, 𝑞̇, 𝑡) −∧ (𝑞, 𝑞̇, 𝑡)] = 0 And hence we again recovered Noether’s Charge. The existence of a conserved quantity for every continuous symmetry is the content of Noether’s theorem. Displacement and Energy Conservation Consider the case that Lagrangian does not explicitly depend on the time therefore, we can write t’ = t - E (time translation equation) Also, the above statement can be written as 𝐿(𝑞, 𝑞̇, 𝑡) = 𝐿(𝑞, 𝑞̇) Sue for the same part say q(t) we can write 𝑞̇(𝑡′) = 𝑞(𝑡) 𝑞̇(𝑡′) 𝑤𝑖𝑡ℎ 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑑 𝑡𝑖𝑚𝑒 𝑖𝑠 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡, 𝑞(𝑡) Using symmetry variation described in this section we say, 𝛿𝑠𝑞(𝑡) = 𝑞′(𝑡) − 𝑞(𝑡) = 𝑞(𝑡′ + 𝑡) − 𝑞(𝑡) = 𝑞(𝑡′)− ∈ 𝑞̇(𝑡′) − 𝑞(𝑡) = ∈ 𝑞̇(𝑡) So, symmetry variation of Lagrangian is 𝛿𝑠𝐿 = 𝐿(𝑞′(𝑡), 𝑞̇′(𝑡)) − 𝐿(𝑞(𝑡) − 𝑞̇(𝑡)) = 𝜕𝐿 𝜕𝑞 𝛿𝑠𝑞(𝑡) + 𝜕𝐿 𝜕𝑞̇ 𝛿𝑠𝑞̇(𝑡) Inserting equation 3 in equation 4 the value of 𝛿𝑠𝑞(𝑡) Equation 10 Equation 11 Equation 1 Equation 2 Equation 3 Equation 4
8. 8. 𝛿𝑠𝐿 = ∈ ( 𝜕𝐿 𝜕𝑞̇ 𝑞̇ + 𝜕𝐿 𝜕𝑞̇ 𝑞̈) = ∈ 𝑑𝐿 𝑑𝑡 this equation is of the form like equation 10 of the previous section with 𝛬 = 𝐿 as we know that time translation is symmetry transformation. so 𝛬 in equation 10 of the previous section concide with Lagrangian. Also, we have Noether’s charge equation as 𝑄 = 𝜕𝐿 𝜕𝑞̇ 𝑞̇ − 𝐿(𝑞, 𝑞̇) → 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛 Let the time-dependent symmetry variation is 𝛿𝑠 𝑡 𝑞(𝑡) = ∈ (𝑡)𝑞̇(𝑡) So corresponding Lagrangian charge is given as 𝛿𝑠 𝑡 𝐿 = 𝜕𝐿 𝜕𝑞 ∈ 𝑞̇ + 𝜕𝐿 𝜕𝑞̇ (∈ 𝑞+∈ 𝑞̈) ̇ = 𝜕𝐿𝑡 𝜕𝑡̇ ∈ + 𝜕𝐿𝑡 𝜕𝑡̇ ∈̇ Where, 𝜕𝐿𝑡 𝜕𝑡̇ = 𝜕𝐿 𝜕𝑞̇ 𝑞̇ 𝜕𝐿𝑡 𝜕𝑡̇ = 𝜕𝐿 𝜕𝑞 𝑞̇ + 𝜕𝐿 𝜕𝑞̇ ∈ 𝑞̈ = 𝑑𝐿 𝑑𝑡 The Noether’s charge coincide with hamiltonian of system (Means time translation fulfil the symmetry condition) Equation 5 Equation 6