Noether's theorem

Noether's theorem is a fundamental principle in physics that connects symmetries and conservation laws. It is named after German mathematician Emmy Noether, who first expressed this theorem in terms of the Euler–Lagrange equations in the Lagrangian framework of classical mechanics. This theorem has profound implications not only in classical mechanics but also in quantum mechanics and general relativity.

The main idea of Noether's theorem is simple but powerful: whenever a system has a continuous symmetry, there is an associated quantity that is conserved. This relation bridges the gap between abstract mathematical symmetries and physical observables, such as the conservation of energy, momentum, and angular momentum.

Understanding Lagrangian mechanics

To fully understand Noether's theorem, we must first understand the basics of Lagrangian mechanics. In this framework, the momentum of a system is derived from a function called the Lagrangian, usually denoted by L The Lagrangian is a function of the generalized coordinates q_i, the generalized velocity dot{q}_i, and possibly time t :

        L = L(q_i, dot{q}_i, t)
    

The principle of least action states that the path taken by the system is the one that minimizes the action S, which is defined as the integral of the Lagrangian over time:

        S = int L(q_i, dot{q}_i, t) , dt
    

The equations of motion are obtained by applying the Euler–Lagrange equations :

        frac{d}{dt} left( frac{partial L}{partial dot{q}_i} right) - frac{partial L}{partial q_i} = 0
    

Symmetries and conservation laws

Symmetry in physics refers to transformations that leave some properties of a system unchanged. For example, if changing the origin of time or space has no effect on a physical system, then it has time or space translation symmetry.

When a system exhibits such a symmetry, Noether's theorem asserts that there is a corresponding conserved quantity. Let's consider some examples of common symmetries:

• Time translation symmetry: if the Lagrangian does not depend explicitly on time, then the energy of the system is conserved.
• Space translation symmetry: if the system is invariant under spatial translation, then linear momentum is conserved.
• Rotational symmetry: If the system remains unchanged under rotation, then angular momentum is conserved.

Examples: free radicals

Consider a free particle moving in one dimension. The Lagrangian is given as follows:

        l = frac{1}{2} m dot{x}^2
    

Here, the Lagrangian does not depend explicitly on t, which reflects time translation symmetry. According to Noether's theorem, energy is conserved.

Similarly, since L depends only on dot{x}, not on x, the space is also a translation symmetry, so momentum is conserved. Here is an intuitive demonstration of how this theorem works.

Illustration of Noether's theorem

Let us consider a rotating system to understand the concept of rotational symmetry and conservation of angular momentum.

Suppose a particle is moving about an axis in a force field. If the field and motion are symmetric about the axis, then the angular momentum of the particle is conserved.

The math behind Noether's theorem

Let's take a deeper look at the mathematical formalism of Noether's theorem. Suppose that the Lagrangian L is invariant under a continuous transformation of this form:

        q_i rightarrow q_i + epsilon f_i(q, dot{q}, t)
    

where epsilon is a small parameter. Noether's theorem states that there exists a conserved quantity J such that:

        J = sum_i left( frac{partial L}{partial dot{q}_i} f_i(q, dot{q}, t) right) - F
    

Here, the function F is derived as part of the symmetry transformation. Its role is to adjust the form of J to ensure conservation.

Example: pendulum

Consider a simple pendulum making small oscillations. Its angle from the vertical is the coordinate theta. The Lagrangian is:

        L = frac{1}{2} ml^2 dot{theta}^2 - mgl(1 - costheta)
    

For small oscillations, costheta approx 1 - frac{1}{2} theta^2, simplifying the Lagrangian:

        L = frac{1}{2} ML^2 dot{theta}^2 - frac{1}{2} MGL theta^2
    

Since the Lagrangian shows no explicit dependence on time t, energy conservation follows from Noether's theorem.

Noether's theorem in quantum mechanics and relativity

Although our discussion has been focused on classical mechanics, Noether's theorem is equally important in more advanced theories such as quantum mechanics and general relativity.

In quantum mechanics, symmetries play a central role in determining the rules that govern systems. Here, Noether's theorem provides a basis for deriving conservation rules in the presence of symmetries in the wave function formulation.

In general relativity, the symmetries of spacetime lead to conservation laws. For example, diffeomorphism invariance leads to conservation of the energy–momentum tensor, which is a cornerstone of the theory.

Conclusion

Noether's theorem is one of the most beautiful and important bridges between mathematics and physics. By linking symmetries to conservation laws, it provides a unified explanation of why certain quantities remain unchanged over time. As theories of physics evolve, Noether's insights continue to inspire and shape our understanding of the universe.

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