Question

The Lagrangian $$\mathcal{L}=\frac{1}{2} \dot{y}^{2}+\frac{1}{2} y^{-2}$$ corresponds to the equation of motion \(\ddot{y}+y^{-3}=0\). This equation of motion has three infinitesimal generators: $$\begin{array}{c}U_{1}=\partial_{t} \\\U_{2}=2 t \partial_{t}+y \partial_{y} \\\ U_{3}=t^{2} \partial_{t}+t y \partial_{y}\end{array}$$ Use Noether's theorem to find the invariants that correspond to each of these infinitesimal generators.

Short answer

The invariants are: 1. Hamiltonian \(E\) for \(U_1\), 2. \(tE - \frac{1}{2} y \dot{y}\) for \(U_2\), 3. \(t^2 E - t y \dot{y} + \frac{1}{2} y^2\) for \(U_3\).

Step-by-step solution

Understand Noether's Theorem

Noether's theorem states that every differentiable symmetry of the action of a physical system corresponds to a conservation law. To apply this theorem, we need to find the conserved quantities (invariants) corresponding to the given infinitesimal generators of symmetries.

Identify the Action Integral

The action integral for the given Lagrangian is defined as \(S = \int \mathcal{L} \, dt = \int \left(\frac{1}{2} \dot{y}^{2} + \frac{1}{2} y^{-2}\right) \, dt.\)The goal is to find transformations that leave this action invariant under changes in the generalized coordinates and time.

Use the First Infinitesimal Generator (U_1)

The first infinitesimal generator is \(U_1 = \partial_t\), which suggests a symmetry under translation in time. Applying Noether's theorem, the conserved quantity associated with time translation symmetry is the Hamiltonian, \(E = \frac{1}{2} \dot{y}^{2} - \frac{1}{2} y^{-2}.\)

Apply the Second Infinitesimal Generator (U_2)

The second infinitesimal generator is \(U_2 = 2t \partial_t + y \partial_y\), indicating a scaling symmetry. The invariant associated with this symmetry is found by substituting into Noether's theorem, resulting in a conserved quantity:\(I_2 = t E - \frac{1}{2} y \dot{y}.\)This represents a dilation in time and space.

Evaluate the Third Infinitesimal Generator (U_3)

The third infinitesimal generator is \(U_3 = t^2 \partial_t + ty \partial_y\), which describes a more complex symmetry combining time translation and scaling. Applying Noether's theorem, the conserved quantity becomes:\(I_3 = t^2 E - t y \dot{y} + \frac{1}{2} y^2.\)This is related to a more advanced symmetry in the system combining temporal and spatial transformations.

Key concepts

Infinitesimal Generators

Infinitesimal generators are mathematical tools used to explore symmetries in continuous transformational systems. They help in understanding how a given system behaves when small changes are applied.

• Each infinitesimal generator corresponds to a symmetry of the system's Lagrangian.
• These generators manifest as operators, which, when applied to variables in a system, show how they influence transformations.

For example, in the given exercise, the infinitesimal generators are:

• \(U_1 = \partial_t\) - This represents time translation symmetry, meaning that the system remains unchanged if we slightly change the time coordinate.
• \(U_2 = 2t \partial_t + y \partial_y\) - This demonstrates a scaling symmetry, referring to the fact that if we scale time and the space variable, the system's properties do not alter.
• \(U_3 = t^2 \partial_t + ty \partial_y\) - Exhibiting a more complex interaction between time and space scaling, suggesting a profound symmetry in the system.

Each of these generators helps identify different conservation laws through Noether's theorem.

Lagrangian Mechanics

Lagrangian mechanics provides a framework to analyze the dynamics of a system. It simplifies understanding of complex systems by focusing on energy rather than forces.

• The Lagrangian \(\mathcal{L}\) of a system expresses the difference between kinetic and potential energy.
• This approach emphasizes finding equations of motion from the principle of least action—minimizing the action integral, which is the integral of the Lagrangian over time.

In this exercise:

• The given Lagrangian is \(\mathcal{L} = \frac{1}{2} \dot{y}^2 + \frac{1}{2} y^{-2}\), combining terms for kinetic energy and potential energy-like terms.
• The equation of motion derived from this is \(\ddot{y} + y^{-3} = 0\), showcasing how Lagrangian mechanics leads to differential equations that predict system behavior.

By utilizing the symmetries and conserved quantities predicted by Noether's theorem, we further simplify analyzing these equations.

Conserved Quantities

Conserved quantities are properties of a system that remain constant over time, providing valuable insights into the system's behavior and integrity.

• Noether's theorem links symmetries of a system to its conserved quantities, which include energy, momentum, and angular momentum, among others.
• Identifying these conserved quantities helps in solving the equations of motion since they reduce complexity and often lead to first integrals of motion.

In the context of this exercise:

• The first conserved quantity corresponds to the time translation symmetry \(U_1\), known as the Hamiltonian or total energy, \(E = \frac{1}{2} \dot{y}^2 - \frac{1}{2} y^{-2}\).
• The second conserved quantity emerges from the scaling symmetry \(U_2\), leading to \(I_2 = t E - \frac{1}{2} y \dot{y}\), which mixes time-energy scaling.
• The third conserved quantity \(I_3 = t^2 E - t y \dot{y} + \frac{1}{2} y^2\) is associated with the symmetry given by \(U_3\), highlighting an intricate relationship between time and spatial transformations.

These conserved quantities, discovered through Noether's theorem, are essential for simplifying and solving the overall system dynamics.