Noether’s Theorem

Explore Noether’s Theorem in analytical mechanics, linking symmetries to conservation laws, and understand its impact on energy, momentum, and angular momentum conservation.

 

Noether’s Theorem

Noether’s theorem is a fundamental result in theoretical physics and analytical mechanics, linking symmetries of a physical system to its conserved quantities. This powerful theorem, formulated by Emmy Noether in 1915, provides deep insights into the nature of conservation laws in both classical and modern physics.

Statement of Noether’s Theorem

Noether’s theorem states that every differentiable symmetry of the action of a physical system corresponds to a conservation law. In simpler terms, if a system’s laws of motion remain unchanged (invariant) under a continuous transformation, there exists a corresponding quantity that is conserved throughout the motion.

The Action and Lagrangian Formulation

In analytical mechanics, the action $S$ is defined as the integral of the Lagrangian $L$ over time:

$$S={\int }_{t_1}^{t_2}L(q_i,{\dot{q}}_i,t)dt,$$

where $q_i$ are the generalized coordinates, $\dot{q}_i$ are the generalized velocities, and $t$ is time.

A symmetry of the action is a transformation of the coordinates and time that leaves the action invariant. Mathematically, if $\delta q_i$ and $\delta t$ represent infinitesimal changes in $q_i$ and $t$, the action is invariant if:

$$\delta S=0.$$

Types of Symmetries and Conserved Quantities

Time Translation Symmetry

If the Lagrangian does not explicitly depend on time, $t$, the system is invariant under time translations $t \rightarrow t + \epsilon$. According to Noether’s theorem, this symmetry corresponds to the conservation of energy. The conserved quantity is the Hamiltonian $H$, representing the total energy of the system:

$$H=\sum _i\left(\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}{\dot{q}}_i\right)-L\mathrm{.}$$

Spatial Translation Symmetry

If the Lagrangian is invariant under spatial translations $q_i \rightarrow q_i + \epsilon$, this symmetry corresponds to the conservation of linear momentum. For a coordinate $q_i$, the conserved momentum $p_i$ is given by:

$$p_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}\mathrm{.}$$

Rotational Symmetry

If the Lagrangian is invariant under rotations, this symmetry corresponds to the conservation of angular momentum. For a system with rotational symmetry about an axis, the conserved angular momentum $L_i$ around that axis is:

$$L_i=\sum _j\left(q_j\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_j}\right)\mathrm{.}$$

Noether’s Theorem in Hamiltonian Mechanics

While Noether’s theorem is often formulated in the context of Lagrangian mechanics, it also has an important role in Hamiltonian mechanics. In this framework, symmetries can be related to Poisson brackets. For a Hamiltonian $H$ and a conserved quantity $Q$, the symmetry implies:

$$\{Q,H\}=0.$$

This relation indicates that $Q$ is conserved if its Poisson bracket with the Hamiltonian $H$ vanishes, reflecting the invariance of the system under the associated symmetry.

Example: Harmonic Oscillator

Consider a simple harmonic oscillator with the Lagrangian:

$$L=\frac{1}{2}m{\dot{q}}^2-\frac{1}{2}k{q}^2\mathrm{.}$$

This system is invariant under time translations, leading to the conservation of energy. The Hamiltonian for this system is:

$$H=\frac{p^2}{2m}+\frac{1}{2}k{q}^2,$$

where $p = m\dot{q}$ is the momentum. The total energy $H$ remains constant over time, illustrating Noether’s theorem.

Conclusion

Noether’s theorem elegantly connects the symmetries of a physical system to its conservation laws, providing a profound understanding of the underlying principles governing motion. In analytical mechanics, this theorem is a vital tool for identifying conserved quantities, simplifying the analysis of dynamical systems, and revealing the deep connections between symmetry and conservation in both classical and modern physics.