Conservation laws

Abstract quantities:

Conservation of energy
- arises from time-independence of quantum physical processes
Conservation of linear momentum
- arises from location-independence of quantum physical processes
Conservation of angular momentum
- arises from orientation-independence of quantum physical processes

Noether's theorem states that every conservation law corresponds to an underlying symmetry in the system.

Other conservation laws concern countable quantities:

conservation of charge
conservation of baryons
conservation of leptons

Conservation laws and symmetry

Lagrangian mechanics and the least action principle allows us to cleanly derive the relationship between symmetries and conservation laws (Noether's theorem).

Conservation of generalized momentum

Firstly, if the Lagrangian of a multi-particle system does not involve a given coordinate $q_{j}$ , then that coordinate is said to be ignorable or cyclic and:

\begin{aligned} \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{j}} & = \frac{\partial L}{\partial q_{j}} = 0 \\ \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{j}} & = \frac{d p_{j}}{d t} = 0 \\ ⟹ p_{j} & = constant \end{aligned}

To obtain the classical conservation of linear and angular momentum, suppose a coordinate $q_{j}$ corresponds respectively to a translation or rotation of the whole system. We begin with a non-zero generalized force along these components but can set them to zero to make $q_{j}$ cyclic and the corresponding $p_{j}$ constant, hence proving the desired results. We also assume for simplicity that the potential does not depend on ${\dot{q}}_{j}$

\begin{aligned} {\dot{p}}_{j} = \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{j}} = \frac{\partial L}{\partial q_{j}} = - \frac{\partial V}{\partial q_{j}} & = Q_{j} = \sum_{i} {\vec{F}}_{i} \cdot \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} \\ \begin{aligned} Translation \\ \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} & = \hat{n} \\ ⟹ Q_{j} & = \sum_{i} {\vec{F}}_{i} \cdot \hat{n} = \hat{n} \cdot \vec{F} \\ and p_{j} & = \frac{\partial T}{\partial {\dot{q}}_{j}} = \sum_{i} m_{i} {\dot{\vec{r}}}_{i} \cdot \frac{{\dot{\partial \vec{r}}}_{i}}{\partial {\dot{q}}_{j}} \\ = \sum_{i} m_{i} {\vec{v}}_{i} \cdot \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} = \hat{n} \cdot \sum_{i} m_{i} {\vec{v}}_{i} \\ = \hat{n} \cdot \vec{P} \end{aligned} & \begin{aligned} Rotation \\ \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} & = \hat{n} \times {\vec{r}}_{i} \\ ⟹ Q_{j} & = \sum_{i} {\vec{F}}_{i} \cdot \hat{n} \times {\vec{r}}_{i} = \sum_{i} \hat{n} \cdot {\vec{r}}_{i} \times {\vec{F}}_{i} = \hat{n} \cdot \vec{τ} \\ and p_{j} & = \frac{\partial T}{\partial {\dot{q}}_{j}} = \sum_{i} m_{i} {\vec{v}}_{i} \cdot \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} \\ = \sum_{i} \hat{n} \cdot {\vec{r}}_{i} \times m_{i} {\vec{v}}_{i} = \hat{n} \cdot \sum_{i} {\vec{L}}_{i} \\ = \hat{n} \cdot \vec{L} \end{aligned} \end{aligned}

The significance of cyclic translation or rotation coordinates in relation to the properties of the system deserves some comment at this point. If a generalized coordinate corresponding to a displacement is cyclic, it means that a translation of the system, as if rigid, has no effect on the problem. In other words, if the system is invariant under translation along a given direction, the corresponding linear momentum is conserved. Similarly, the fact that a generalized rotation coordinate is cyclic (and therefore the conjugate angular momentum conserved) indicates that the system is invariant under rotation about the given axis. Thus, the momentum conservation theorems are closely connected with the symmetry properties of the system. If the system is spherically symmetric, we can say without further ado that all components of angular momentum are conserved. Or, if the system is symmetric only about the $z$ axis, then only $L_{z}$ will be conserved, and so on for the other axes.

- Herbert Goldstein, Charles Poole and John Safko

Conservation of energy

Consider a general Lagrangian $L (\vec{q}, \dot{\vec{q}}, t)$ . Taking it's total time derivative gives

\begin{aligned} \frac{d L}{d t} & = \sum_{j} \frac{\partial L}{\partial q_{j}} \frac{d q_{j}}{d t} + \sum_{j} \frac{\partial L}{\partial {\dot{q}}_{j}} \frac{d {\dot{q}}_{j}}{d t} + \frac{\partial L}{\partial t} \\ = \sum_{j} \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) {\dot{q}}_{j} + \sum_{j} \frac{\partial L}{\partial {\dot{q}}_{j}} \frac{d {\dot{q}}_{j}}{d t} + \frac{\partial L}{\partial t} \\ = \sum_{j} \frac{d}{d t} ({\dot{q}}_{j} \frac{\partial L}{\partial {\dot{q}}_{j}}) + \frac{\partial L}{\partial t} \end{aligned}

Therefore, we have:

\frac{d}{d t} (\sum_{j} {\dot{q}}_{j} \frac{\partial L}{\partial {\dot{q}}_{j}} - L) + \frac{\partial L}{\partial t} = 0

This is often written as:

\frac{d h}{d t} = - \frac{\partial L}{\partial t}

where

h = \sum_{j} {\dot{q}}_{j} \frac{\partial L}{\partial {\dot{q}}_{j}} - L

is termed the energy function and is equivalent to the total energy of the system in many cases of interest. Importantly, the energy function is conserved whenever the Lagrangian doesn't explicitly depend on time, i.e., the system is symmetric wrt temporal translations.

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