# Systematic way of obtaining conservation laws in dynamical systems

Motivation

Consider a point particle of mass $$m$$ moving in $$\mathbb{R}^3$$ under the influence of some force field $$\vec{F}(\vec{r},t)$$. The fundamental equation governing the dynamics of this system is Newton's second law stating $$\boxed{\vec{F} = \frac{d \vec{p}}{dt}}$$ where $$\vec{p} = m \vec{v}$$ is the (linear) momentum of the particle. This is also an example of a conservation law. It states that a free particle, i.e. one for which there is no force acting on it, moves in a way such that its (linear) momentum is conserved $$\vec{F} = \vec{0} \quad \iff \quad \vec{p} = constant.$$ In this question, I am interested in a systematic way of constructing all such conserved quantities.

My approach

My approach, in a nutshell, is to multiply the Newton's law with various quantities and, after some manipulation, recognize that some quantities are constant when $$\vec{F} = \vec{0}.$$ In the following, I will list some examples.

1. Dot product $$\vec{v} \cdot \vec{F}$$ leads to the conservation of kinetic energy $$T = \frac{1}{2}m \vec{v}^2$$. We have $$\boxed{\vec{v} \cdot \vec{F} = \frac{d}{dt} \left( \frac{1}{2}m \vec{v}^2 \right)}$$ so that $$\vec{F} = \vec{0} \quad \iff \quad T = constant.$$
2. Cross product $$\vec{v} \times \vec{F}$$ apparently does not lead to any conservation laws because we trivially have $$\boxed{\vec{v} \times \vec{F} = m \vec{v} \times \vec{a}}$$ and the RHS cannot be written as a time derivative. Therefore, it looks as though $$\vec{v} \times \vec{F}$$ is not a useful quantity. (I could be wrong about this, so please correct me.)
3. Dot product $$\vec{r} \cdot \vec{F}$$ does not lead to a conservation law due to an extra term on the RHS, $$\boxed{\vec{r} \cdot \vec{F} = \frac{d}{d t}(\vec{r} \cdot \vec{p})+2T}$$ However, this equation is the starting point in the derivation of Virial theorem, so it's not totally useless.
4. Cross product $$\vec{r} \times \vec{F}$$ leads to the conservation of angular momentum $$\vec{L} = \vec{r} \times \vec{p}$$ because $$\boxed{\vec{r} \times \vec{F} = \frac{d}{d t} (m \vec{r} \times \vec{v})}$$ so that $$\vec{F} = \vec{0} \quad \iff \quad \vec{L} = constant.$$
5. Multiplication with time leads to the conservation of the center of mass motion, $$\boxed{t \vec{F} = \frac{d}{dt} (t \vec{p} - m \vec{r})}$$ so that $$\vec{F} = \vec{0} \quad \iff \quad t \vec{p} - m \vec{r} = constant.$$

The question

In my approach, I have been able to reproduce the 10 well known integrals of motion simply by guesswork. However, I have no guarantee that there aren't any more integrals of motion. So, is there a systematic way of finding all integrals of motion for a given dynamical system?

• See the Noether theorems of Hamiltonian mechanics. In short, any symmetry has associated conserved quantities, one for each symmetry generator. Rotational symmetry leads to conservation of angular momentum etc. Mar 12, 2019 at 22:40
• @LutzL: Is there a way to find all symmetries of a given Hamiltonian? Mar 12, 2019 at 22:44