Energy

Prerequisites: Newtonian Mechanics. Full path (5 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Energy is a number we can compute from the state of a system that keeps the same value while the system changes. A ball thrown upwards slows down, stops and falls back. Its speed changes all the time, and so does its height. If we add the energy of motion (kinetic energy) to the energy of position in the gravitational field (potential energy), the sum stays the same during the whole flight, as long as air resistance can be neglected.

When friction is present, the sum of kinetic and potential energy goes down, but the lost amount shows up elsewhere, for example as heat in the ball and the air. Physicists have found that once all forms are included (motion, position, heat, chemical bonds, radiation, the mass of a body itself), the total never changes in an isolated system. This makes energy a bookkeeping tool: if we know how much there is at the start, we know how much there is at the end, without following every detail in between.

The reason energy is conserved is that the laws of physics are the same today as they were yesterday. This link between a symmetry in time and a conserved quantity is the content of Noether's theorem. It also explains why energy is tied to change in time: in quantum mechanics the energy of a state sets how fast its phase rotates.

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Concrete

Energy is the conserved quantity that we derive using Noether's theorem if our system is symmetric under temporal translations.

In addition, energy is responsible for temporal translations. We say energy generates temporal translations.

The total energy is defined as $$\begin{equation} E(t) \equiv K(t) + V(q(t)), \end{equation}$$

where $K$ denotes the kinetic energy and $V$ the potential energy.

Kinetic Energy

Kinetic energy is defined as $$\begin{equation} K(t) \equiv \frac{1}{2}m\,v(t)\cdot v(t). \end{equation}$$ This quantity is useful because $$\begin{split} \frac{d}{dt}K(t) &= m\,v(t)\cdot a(t) \\ &= F(q(t))\cdot v(t). \end{split}$$ We can see here that the kinetic energy goes up whenever we push an object in the direction of its velocity. Moreover, it goes down whenever we push it in the opposite direction.

In addition, we have $$\begin{split} K(t_1)-K(t_0) &= \int_{t_0}^{t_1} F(q(t))\cdot v(t)\,dt \\ &= \int_{t_0}^{t_1} F(q(t))\cdot \dot{q}(t)\, dt. \end{split}$$ This tells us that the change of kinetic energy is equal to the work done by the force. The work is defined as the integral of $F$ along the trajectory.

Potential Energy

$\nabla V \equiv - F,$

where $F$ denotes the force.

Abstract

Hamiltonian mechanics. On a symplectic manifold $(M,\omega)$ the energy is the Hamiltonian $H : M \to \mathbb R$, and time evolution is the flow of the Hamiltonian vector field $X_H$ defined by

$$\iota_{X_H}\omega = dH .$$

With $\omega = dq^i\wedge dp_i$ this reproduces Hamilton's equations $\dot q^i = \partial H/\partial p_i$, $\dot p_i = -\partial H/\partial q^i$. Along the flow, $\frac{dH}{dt} = dH(X_H) = \omega(X_H, X_H) = 0$ by antisymmetry of $\omega$. For a time-dependent Hamiltonian one finds instead $dH/dt = \partial H/\partial t$. Energy conservation is therefore the statement that the generator of time evolution is itself invariant under time evolution.

Quantum mechanics. By Stone's theorem a strongly continuous one-parameter unitary group $U(t)$ on a Hilbert space has the form $U(t) = e^{-iHt/\hbar}$ with a unique self-adjoint operator $H$. The energy is this generator. Its spectrum is the set of possible energy values, and physical stability requires that it be bounded from below. Stationary states are eigenvectors of $H$, and their time dependence is the phase $e^{-iEt/\hbar}$.

Special relativity. Energy is the time component of the four-momentum $P^\mu$, which generates spacetime translations in the Poincaré group. For a field theory with conserved energy-momentum tensor, $\partial_\mu T^{\mu\nu} = 0$, the charges $P^\nu = \frac{1}{c}\int d^3x\, T^{0\nu}$ are conserved, and $E = cP^0 = \int d^3x\, T^{00}$. Energy alone is not a Lorentz scalar. The invariant combination is $P_\mu P^\mu = m^2c^2$ (signature $(+,-,-,-)$), which defines the mass.

General relativity. The local law $\nabla_\mu T^{\mu\nu} = 0$ does not by itself give a conserved global energy, because integrating a covariant divergence of a tensor does not give a boundary term. If the spacetime has a Killing vector field $\xi$, the current $J^\mu = T^{\mu\nu}\xi_\nu$ satisfies $\nabla_\mu J^\mu = 0$ (using the symmetry of $T^{\mu\nu}$ and Killing's equation $\nabla_{(\mu}\xi_{\nu)} = 0$), and its flux through a Cauchy surface is conserved. For a timelike Killing field this is the conserved energy of matter. For asymptotically flat spacetimes a total energy including the gravitational field can be defined at spatial infinity, the ADM energy. There is no general local energy density for the gravitational field.

Why is it interesting?

Energy is the most important concept in physicsBrian Skinner

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus
5. Newtonian Mechanics

Leads to: Thermodynamics, Work-Energy theorem.