Brian Bi

## Why isn't kinetic energy a vector instead of a scalar?

Kinetic energy is a scalar because it is defined to be a scalar.

The reason why scientists define certain words to mean certain things is that some definitions are more useful than others. In particular, if a quantity is useful, it is likely to have a name. The quantity $$T = \frac{1}{2}mv^2$$, a scalar, is useful, so it gets a name: kinetic energy. The quantity $$\frac{1}{2}mv^2 \hat{\mathbf{v}}$$, which is a vector whose magnitude is the kinetic energy and whose direction is the direction of motion, is not useful, so it doesn't have a name.

Now, there are at least three different reasons why kinetic energy, as a scalar, is a useful quantity in physics:
1. It's part of the total energy—kinetic plus potential—which is very important because of conservation of energy. $$E = T + V$$ would have to be changed to $$E = \|\mathbf{T}\| + V$$ if we defined kinetic energy as a vector. That is, we'd just throw away the direction. There is no vector conservation law for energy, the way there is for momentum.
2. It's a term in the Hamiltonian, which tells you how a physical system in general evolves in time. The Hamiltonian's value is usually the same as the total energy, $$H = T + V$$, although for some systems it's different. Again, if we defined kinetic energy as a vector, we'd just have to throw away the direction, $$H = \|\mathbf{T}\| + V$$.
3. It's a term in the Lagrangian, which, like the Hamiltonian, governs the behaviour of a physical system, and is very important theoretically because its mathematical form makes certain properties of physical systems obvious. The Lagrangian for a particle is $$L = T - V$$. Again, if we defined kinetic energy as a vector, we'd have to throw away the direction, and write $$L = \|\mathbf{T}\| - V$$.
There's no point in defining kinetic energy as a vector when we'd just have to throw away the direction every time we wanted to use it.