These are some study notes I made for the excellent Classical Mechanics course from The Theoretical Minimum program. They’re mostly for my personal use, but I’m putting them here in case someone else finds these useful.
A physical law constitutes of a phase space and a transition function . In a discrete setting, the physical system evolves as with . The transition function needs to be:
- Deterministic – knowing the state at time is enough to compute the state at an arbitrary time in future. This just follows from being a function.
- Reversible – knowing the state at time is enough to compute the state at an arbitrary time in the past (“information preserving”, in other words). See Liouville’s theorem below for a more formal statement.
Example of an inadmissible law
Aristotle’s law of motion is inadmissible. To see why, consider what happens if we attach a block of mass to a spring with spring constant . Then the law of motion becomes , which solves to . This transition function is not reversible – the final state for every initial state is .
Every physical system has an associated Lagrangian = where is the state of the system (coordinates in phase space) and is time. For many (but not all!) simple systems where is the kinetic energy of the system and is the potential energy of the system. does not always cleanly separate into kinetic and potential energies this way.
, the generalized (or canonical) momentum with respect to the coordinate, is defined to be .
The Euler-Lagrange equation dictates the laws of motion:
These are derived by applying the the principle of least action, though I could not grasp why the principle of least action is alluded to be more fundamental than .
Symmetries and Conservation
A “symmetry” is a change in the inputs to the Lagrangian through which the value of the Lagrangian does not change.
Co-ordinate translation symmetry
If is invariant under the change then the quantity is conserved.
Time translation symmetry
The Hamiltonian of a system, , is defined as:
Then using the Euler-Lagrange equations, we can show that . If does not depend directly on then we say has time-translation symmetry, and it follows that does not change with time. is the definition of energy, so time-translation symmetry implies energy conservation.
can be written as a function of and by solving for in terms of . In cases where we can’t solve for the Hamiltonian is invalid (disallowed by quantum mechanics in ways not covered in this course).
Hamilton’s equations (as opposed to Euler-Lagrange) are symmetric in how they treat momentum and position:
In Hamiltonian mechanics the phase space is a set of points with coordinates and , and each point moves around with time. Liouville’s theorem (formal and continuous analog of “physical laws need to be reversible”) states that there are no sinks and sources in this “fluid”.
Proof sketch: let be and be . Then and form a vector field , such that (using algebra + Hamilton’s equations)1.
Poisson brackets are a notation defined as:
With this notation, Hamilton’s equation can stated as: for any function over the phase space of a system, .
Given a quantity which is conserved, gives us the small change in under the symmetry operation that conserves . So, e.g., (where is the angular momentum) gives us the change in due to a small rotation. is just an application of this principle since is conserved on time translation.
Magnetic and Electrostatic Forces
Gradient, divergence and curl
Let be the Levi-Civita symbol. Then:
Two pertinent algebraic facts:
- The divergence of a vector field is iff that field is a curl of some field.
- The curl of a field is iff the field is a gradient of some (scalar) field.
Experimentally, we know that magnets do not have mono-poles, which is another way of saying that the divergence of a magnetic field, , is always . Therefore must be the curl of another field; call it (vector potential). is not unique – adding a gradient (of some scalar) to it keeps the same while changing . This operation of changing by adding a gradient is called a “gauge transformation”.
is measurable while is not, but is necessary to write out the Lagrangian.
Let be the electric potential. Then
The second term in is the Lorentz force. This is similar to the Coriolis force.
Lagrangian and Hamiltonian
The electric potential energy is simple, so let’s focus on the part of the Lagrangian due to the magnetic field (i.e. assume electric field is ):
We can use the principle of least action to show that gauge transformations do not affect observable behavior of a system.
From we get . This means the Hamiltonian is
Note that is basically just .
The divergence of a vector field is defined as . ↩