Hamiltonian & Symmetries

Hamilton’s Equations

Hamiltonian

Recall that the canonical momentum is defined as $p_j=\frac{\partial L}{\partial \dot{q_j}}$, we define the hamiltonian as the Legendre transform of the Lagrangian, viewed as a function of $\dot{\mathbf{q}}$: $H(\mathbf{q},\mathbf{p},t)=\dot{\mathbf{q}}\cdot \mathbf{p}-L(\mathbf{q},\dot{\mathbf{q}},t)$

Hamilton’s Equations

We can easily derive the following Hamilton’s equations from the Euler-Lagrange equation: ${\dot{p}}_j=-\frac{\partial H}{\partial q_j},{\dot{q}}_j=\frac{\partial H}{\partial p_j},-\frac{\partial L}{\partial t}=\frac{\partial H}{\partial t}.$

Theorem

The hamiltonian in a conservative potential is the total energy.

Proof Under the assumption that $L=T-U$, and $T$ is a quadratic form with respect to $\dot{\mathbf{q}}$, in a conservative system, $U$ only depends on $\mathbf{q}$. We invoke the lemma, and obtain $H=2T-(T-U)=T+U$. $\square$

Corollary

$\text{ddf}Ht=\text{pddf}Ht.$In particular, for a system whose Hamiltonian does not dependent explicitly on time, the Hamiltonian is conserved.

Proof We have the following equations hold: $\begin{array}{rl}\text{ddf}Ht& =\text{pddf}H\mathbf{p}\dot{\mathbf{p}}+\text{pddf}H\mathbf{q}\dot{\mathbf{q}}+\text{pddf}Ht\\ & =-\text{pddf}H\mathbf{p}\text{pddf}H\mathbf{q}+\text{pddf}H\mathbf{q}\text{pddf}H\mathbf{p}+\text{pddf}Ht\\ & =\text{pddf}Ht.\end{array}$ $\square$

Symmetries

Proposition

One coordinate is cyclic if and only if $\frac{\partial H}{\partial q_i}=0$, which leads to a constant momentum.

Noether's Theorem

Every continuous symmetry of the action corresponds to a conserved quantity and vice versa. $\text{ddf}L(S,Q,\dot{Q},t)S=0⟹\text{pddf}L\dot{Q}\text{pddf}QS=0$where $L(S,Q,\dot{Q},t)$ is the Lagrangian, $S$ is the action and $Q$ is some generalized coordinates.

Proof A proof of Noether’s theorem using symplectic geometry is given here. $\square$