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Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Edit: Hamilton equations and energy conservation. $$H(q,p,t) := p \dot{q} - L(\dot{q}(q,p,t),q(t),t)$$

$$\begin{aligned} d H & = \left(\frac{\partial H}{\partial q}\right) d q + \left(\frac{\partial H}{\partial p}\right) d p + \left(\frac{\partial H}{\partial t}\right) dt = \\ & = d p \dot q + \underbrace{ p d \dot{q} - \frac{\partial L}{\partial \dot q} d \dot q}_{=0 \text{ since } p:=\frac{\partial L}{\partial \dot q}} - \frac{\partial L}{\partial q} d q - \frac{\partial L}{\partial t} dt \ , \\ \end{aligned}$$

so that by comparison, and Lagrange equation $0 = \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q}\right) - \left( \frac{\partial L}{\partial q}\right) = \dot{p} -\left( \frac{\partial L}{\partial q}\right) $, you get

$$\begin{cases} \dot p = \left( \frac{\partial L}{\partial q} \right) = - \left( \frac{\partial H}{\partial q} \right) \\ \dot q = \left( \frac{\partial H}{\partial p} \right) \\ \frac{\partial H}{\partial t} = - \frac{\partial L}{\partial t} \end{cases}$$

Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Edit: Hamilton equations and energy conservation. $$H(q,p,t) := p \dot{q} - L(\dot{q}(q,p,t),q(t),t)$$

$$\begin{aligned} d H & = \left(\frac{\partial H}{\partial q}\right) d q + \left(\frac{\partial H}{\partial p}\right) d p + \left(\frac{\partial H}{\partial t}\right) dt = \\ & = d p \dot q + \underbrace{ p d \dot{q} - \frac{\partial L}{\partial \dot q} d \dot q}_{=0 \text{ since } p:=\frac{\partial L}{\partial \dot q}} - \frac{\partial L}{\partial q} d q - \frac{\partial L}{\partial t} dt \ , \\ \end{aligned}$$

so that by comparison, and Lagrange equation $0 = \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q}\right) - \left( \frac{\partial L}{\partial q}\right) = \dot{p} -\left( \frac{\partial L}{\partial q}\right) $, you get

$$\begin{cases} \dot p = \left( \frac{\partial L}{\partial q} \right) = - \left( \frac{\partial H}{\partial q} \right) \\ \dot q = \left( \frac{\partial H}{\partial p} \right) \\ \frac{\partial H}{\partial t} = - \frac{\partial L}{\partial t} \end{cases}$$

Now, knowing these relations, we can evaluate the time derivative of a function $f(q(t),p(t),t)$ as

$$\begin{aligned} \frac{d f}{dt} & = \frac{\partial f}{\partial q} \dot q + \frac{\partial f}{\partial p} \dot p + \frac{\partial f}{\partial t} = \\ & = \frac{\partial f}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial H}{\partial q} + \frac{\partial f}{\partial t} = \left\{ f, H \right\} + \frac{\partial f}{\partial t} \ , \end{aligned}$$

having introduced the Poisson brackets for completeness.

Now, if $f = H$, the first term is zero and we get

$$\dfrac{d H}{d t} = \frac{\partial H}{\partial t} \ ,$$

and if the Hamiltonian doesn't explicitly depend on time $t$, $\dfrac{\partial H}{\partial t} = 0$ and energy conservation follows

$$\frac{d H}{dt} = 0 \ .$$

Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Edit: Hamilton equations and energy conservation. $$H(q,p,t) := p \dot{q} - L(\dot{q}(q,p,t),q(t),t)$$

$$\begin{aligned} \dfrac{d}{dt} H & = \left(\frac{\partial H}{\partial q}\right) \dot q + \left(\frac{\partial H}{\partial p}\right) \dot p + \left(\frac{\partial H}{\partial t}\right) = \\ & = \dot{p} \dot{q} + \underbrace{ p \ddot{q} - \frac{\partial L}{\partial \dot q} \ddot q}_{=0 \text{ since } p:=\frac{\partial L}{\partial \dot q}} - \frac{\partial L}{\partial q} \dot q - \frac{\partial L}{\partial \dot t} = \\ \end{aligned}$$

Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Edit: Hamilton equations and energy conservation. $$H(q,p,t) := p \dot{q} - L(\dot{q}(q,p,t),q(t),t)$$

$$\begin{aligned} d H & = \left(\frac{\partial H}{\partial q}\right) d q + \left(\frac{\partial H}{\partial p}\right) d p + \left(\frac{\partial H}{\partial t}\right) dt = \\ & = d p \dot q + \underbrace{ p d \dot{q} - \frac{\partial L}{\partial \dot q} d \dot q}_{=0 \text{ since } p:=\frac{\partial L}{\partial \dot q}} - \frac{\partial L}{\partial q} d q - \frac{\partial L}{\partial t} dt \ , \\ \end{aligned}$$

so that by comparison, and Lagrange equation $0 = \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q}\right) - \left( \frac{\partial L}{\partial q}\right) = \dot{p} -\left( \frac{\partial L}{\partial q}\right) $, you get

$$\begin{cases} \dot p = \left( \frac{\partial L}{\partial q} \right) = - \left( \frac{\partial H}{\partial q} \right) \\ \dot q = \left( \frac{\partial H}{\partial p} \right) \\ \frac{\partial H}{\partial t} = - \frac{\partial L}{\partial t} \end{cases}$$

Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Hamiltonian is the energy of the system, as a function of generalized coordinate $\mathbf{q}$ and momentum $\mathbf{p}$, $H(\mathbf{q}, \mathbf{p}, t)$.

For a simple 1D harmonic oscillator,

$$q = x \qquad , \qquad p = m \dot{x} = m \dot q \qquad, \qquad H(q,p) = \frac{1}{2}\frac{p^2}{m} + \frac{1}{2} k q^2$$

whose time derivative reads,

$$0 = \frac{dH}{dt} = \frac{\partial H}{\partial p} \dot p + \frac{\partial H}{\partial q} \dot q = \frac{p}{m} \dot p + k q \dot q = \dot q ( m \ddot q + k q ) \ , $$

and thus, either the system is at rest $\dot q = 0$, or the system is governed by the 2-nd order harmonic differential equation $m\ddot q + k q = 0$.

Edit: Hamilton equations and energy conservation. $$H(q,p,t) := p \dot{q} - L(\dot{q}(q,p,t),q(t),t)$$

$$\begin{aligned} \dfrac{d}{dt} H & = \left(\frac{\partial H}{\partial q}\right) \dot q + \left(\frac{\partial H}{\partial p}\right) \dot p + \left(\frac{\partial H}{\partial t}\right) = \\ & = \dot{p} \dot{q} + \underbrace{ p \ddot{q} - \frac{\partial L}{\partial \dot q} \ddot q}_{=0 \text{ since } p:=\frac{\partial L}{\partial \dot q}} - \frac{\partial L}{\partial q} \dot q - \frac{\partial L}{\partial \dot t} = \\ \end{aligned}$$