# 数学代写|数学建模代写Mathematical Modeling代考|The general scheme of the Hamiltonian principle

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## 数学代写|数学建模代写Mathematical Modeling代考|The general scheme of the Hamiltonian principle

The general scheme of the Hamiltonian principle. Consider a mechanical system, without giving its formal and rigorous definition, taking into account that the interaction between all its elements is determined by the laws of mechanics (one of the simplest examples is the “ball-spring” system considered in section 2.4). Introduce the concept of a generalized coordinate $Q(t)$, completely defining the position of the mechanical system in space. The quantity $Q(t)$ can coincide with the Cartesian coordinate (for example, the coordinate $r$ of the system “ball-spring”), the radius-vector, the angular coordinate, the set of coordinates of mass points, etc. It is natural to label the quantity $d Q / d t$ as generalized velocity of the mechanical system in the moment of time $t$. The set of magnitudes $Q(t)$ and $d Q / d t$ determines the state of a mechanical system at all moments in time.

To describe the mechanical system the Lagrange function is introduced; its derivation is a separate problem, considered in more detail in chapter III. In the simplest cases the Lagrange function has a clear content and is denoted as
$$L(Q, d Q / d t)=E_k-E_p,$$
where $E_k, E_p$ are the kinetic and potential energies of the system respectively. For the purposes of the present section there is no need to give the general definition of quantities $E_k, E_p$, in so far as they are calculated in obvious manner in the considered examples.
Let us introduce the quantity $S[Q]$, named an action:
$$S[Q]=\int_{t_1}^{t_2} L\left(Q, \frac{d Q}{d t}\right) d t$$

## 数学代写|数学建模代写Mathematical Modeling代考|The third way of deriving the model of the system “ballspring”

We use the Hamiltonian principle to construct the model of the motion of the ball connected with a spring (section 2.4). As generalized coordinates of the system it is natural to chose the usual Eulerian coordinate of the ball $r(t)$. Then the generalized velocity $d r / d t=v(t)$ is the usual velocity of the ball. The Lagrangian function (1), $L=E_k-E_p$, is rewritten using values of kinetic and potential energies of the system (already found in section 2.4):
$$L=\frac{m(d r / d t)^2}{2}-k \frac{r^2}{2}$$
For the action we obtain
$$S[r]=\int_{t_1}^{t_2} L\left(r, \frac{d r}{d t}\right) d t=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d r}{d t}\right)^2-\frac{k}{2} r^2\right] d t$$
Now, in view of the correspondence with the scheme seen in subsection 1 , we calculate the action over the variations $\varepsilon \varphi(t)$ of the coordinates $r(t)$ :
$$S[r+\varepsilon \varphi]=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d(r+\varepsilon \varphi)}{d t}\right)^2-\frac{k}{2}(r+\varepsilon \varphi)^2\right] d t$$

Now, in view of the correspondence with the scheme seen in subsection 1 , we calculate the action over the variations $\varepsilon \varphi(t)$ of the coordinates $r(t)$ :
$$S[r+\varepsilon \varphi]=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d(r+\varepsilon \varphi)}{d t}\right)^2-\frac{k}{2}(r+\varepsilon \varphi)^2\right] d t .$$

## 数学代写|数学建模代写MATHEMATICAL MODELING代考|THE GENERAL SCHEME OF THE HAMILTONIAN PRINCIPLE

$$L(Q, d Q / d t)=E_k-E_p$$

$$S[Q]=\int_{t_1}^{t_2} L\left(Q, \frac{d Q}{d t}\right) d t$$

## 数学代写|数学建模代写MATHEMATICAL MODELING代考|THE THIRD WAY OF DERIVING THE MODEL OF THE SYSTEM “BALLSPRING”

$$L=\frac{m(d r / d t)^2}{2}-k \frac{r^2}{2}$$

$$S[r]=\int_{t_1}^{t_2} L\left(r, \frac{d r}{d t}\right) d t=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d r}{d t}\right)^2-\frac{k}{2} r^2\right] d t$$

$$S[r+\varepsilon \varphi]=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d(r+\varepsilon \varphi)}{d t}\right)^2-\frac{k}{2}(r+\varepsilon \varphi)^2\right] d t$$

$$S[r+\varepsilon \varphi]=\int_{t_1}^{t_2}\left[\frac{m}{2}\left(\frac{d(r+\varepsilon \varphi)}{d t}\right)^2-\frac{k}{2}(r+\varepsilon \varphi)^2\right] d t$$

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