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# 物理代写|电动力学作业代写Electrodynamics代考|Degrees of Freedom in Classical Mechanics

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## 物理代写|电动力学作业代写Electrodynamics代考|Hypersurfaces in Minkowski Space-Time

In the context of Newtonian mechanics the concept of degree of freedom refers to the number of Lagrangian coordinates describing the system. A particle moving in a three-dimensional space, for example, carries three degrees of freedom in that, one would say, at any instant $t$ its position is specified by the three coordinates $\mathbf{y}(t)=(x(t), y(t), z(t))$. We can actually analyze the system from a different point of view, asking the question: how many initial data, say at time $t=0$, must we fix in order to predict the position of the particle at any time $t$ ? The answer – six, not three – is strictly tied to the dynamics of the system, specified by Newton’s second law
$$m \ddot{\mathbf{y}}=\mathbf{F}(\mathbf{y}, \dot{\mathbf{y}}, t) .$$
This equation corresponds indeed to a system of three second-order differential equations and admits, hence, a unique solution, once we impose the six initial data $\mathbf{y}(0)$ and $\dot{\mathbf{y}}(0)$. An equivalent manner to describe the dynamics of the system is offered by the Hamiltonian formalism, a framework complementary to the Lagrangian formalism implicitly employed above, in which the position $\mathbf{y}(t)$ and the velocity $\mathbf{v}(t)$, spanning the phase space, are considered as independent variables. In this framework one imposes the six first-order differential equations
$$m \dot{\mathbf{v}}=\mathbf{F}(\mathbf{y}, \mathbf{v}, t), \quad \dot{\mathbf{y}}=\mathbf{v},$$
which admit a unique solution once the initial data $\mathbf{y}(0)$ and $\mathbf{v}(0)$ have been specified. In the Hamiltonian formalism the system appears, thus, as a system of $s i x$ degrees of freedom. We realize so that the common statement “a particle carries three degrees of freedom” actually means “three degrees of freedom of the second order”. Equivalently we could, in fact, say that the particle carries six degrees of freedom of the first order. The preference for the first convention – commonly adopted in physics – stems primarily from the peculiar relation existing between the degrees of freedom of a classical field and the particles associated to it at the quantum level, see the end of Sect. 5.1.2. Henceforth, with the term degree of freedom we will always understand degree of freedom of the second order, if not differently specified.

## 物理代写|电动力学作业代写Electrodynamics代考|Degrees of Freedom in Field Theory

In field theory the fundamental variables are the fields, and from a mechanical point of view each field corresponds to a system of infinite degrees of freedom. Preserving the analogy with Newtonian mechanics based on Eq. (5.1) – while adapting the perspective – we give the following definition.

Definition. We say that a real field $\varphi(t, \mathbf{x})$ carries one degree of freedom if the equations of motion governing its dynamics are such that, once the data $\varphi(0, \mathbf{x})$ and $\partial_{0} \varphi(0, \mathbf{x})$ are known for every $\mathbf{x}$, they determine the field $\varphi(t, \mathbf{x})$ for any $t$ and for any $x$.

As prototype of such an equation we consider the partial differential equation for a scalar field
$$\square \varphi=P(\varphi)$$
where
$$\square=\partial_{\mu} \partial^{\mu}=\partial_{0}^{2}-\nabla^{2}$$
is the d’Alembertian operator – a relativistic completion of the Laplacian – and $P(\varphi)$ is a polynomial in $\varphi$. Equation (5.2) is of second order in the time derivatives and we expect, hence, that it confers to $\varphi$ one degree of freedom. To confirm this expectation explicitly we fix the initial data
$$\varphi(0, \mathbf{x}), \quad \partial_{0} \varphi(0, \mathbf{x})$$

and attempt to determine $\varphi(t, \mathbf{x})$ enforcing Eq. (5.2). Assuming that the solution is an analytic function of $t$ we perform a Taylor expansion around $t=0$
$$\varphi(t, \mathbf{x})=\sum_{n=0}^{\infty} \frac{\partial_{0}^{n} \varphi(0, \mathbf{x})}{n !} t^{n}$$
and try to determine its coefficients using (5.2). The coefficients relative to $n=0$ and $n=1$ correspond to the initial data (5.3). The coefficient relative to $n=2$ can be determined evaluating $(5.2)$ at $t=0$
$$\partial_{0}^{2} \varphi(0, \mathbf{x})=\nabla^{2} \varphi(0, \mathbf{x})+P(\varphi(0, \mathbf{x}))$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|Cauchy Problem for Maxwell’s Equations

In this section we formulate the Cauchy problem relative to the Maxwell equations (2.21) and (2.22). In doing so we will establish how many, and which ones, are the degrees of freedom associated with the electromagnetic field. For this purpose it is convenient to adopt the strategy outlined in Sect. 2.2.4, consisting in the solution of the Bianchi identity in terms of a vector potential $A^{\mu}$, see (2.46). Accordingly, the system of equations we must solve can be written schematically as
$$\partial_{\mu} F^{\mu \nu}=j^{\nu}, \quad F^{\mu \nu}=\partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu}, \quad A^{\mu} \approx A^{\mu}+\partial^{\mu} \Lambda$$
where, we recall, the last relation signals that $A^{\mu}$ is defined modulo gauge transformations.

Asymptotic conditions. Before proceeding any further, we specify the class of vector potentials and currents that we consider physically acceptable. First of all we assume that the four-current is known and fulfils the conservation law $\partial_{\mu} j^{\mu}=0$. Furthermore, we suppose that $j^{\mu}$ has compact spatial support, as it happens for all charge distributions realizable in nature. More precisely we require that
$$j^{\mu}(t, \mathbf{x})=0, \quad \text { for all }|\mathbf{x}|>l$$
where the radius $l$ may depend on $t$, as it happens, for example, for a charged particle following an unbounded trajectory. Correspondingly, we will accept as physical solutions of Maxwell’s equations only the vector potentials which for any fixed $t$ vanish (sufficiently fast) at spatial infinity
$$\lim _{|\mathbf{x}| \rightarrow \infty} A^{\mu}(t, \mathbf{x})=0$$

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|DEGREES OF FREEDOM IN FIELD THEORY

=∂μ∂μ=∂02−∇2

∂02披(0,X)=∇2披(0,X)+磷(披(0,X))

## 物理代写|电动力学作业代写ELECTRODYNAMICS代考|CAUCHY PROBLEM FOR MAXWELL’S EQUATIONS

∂μFμν=jν,Fμν=∂μ一种ν−∂ν一种μ,一种μ≈一种μ+∂μΛ

$$j^{\mu}(t, \mathbf{x})=0, \quad \text { for all }|\mathbf{x}|>l$$

$$\lim _{|\mathbf{x}| \rightarrow \infty} A^{\mu}(t, \mathbf{x})=0$$

## Matlab代写

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