# تفاضل وتكامل التغيرات

طرق رياضية لإيجاد دالة تعطي القيمة الكبرى (أو الصغرى) لتكامل محدد.

## Mathematical background

A 'function' is a mapping from a space of numbers to another space of numbers. A mathematical object known as a 'functional' is a mapping of a space of functions
to a number. A definite integral is one such example of a functional.^{[1]} ^{[2]}

## Physics background

In mathematical physics there are problems which require that a path or surface represents the minimization of conserved physical quantities.

Four classic problems of variational calculus are the catenary, brachistochrone, geodesic, and the minimal surface problems.

### Lagrangian formulation of the classical action

In general, all such problems are reduced to the minimization of the *classical action* **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S }**
is defined by the integral

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S = \int_A^B L(x,\dot{x}, t) \; dt. }**

in such a way that the optimal function **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x(t) }**
is produced. Here **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x} \equiv dx/dt }**
.
The function **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(x,\dot{x}, t) }**
is a scalar-valued function, but **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x }**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x} }**
may be in **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^n}**
where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n }**
is
greater than 1.

### Application of Taylor's theorem

Regarding the application of Taylor's theorem, we may write **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(x + \delta x ,\dot{x} + \delta \dot{x}, t) }**
as

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L(x + \delta x, \dot{x} + \delta \dot{x}, t) = L(x,\dot{x}, t) + \left[ \frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot{x}} \delta \dot{x} \right] + \frac{1}{2!} \left\{ \frac{\partial^2 L }{\partial x^2} \left(\delta x\right)^2 + 2 \frac{\partial^2 L }{\partial x \partial \dot{x} } \delta x \delta \dot{x} + \frac{\partial^2 L}{\partial \dot{x}^2} \left(\delta \dot{x} \right)^2 \right\} + O(\epsilon^3).}**

Here **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta x = x(t) - h(t) }**
such that **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle | x - h | < \epsilon << 1 }**
. Terms in square brackets
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [...]}**
is commonly called the *first variation* and the part in braces **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{ ... \} }**
is
commonly called the *second variation.*

The process of solving a given problem is to minimize the first variation of the classical action which is given formally as

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta S = \int_A^B \left[ L(x + \delta x, \dot{x} + \delta \dot{x}, t) - L(x,\dot{x}, t) \right] \; dt \sim \int_A^b \left[ \frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot{x}} \delta \dot{x} \right] \; dt . }**

The problem then is to find the curve **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x(t) }**
which satisfies the condition that **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta S = 0 }**
and that all of the potential solution curves have the same values at the endpoints of integration **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A }**
and
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B }**
. This means that the variations of the solution **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta x(A) = 0 }**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta x(B) = 0. }**

### The Euler-Lagrange equations

The integral representation of the first variation of the action

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta S = \int_A^B \left[ \frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot{x}} \delta \dot{x} \right] \; dt }**

may be simplified by performing integration by parts on the second term in the integrand, wherein the **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta \dot{x} }**
factor is integratied to yield

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial \dot{x}} \delta x(t) \bigg|_A^B - \int_A^B \frac{d}{dt} \left[\frac{\partial L}{\partial \dot{x}}\right] \delta x(t) \; dt }**

.

The endpoint evaluation vanishes because the endpoints of all of the solution curves are taken to be the same
for all possible solutions, hence **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta x(A) = \delta x(B) = 0. }**
Substituting the result back into
the integral for **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta S }**
we have

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta S = \int_A^B \left\{ \frac{\partial L}{\partial x} - \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}}\right]\right\} \delta x \; dt =0, }**

implying that the integrand vanishes

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial x} - \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}}\right] = 0. }**

Without loss of generality, we may consider **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x }**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x} }**
to be in **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}**
-dimensions,
permitting us to write

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial x_i} - \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}_i }\right] = 0, }**

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i = 1,2,...,n}**
.

Solving the Euler-Lagrange equations is equivalent to finding the gradient of the Lagrangian in all of its variables, and setting this equal to zero. The
solution curve **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x(t) }**
is the projection of the minimum path in phase space onto the spatial coordinates (the configuration space), depending on the problem.

### Properties of the Lagrangian

The Lagrangian formulation is linear. The Lagrangian of a system is the sum of the Lagrangians of the subsystems that make up that system.

### Case I

Case I: **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = L(x, \dot{x} ) }**

In this case, the Lagrangian is not explicitly a function of the integration variable.

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \left[ \frac{\partial L}{\partial \dot{x} } \right] = \frac{\partial^2 L }{\partial t \partial x } + \frac{\partial^2 L}{\partial x \partial \dot{x} } \frac{d x}{dt} +\frac{\partial^2 L}{\partial \dot{x}^2 } \frac{d^2 x}{dt^2} = \frac{\partial^2 L }{\partial x \partial \dot{x} } \frac{d x}{dt} +\frac{\partial^2 L }{\partial \dot{x}^2 } \frac{d^2 x}{dt^2} }**

#### Alternately

We may write the full time derivative of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L }**

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d L}{ dt} = \frac{\partial L}{\partial x }\frac{d x}{dt} + \frac{\partial L}{\partial \dot{x} }\frac{d^2 x}{dt^2} }**

and

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{d t} \left[\dot{x} \frac{\partial L}{\partial \dot{x} } \right] = \frac{\partial L}{\partial \dot{x}} \frac{d^2 x}{d t^2} + \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}} \right] \frac{dx}{dt} }**

Combining the two identities we have

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \left[ L - \dot{x} \frac{d L}{\partial x }\right] = \frac{\partial L}{\partial x} \frac{d x}{dt} + \frac{\partial L}{ \partial \dot{x} } \frac{d^2 x }{ dt^2} - \frac{\partial L}{ \partial \dot{x} } \frac{d^2 x }{ dt^2} - \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}} \right] \frac{dx}{dt} = \left( \frac{\partial L}{\partial x} - \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}}\right] \right) \frac{dx}{dt} = 0 }**

.

The term in parentheses **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (...) ,}**
is Euler-Lagrange.

Hence

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \left[ L - \dot{x} \frac{d L}{\partial \dot{x} }\right] =0 }**

is equivalent to the Euler-Lagrange equations when **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L }**
does not depend on the integration variable.

We may immediately write

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L - \dot{x} \frac{d L}{\partial \dot{x}} = c }**

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c }**
is a constant.

### Case II

Case II - **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = L(\dot{x}) }**

The Lagrangian does not depend on **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x }**
or on the integration variable **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t }**
.

With no explicit dependence of the Lagrangian on

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}\left[ \frac{\partial L}{\partial \dot{x}_j }\right] = 0 }**

implying that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial \dot{x}_j} = \mbox{a constant vector}. }**

Also, because **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L }**
does not depend on space

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial x_j} = 0 }**

implying that the Lagrangian **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = \mbox{const.} }**
.

## Physics Ramifications

If we consider the problems of mechanics that different forms of the Lagrangian represent, we see important ramifications regarding the nature of symmetries that lead to the equations of classical mechanics.

### Free space, no potential, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = L(\dot{x} ) }**

As above this implies that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L }{ \partial x } = 0 }**
meaning that

We also have from the Euler-Lagrange equations that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} \left[ \frac{\partial L}{\partial \dot{x}_j } \right] = 0 }**

implying that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial \dot{x}_j } = w_j }**

,

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w_j }**
is a constant vector.

The last result follows because the partial derivative with respect to **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x} }**
is the gradient with respect to the velocity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{x}_j = v_j }**
.

Thus, the Lagrangian cannot be a function of the direction of the velocity vector, meaning that the Lagrangian can only be a function of the magnitude of velocity, but not its
direction. This is a statement of the *isotropy of space*.

The result implies that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = v_j w_j + c_1 }**

,

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_1 }**
is a constant of integration.

By the isotropy of space

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_j w_j = |v||w| \cos \theta }**

But because there can be no angular dependence, we may take **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w_j = K v_j }**
and thus the expression for the Lagrangian becomes

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = K v_j v_j + c_1 }**

.

The simplest model is of a particle traveling at a constant velocity, in a constant direction, which is the definition of an inertial frame.

If we define **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K = m/2 }**
, where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m }**
is mass, then the Lagrangian takes on the form of kinetic energy, to within a constant

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = 1/2 m v_j v_j + c_1 }**

.

### Conservation of linear momentum

Substituting this form of the Lagrangian into

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial L}{\partial v_j} = m v_j \equiv p_j }**

where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_j }**
is the momentum, which must be the same as the constant vector **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w_j }**
defined in the previous discussion. This
is a statement of *conservation of linear momentum.*

### Free Space with a potential field **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(x) }**

By the linearity of the Lagrangian, we need only add the potential **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -U(x) }**
to the previous expression of the Lagrangian to obtain

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L = \frac{1}{2} m v_j v_j - U(x_j) }**

.

Solving the Euler-Lagrange equations (here with **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_j }**
we have

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ d }{dt} \left[ \frac{\partial L }{\partial v_j} \right] = \frac{\partial L}{\partial x_j } }**

.

Invoking the equivalence of the velocity gradient of L and momentum, we have

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{ d }{dt} \left[ m v_j \right] = - \frac{\partial U}{\partial x_j } = - \nabla_j U(x_j). }**

.

Applying the time derivative to the velocity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_j }**
we obtain the acceleration **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_j }**
, we have Newton's equation of motion

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m a_j = - \nabla_j U \equiv F_j }**

.

Thus, Newton's laws of mechanics follow from the consideration of the homogeneity and isotropy of space and time.