The study of Classical Mechanics usually starts with the three laws of motion given by Sir Isaac Newton. One knew from these three laws that, if it is possible to completely determine the net force acting on a particle at each instant, the motion of the particle is uniquely determined by the solution of the second order differential equation
\[
{\vec{F}=m\vec{a}}\tag{1.1}
\]
(where it should be provided with the necessary initial conditions).
These laws work so fine that all the celestial and terrestrial motions are perfectly described using them.
But, it continues to be a matter of curiosity to ask questions such as "why the particle follows Newton's laws of motion?"
For example, In the general motion of a classical particle,
"why doesn't the particle take any other path, other than the prescribed path given by Newton's laws of motion?"
Instead of directly going from an initial point A to some final point B, "why doesn't it take an infinite path all around the universe and come to point B ?".
These questions asked by some of the greatest Physicists led to the so called Action principle.
It states that in any classical motion, there is a quantity defined as the "action" which is always either a maximum or a minimum.
Though it may seem an abstract concept at the first glance, it has a pure logical reasoning.
For example, if it is observed the particle to follow a path with some finite value of action other than the extremum value, the question then arises, "why this path and not the nearby path with a different value of action".
This question applies symmetrically to everywhere, which affects all other paths nearby to the original considered path. As a consequence it is inferred that, the resultant path can never be an ordinary one like others but it should have some unique features that always distinguishes it from others. And thus it is arrived at the extremum paths, the only one that follows these unique features.
The significance of the extremum paths can be illustrated from a famous quote by Euler, (the English translation)
"For since the fabric of the Universe is most perfect and the work of a most wise creator, nothing at all takes place in the Universe in which some rule of maximum or minimum does not appear"
So far, it is simplified the fundamental question from "why not all other paths" to "why the extremum path". But it is left with a new quantity named "Action" that needs to be defined and quantified in a mathematical form.
For this, it is begun with the Variational principle that states that, the motion of a system from time $t_1$ to time $t_2$ will be such that the line integral "S" called action or action integral has stationary value for the actual path of the motion. And Action "S" is defined as
\[
S = \int_{t_1}^{t_2} L \,dt \tag{1.2}
\]
The function $L$ which will be later called Lagrangian, is a function of position and velocity and it can also be an explicit function of time,
\[ L = L(x(t),\dot{x}(t),t) \tag{1.3}\]
It is normally used $x(t), \dot{x}(t)$ to denote the generalized position and generalized velocity.
By the term "stationary value" for a line integral, it is meant that the integral along the given path has the same value to within first-order infinitesimals as that along all neighbouring paths. The notion of a stationary value for a line integral thus corresponds in ordinary function theory to the vanishing of the first derivative.
The statement is equivalent to saying that the action should be an extremum for the actual path. Out of all possible paths by which the system point in configuration space could travel from its position at time $t_1$ to its position at time $t_2$, it will actually travel along the path where the action integral is an extremum or has a stationary line integral.
Euler Lagrange Equations:
From the definition, the action integral gives some definite value for every possible path, out of which the maximum value will correspond to the actual path. So, the general path is taken as,
\[x(t,\alpha) = x(t,0) + \alpha\eta(t)\tag{1.4}\]
where $\eta(t)$ is an arbitrary function defined to create an arbitrary variation to the initial path. It is well behaved, continuous in first and second derivatives and non-singular between $t_1$ and $t_2$.
Therefore, any path $x(t,\alpha)$can be expressed from $x(t,0)$ as the variation due to $\eta(t)$.
This variation is chosen such that it should vanish at the end points $t_1$ and $t_2$. i.e.
$\eta(t) = \frac{\partial{x}}{\partial{\alpha}}$ and
\[\eta(t_1)=\eta(t_2) = 0\]
The action for the general path is,
\[S(\alpha) = \int_{t_1}^{t_2} L \,dt = \int_{t_1}^{t_2}L(x(t,\alpha),\dot{x}(t,\alpha),t)\,dt \tag{1.5}\]
The condition for this quantity to have a stationary value is that,
\[ \left.\frac{\partial{S}}{\partial\alpha} \right\vert_{\alpha=0} = 0\,\,\,\tag{1.6}\]
Substituting for S and differentiating under the integral sign,
\[\frac{\partial{S}}{\partial\alpha} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}\frac{\partial{x}}{\partial\alpha} + \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{\dot{x}}}{\partial\alpha} \right)\,dt \,\,\,\tag{1.7}\]
(time is an independent parameter).
The last term (using integral by parts),
\[\int_{t_1}^{t_2} \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{\dot{x}}}{\partial\alpha} \,dt = \int_{t_1}^{t_2} \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial^2{x}}{\partial\alpha\partial{t}} \,dt \\=\left.\left(\frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{x}}{\partial\alpha}\right)\right\vert_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{\partial{x}}{\partial{\alpha}} \frac{d}{dt}\left(\frac{\partial{L}}{\partial\dot{x}}\right) \,dt \,\,\,\tag{1.8}\]
Using the end point conditions of $\eta(t)$,
\[ \frac{\partial{S}}{\partial\alpha} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right)\right)\frac{\partial{x}}{\partial\alpha}\,dt\,\tag{1.9}\]
Using the arbitrariness of $\eta(t)$, one can choose it as well as a positive quantity through out the domain ($t_1$,$t_2$). But, the condition must hold true for all types of $\eta(t)$ which implies that the other integrand should always be zero. Thus it gives,
\[ \frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right) = 0 \,\,\tag{1.10}\] which is known as the Euler-Lagrange equation. Using variational differentials notation,
\[\frac{\partial{x}}{\partial\alpha} d\alpha = \delta{x} \,\tag{1..11}\]
\[\frac{\partial{S}}{\partial\alpha} d\alpha = \delta{S} \,\tag{1.12}\]
\[\delta{S} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right)\right)\,dx\,dt = 0 \,\tag{1.13}\]
Thus, for every problem in classical mechanics there exists a function called Lagrangian from which it can be extracted everything that needs to be known about the system using Euler-Lagrange equations. It is completely equivalent to Newton's laws of motion except for the fact that there is no need to deal with the force explicitly. Instead, together with the concept of path and action, the new Lagrangian plays the most fundamental role.
Therefore in all the classical problems, determining this Lagrangian for a specific problem is the only task one needs to accomplish.
\[
{\vec{F}=m\vec{a}}\tag{1.1}
\]
(where it should be provided with the necessary initial conditions).
These laws work so fine that all the celestial and terrestrial motions are perfectly described using them.
But, it continues to be a matter of curiosity to ask questions such as "why the particle follows Newton's laws of motion?"
For example, In the general motion of a classical particle,
"why doesn't the particle take any other path, other than the prescribed path given by Newton's laws of motion?"
Instead of directly going from an initial point A to some final point B, "why doesn't it take an infinite path all around the universe and come to point B ?".
These questions asked by some of the greatest Physicists led to the so called Action principle.
It states that in any classical motion, there is a quantity defined as the "action" which is always either a maximum or a minimum.
Though it may seem an abstract concept at the first glance, it has a pure logical reasoning.
For example, if it is observed the particle to follow a path with some finite value of action other than the extremum value, the question then arises, "why this path and not the nearby path with a different value of action".
This question applies symmetrically to everywhere, which affects all other paths nearby to the original considered path. As a consequence it is inferred that, the resultant path can never be an ordinary one like others but it should have some unique features that always distinguishes it from others. And thus it is arrived at the extremum paths, the only one that follows these unique features.
The significance of the extremum paths can be illustrated from a famous quote by Euler, (the English translation)
"For since the fabric of the Universe is most perfect and the work of a most wise creator, nothing at all takes place in the Universe in which some rule of maximum or minimum does not appear"
So far, it is simplified the fundamental question from "why not all other paths" to "why the extremum path". But it is left with a new quantity named "Action" that needs to be defined and quantified in a mathematical form.
For this, it is begun with the Variational principle that states that, the motion of a system from time $t_1$ to time $t_2$ will be such that the line integral "S" called action or action integral has stationary value for the actual path of the motion. And Action "S" is defined as
\[
S = \int_{t_1}^{t_2} L \,dt \tag{1.2}
\]
The function $L$ which will be later called Lagrangian, is a function of position and velocity and it can also be an explicit function of time,
\[ L = L(x(t),\dot{x}(t),t) \tag{1.3}\]
It is normally used $x(t), \dot{x}(t)$ to denote the generalized position and generalized velocity.
By the term "stationary value" for a line integral, it is meant that the integral along the given path has the same value to within first-order infinitesimals as that along all neighbouring paths. The notion of a stationary value for a line integral thus corresponds in ordinary function theory to the vanishing of the first derivative.
The statement is equivalent to saying that the action should be an extremum for the actual path. Out of all possible paths by which the system point in configuration space could travel from its position at time $t_1$ to its position at time $t_2$, it will actually travel along the path where the action integral is an extremum or has a stationary line integral.
Euler Lagrange Equations:
From the definition, the action integral gives some definite value for every possible path, out of which the maximum value will correspond to the actual path. So, the general path is taken as,
\[x(t,\alpha) = x(t,0) + \alpha\eta(t)\tag{1.4}\]
where $\eta(t)$ is an arbitrary function defined to create an arbitrary variation to the initial path. It is well behaved, continuous in first and second derivatives and non-singular between $t_1$ and $t_2$.
Therefore, any path $x(t,\alpha)$can be expressed from $x(t,0)$ as the variation due to $\eta(t)$.
This variation is chosen such that it should vanish at the end points $t_1$ and $t_2$. i.e.
$\eta(t) = \frac{\partial{x}}{\partial{\alpha}}$ and
\[\eta(t_1)=\eta(t_2) = 0\]
The action for the general path is,
\[S(\alpha) = \int_{t_1}^{t_2} L \,dt = \int_{t_1}^{t_2}L(x(t,\alpha),\dot{x}(t,\alpha),t)\,dt \tag{1.5}\]
The condition for this quantity to have a stationary value is that,
\[ \left.\frac{\partial{S}}{\partial\alpha} \right\vert_{\alpha=0} = 0\,\,\,\tag{1.6}\]
Substituting for S and differentiating under the integral sign,
\[\frac{\partial{S}}{\partial\alpha} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}\frac{\partial{x}}{\partial\alpha} + \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{\dot{x}}}{\partial\alpha} \right)\,dt \,\,\,\tag{1.7}\]
(time is an independent parameter).
The last term (using integral by parts),
\[\int_{t_1}^{t_2} \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{\dot{x}}}{\partial\alpha} \,dt = \int_{t_1}^{t_2} \frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial^2{x}}{\partial\alpha\partial{t}} \,dt \\=\left.\left(\frac{\partial{L}}{\partial{\dot{x}}} \frac{\partial{x}}{\partial\alpha}\right)\right\vert_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{\partial{x}}{\partial{\alpha}} \frac{d}{dt}\left(\frac{\partial{L}}{\partial\dot{x}}\right) \,dt \,\,\,\tag{1.8}\]
Using the end point conditions of $\eta(t)$,
\[ \frac{\partial{S}}{\partial\alpha} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right)\right)\frac{\partial{x}}{\partial\alpha}\,dt\,\tag{1.9}\]
Using the arbitrariness of $\eta(t)$, one can choose it as well as a positive quantity through out the domain ($t_1$,$t_2$). But, the condition must hold true for all types of $\eta(t)$ which implies that the other integrand should always be zero. Thus it gives,
\[ \frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right) = 0 \,\,\tag{1.10}\] which is known as the Euler-Lagrange equation. Using variational differentials notation,
\[\frac{\partial{x}}{\partial\alpha} d\alpha = \delta{x} \,\tag{1..11}\]
\[\frac{\partial{S}}{\partial\alpha} d\alpha = \delta{S} \,\tag{1.12}\]
\[\delta{S} = \int_{t_1}^{t_2} \left(\frac{\partial{L}}{\partial{x}}-\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{x}}}\right)\right)\,dx\,dt = 0 \,\tag{1.13}\]
Thus, for every problem in classical mechanics there exists a function called Lagrangian from which it can be extracted everything that needs to be known about the system using Euler-Lagrange equations. It is completely equivalent to Newton's laws of motion except for the fact that there is no need to deal with the force explicitly. Instead, together with the concept of path and action, the new Lagrangian plays the most fundamental role.
Therefore in all the classical problems, determining this Lagrangian for a specific problem is the only task one needs to accomplish.
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