— 1. Variational Calculus —
Definition 1 A functional is a mapping from vector spaces to into real numbers. |
Let . Suppose that and are constants, the functional form of is known.
According to definition 1 is a functional and the goal of the Calculus of Variations is to determine such that the value of is an extremum.
Let be a parametric representation of such that is the function that makes an extremum.
We can write , where is a function of of the class (that means that is a continuous function whose derivative is also continuous) with .
Now is of the form
Therefore the condition for to be an extremum is
Exercise 1 Given the points and , calculate the equation of the curve that minimizes the distance between the points
Now . It is , And it is with . The rest is left as an exercise for the reader. |
— 2. Euler Equations —
In the following section we’ll analyze the condition for to be an extremum:
Since it is and it follows
Now .
For the first term it is since by hypothesis.
Hence
Remembering that and taking into account the fact that is an arbitrary function one can conclude that
The previous equation is known as the Euler’s Equation
To close our thoughts on the Euler equation let us say that there also is a second form for the Euler equation. The second form is
and is used in the cases where doesn’t depend explicitly on .
— 3. Euler Equation for variables —
Let be of the form .
Now we have and for each of the values of . Since are independent functions it follows that for
That is to say we have independent Euler equations.
— 4. Hamilton’s Principle —
Minimum principles have a long history in the history of Physics:
- Heron explained the law of light reflection using a principle of minimum time.
- Fermat corrected Heron’s Principle by stating that light travels between two points in the shortest time available.
- Maupertuis stated his minimum action principle that postulated that a particle’s dynamics always minimized the action.
- Gauss postulated his principle of least constraint.
- Hertz postulaed his principle of minimum curvature.
In modern Physics one uses a more general extremum principle and the focus of this section will be to state this principle and flesh out its consequences.
Definition 2 The lagrangian (sometimes called the lagrangian function), , of a particle is the difference between its kinetic and potential energies. |
Definition 3 The action, , of a particle’s movement (be it a real or virtual one) is: |
Axiom 1 Given a collection of paths that a particle can take between points and in the the time interval the actual path that the particle takes is the one that makes the action stationary |
For rectangular coordinates it is , , so (where is called Newton’s notation).
The function can be identified with the function that we saw on Newtonian Mechanics 04 if one makes the obvious analogies
In this case the Euler equations are called the Euler-Lagrange equations and it is
Example 3 Let us study the harmonic oscillator under Langrangian formalism
First it is . Then we have . Hence it is which is just the harmonic oscillator dynamic equation that we already know. |
Example 4 Consider a planar pendulumplanar pendulum write its Lagrangian and derive its equation of motion.
The Lagrangian for the planar pendulum is If we consider to be a rectangular coordinate (which it isn’t!) it follows that the equation of motion is: This is precisely the equation of motion of a planar pendulum and this result is apparently unexpected since we only analyzed the Lagrangian for rectangular coordinates. |
— 5. Generalized coordinates —
Consider a mechanical system constituted by particles. In this case one would need quantities to describe the position of all particles (since we have 3 degrees of freedom). In the case of having any kind of restraints on the motion of the particles the number of quantities needed to describe the motion of particle is less than . Suppose that one has restrictions than the degrees of freedom are .
Let . These coordinates don’t need to be rectangular, polar, cylindrical nor spherical. These coordinates can be of any kind provided that they completely specify the mechanical state of the system.
Definition 4 The set of coordinates that totally specify the mechanical state of particles is defined to be the set of generalized coordinates.
The generalized coordinates are represented by |
Since we defined the generalized coordinates of a system of particles one can also define its set of generalized velocities.
Let denote the particle, , represent the degrees of freedom , and the number generalized coordinates .
For the generalized velocities it is
The inverse transformations are
and
Finally let us note that we also need equations of constraint
with .
Definition 6 Configuration space is the vector space defined by the generalized coordinates |
The time evolution of a mechanical system can be represented as a curve in the configuration space.
— 6. Euler-Lagrange Equations in generalized coordinates —
Since and are scalar functions is also a scalar function. Therefore is an invariant for coordinate transformations.
Hence it is
and .
Hence we can write Hamilton’s Principle (Section 4) in the form
That is
are the analogies to be made now.
Finally the Euler-Lagrange Equations are
for
To finalize this section let us note the conditions of validity for the Euler-Lagrange equations:
- The system is conservative.
- The equations of constraint have to be functions between the coordinates of the particles and can also be a function of time.