Action Principle: UNIZOR.COM - Physics+ 4 All - Lagrangian

Notes to a video lecture on UNIZOR.COM

From Newton to the Action Principle
Logical Foundations of Lagrangian Mechanics

The purpose of this lecture is to highlight the core principles of Lagrangian Mechanics, marking the transition from the Newtonian model, which is strongly tied to Cartesian coordinates, to a concept of a trajectory of a motion as an extremum of the action functional.

1. Motion is Objective Reality
Consider a conservative ⁽¹⁾ system of N point-mass components in three-dimensional space each moving along its path with all these paths together as a set representing the whole system's trajectory. Within classical non-relativistic mechanics and inertial frames this trajectory represents an objective physical reality, regardless of how we view this motion, what instruments we use to observe it or what system of coordinates we prefer.
We will use the word “trajectory” in three closely related senses: (i) the physical paths traced by system components in three-dimensional space; (ii) the collection of these paths forming the system’s physical trajectory; and (iii) a curve in configuration space representing this motion mathematically. The first two are objective physical realities; the third depends on the chosen coordinates.

2. Physical Reality vs. Mathematical Representation
The same physical reality can be mathematically represented in different ways by using different coordinate systems, depending on our choice.
We will only consider purely geometrical time-independent transformation of coordinates.
Assume, we describe a position of all components of a system using 3N-dimensional configuration space by time-dependent parameters s(t)={s₁(t),...,s_n(t)} where n≡3N.
Let's start with Cartesian coordinates in an inertial reference frame in 3D Euclidean space.
Consider a time-independent smooth one-to-one transformation of coordinates
q_i = Q_i(s) where i∈[1,n]
Now we can describe the trajectory of our mechanical system in terms of new coordinates q(t)={q₁(t),...,q_n(t)}.
These two different coordinate systems, mathematically representing the whole system's trajectory, would present different sets of n coordinate functions of time to define positions of all system's components, but they describe the same physical trajectory of a system.
When we use the word trajectory, we refer to both physical traces in 3D space of all N point-masses composing a mechanical system (objective physical reality independent of our choice to represent it mathematically) and their mathematical representation as a set of n time-dependent functions related to a particular configuration space.
Sometimes, to differentiate between these two meaning, we will use physical trajectory (N traces in 3D space, coordinate-independent physical entity) or mathematical trajectory (a set of n time-dependent coordinate functions).
Not only a trajectory, but also other coordinate-independent physical quantities of classical mechanics, like kinetic energy, can be mathematically represented in different form depending on our choice of a system of coordinates. But our mathematical calculations of any objective physical characteristic of an object traveling along its trajectory at any moment of time must be the same in any two systems of coordinates transformable into each other via some set of time-independent smooth transformation functions.
Mathematically, this follows from the fact that kinetic energy is a quadratic form induced by the Euclidean metric on configuration space, and smooth coordinate changes preserve its scalar value.
For example, consider Cartesian (x,y) and polar (r,θ) coordinates on a plane. They are mutually transformable into each other as follows.
(r,θ) → (x,y):
x = r·cos(θ)
y = r·sin(θ)
(x,y) → (r,θ):
r = √x²+y²
Transformation for θ is different in different quarters:
θ = arctan(y/x) for x > 0
θ = arctan(y/x) + π for x < 0, y ≥ 0
θ = arctan(y/x) − π for x < 0, y < 0
θ = π/2 for x = 0, y > 0
θ = −π/2 for x = 0, y < 0
θ is undefined for x = 0, y=0
A uniform circular motion of some point-mass on a two-dimensional plane can be represented in Cartesian coordinates as
x(t) = R·cos(ω·t)
y(t) = R·sin(ω·t)
where R is the radius of an orbit and ω is the angular speed of rotation.
The same trajectory can be represented on that two-dimensional plane in polar coordinates as
r(t) = R
θ(t) = ω·t
These two representations of a circular motion as formulas look totally different but describe the same physical trajectory.

3. Newton's Second Law is Tied to Cartesian Coordinates
The structure of equations of motion based on the Newton's Second law explicitly depends on time-dependent Cartesian coordinates.
Consider a system of only one component and Cartesian coordinates {x(t),y(t),z(t)}.
A vector of force F(t) acting on this object has three Cartesian coordinates
F(t) = {F_x(t),F_y(t),F_z(t)}
The Newton's Second law relates this force to a vector of acceleration
a(t)={x"(t),y"(t),z"(t)}
(a second derivative of a position by time)
F_x(t) = m·x"(t)
F_y(t) = m·y"(t)
F_z(t) = m·z"(t)
The differential equations of motion above mathematically represent a motion in Cartesian coordinates. This is a requirement for using the Newton's Second law.
In a general case a system of N point-mass components in three-dimensional space the trajectory of a system is described by n=3N equations
F_kx(t) = m_k·x_k"(t)
F_ky(t) = m_k·y_k"(t)
F_kz(t) = m_k·z_k"(t)
for k∈[1,N]
or, using uniform Cartesian coordinates ⁽³⁾
s(t)={s₁(t),...,s_n(t)}
instead of classical
{x₁(t),y₁(t),...,x_N(t),y_N(t),z_N(t)},
F_i(t) = m_i·s_i"(t) for i∈[1,n] where n=3N.

4. Euler-Lagrange Equation in Cartesian Coordinates ≡ Newton's Second Law
We have analytically proven in previous lectures ⁽⁴⁾ that the Euler-Lagrange equations
d/dt ∂L/∂s'_i − ∂L/∂s_i = 0
for all i∈[1,n]
with a Lagrangian L(s(t),s'(t)) of a mechanical system expressed in Cartesian coordinates are equivalent to Newton's Second law equations above in the same Cartesian coordinates.
Their equivalency means that their solution, a physical trajectory of a mechanical system as a function of time, is the same. In other words, if trajectory s_NS(t) is a solution to Newton's Second law equations in Cartesian coordinates and trajectory s_EL(t) is a solution to Euler-Lagrange equations in the same Cartesian coordinates with the same initial conditions, then
s_NSi(t) = s_ELi(t) for all i∈[1,n]
It should be noted, however, that Newton’s Second Law equates vectors and therefore requires a coordinate-dependent notion of direction, while Lagrangian mechanics is formulated entirely in terms of scalar quantities, which naturally survive coordinate transformations.

5. Lagrangian is Invariant to Coordinate Transformation
Kinetic K and potential U energies of a system, as physical quantities evaluated for each moment of time for a system moving along its trajectory, do not change under smooth, one-to-one, time-independent coordinate transformations.
Therefore, Lagrangian L=K−U represents a scalar physical quantity, whose numerical value (not a mathematical formula) along a given trajectory at a fixed moment of time is invariant within a scope of coordinate transformation delineated above.
Lagrangian's mathematical representation is based on our choice of coordinates. It is expressed in terms of time-dependent coordinate functions s(t)={s₁(t),...,s_n(t)} describing an object's positions and time derivatives of these functions s'(t)={s₁'(t),...,s_n'(t)} describing an object's velocities:
L(s(t),s'(t)) =
= K(s(t),s'(t)) − U(s(t),s'(t))
Consider two inertial frames with two systems of coordinates s (assume, it's Cartesian) and q (called generalized) transformable into each other by smooth, one-to-one, time-independent coordinate transformations.
Coordinate system s describes a trajectory as s(t)={s₁(t),...,s_n(t)}, and Lagrangian in this system looks like L_s(s(t),s'(t)), while coordinate system q describes this same trajectory as q(t)={q₁(t),...,q_n(t)}, and Lagrangian in this system looks like L_q(q(t),q'(t)).
Expressions L_s and L_q, as formulas of their arguments s(t), s'(t) and q(t), q'(t), look differently.
But for the same physical trajectory and time the calculated numerical values of these two Lagrangians are the same because they represent a physical quantity specific for a trajectory and system's movement along it.
In more general case, when transformation between coordinates is time-dependent or potential energy depends on velocity, the Lagrangian, strictly speaking, is not an invariant to transformations of coordinate system and might not be a scalar; however, in the scope of this presentation for conservative systems under time-independent smooth one-to-one coordinate transformations, its numerical value (not a coordinates-dependent formula) evaluated along a given physical trajectory at a fixed time is invariant.
For example, kinetic energy in Cartesian coordinates (x,y) depends on a magnitude of a velocity (x',y') (we use apostrophe to indicate a derivative by time and, for brevity, we omit time-dependency (t) of coordinates and velocities)
K_C = ½m·(x'²+y'²)
In polar coordinates (r,θ) the velocity vector in projection to radial and tangential axes is (r',rθ'), so the kinetic energy is
K_P = ½m·(r'²+r²·θ'²)
As we see, K_C and K_P, as formulas of their arguments, look differently.
To represent the same point in space at the same time, Cartesian and polar coordinates must be related by transformation functions.
The coordinate transformation from polar to Cartesian coordinates is
x = r·cos(θ)
y = r·sin(θ)
This transforms K_C as follows:
x' = r'·cos(θ)−r·sin(θ)·θ'
y' = r'·sin(θ)+r·cos(θ)·θ'
x'² = r'²·cos²(θ) + r²·sin²(θ)·θ'² −
− 2r'·cos(θ)·r·sin(θ)·θ'
y'² = r'²·sin²(θ) + r²·cos²(θ)·θ'² +
+ 2r'·sin(θ)·r·cos(θ)·θ'
Since sin²(θ)+cos²(θ)=1
x'² + y'² = r'² + r²θ'²
Therefore, K_C = K_P
The calculated numerical value of the kinetic energy, as we see, is the same in both Cartesian and polar coordinates.
In general, any real physical characteristic of a mechanical system, including numeric value of kinetic and potential energies and, consequently, numeric value of Lagrangian, evaluated along a given physical trajectory for any particular moment in time is an invariant relative to smooth one-to-one time-independent transformations of coordinates.

6. Action Extremals are Invariant to Coordinate Transformation
Consider now all possible trajectories of a mechanical system that moves from position A in space at time t₁ to position B at time t₂.
Mathematically, one trajectory might be represented in Cartesian coordinates by
s⁽¹⁾(t)={s⁽¹⁾₁(t),...,s⁽¹⁾_n(t)}
with s⁽¹⁾(t₁) = A and s⁽¹⁾(t₂) = B
Another trajectory might be
s⁽²⁾(t)={s⁽²⁾₁(t),...,s⁽²⁾_n(t)}
with s⁽²⁾(t₁) = A and s⁽²⁾(t₂) = B
Consider now an action functional ⁽⁵⁾ in these Cartesian coordinates
∫_[t1,t2] L(s(t),s'(t))·dt
(where s(t)={s₁(t),...,s_n(t)}).
It is an integral of Lagrangian L(s(t),s'(t)) along a trajectory by time.
Action functional will have some (generally speaking , different) value on each of these trajectories. On some of these trajectories this functional will have an extremum in a sense described in the Theory of Variations ⁽⁶⁾ (in many practical cases, but not generally, a particular trajectory would bring this functional to its absolute minimum).

As we stated above, numerical value of Lagrangian, as a physical characteristic, is defined by the dynamics of movement, not by a system of coordinates. So, for an object moving along its trajectory it has a value independent of the described above transformations of coordinates.
Consequently, the value of the action functional evaluated along any trajectory is an invariant to these transformations. The set of action's extremal trajectories is, therefore, also an invariant to transformations of coordinates.
Since the coordinate transformation establishes a one-to-one correspondence between admissible trajectories with fixed endpoints, extremality is preserved.
As an analogy, the shortest in meters road from point A to point B is the shortest in feet or miles.

In a way, it's the same logic as when you compare different roads from A to B. If one particular road is the shortest among all roads in meters, it's the shortest in feet or miles.

If a particular trajectory is an action functional's extremal in Cartesian coordinates, it is an extremal in any other coordinate system obtained by time-independent one-to-one smooth transformation of coordinates.
This invariance relies crucially on time-independent coordinate transformations; time-dependent transformations introduce additional terms in the Lagrangian and require separate treatment.

7. Real Trajectory is a Solution to Euler-Lagrange Equation in any System of Coordinates
Consider any extremal trajectory in the variational sense described above.
Let's establish some Cartesian coordinates and consider a mathematical representation of this trajectory s(t)={s₁(t),...,s_n(t)}.
Physical trajectory that is represented by s(t) in Cartesian coordinates is a frame-invariant physical entity, it will bring to extremum an action functional regardless of the way we represent it mathematically.
From the Theory of Variations ⁽⁶⁾ follows that functions s_i(t) that mathematically comprise this trajectory with fixed ends satisfy the Euler-Lagrange equations
d/dt ∂L/∂s'_i − ∂L/∂s_i = 0
for all i∈[1,n]
In the item #4 above we mentioned that in Cartesian coordinates Euler-Lagrange equation produces exactly the same solution as Newton's Second law, that we postulate as the law describing the real motion in Cartesian coordinates.
Therefore, a trajectory s(t) in Cartesian coordinates is
(a) the extremal trajectory for action functional independent of coordinates;
(b) ⇒ the solution to Euler-Lagrange equation in Cartesian coordinates;
(c) ⇒ the solution to Newton's Second law equations;
(d) ⇒ the real trajectory of motion of a mechanical system that moves from position A in space at time t₁ to position B at time t₂.
Hence, the real trajectory of a system's motion has an important property independent of coordinate system - it extremizes the action functional.
To find a real trajectory from A to B in Cartesian coordinates we can use Newton's Second law. But finding a trajectory that extremizes the action functional using Euler-Lagrange equation can be done in any coordinate system.
Different Euler–Lagrange equations written in different coordinate systems are not different laws of motion; they are different mathematical descriptions of the same physical trajectory.

8. Solving Euler-Lagrange Equation in Any Generalized Coordinates Produces a Real Trajectory in These Coordinates
For some reason, to find a mechanical system's trajectory, we might consider it's more convenient to work in non-Cartesian coordinates q(t)={q₁(t),...,q_n(t)} transformable into and from Cartesian by a smooth time-independent one-to-one transformation.
To accomplish that, we should express a Lagrangian in this coordinate system as a function of positions and velocities and solve a system of Euler-Lagrange equations.

In the next lecture we will examine how physical constraints restrict the set of admissible trajectories and how generalized coordinates arise naturally as coordinates on the constraint manifold.

It should be noted, however, that in Lagrangian mechanics the mathematical expression for the Lagrangian, generally speaking, is not unique: adding a total time derivative of an arbitrary smooth function F(q,t) modifies the numerical value of the action by boundary terms depending only on the endpoints. However, it does not change the Euler–Lagrange equations and does not alter the extremal trajectories, which are the solutions to Euler-Lagrange equations and, therefore, are physically equivalent descriptions of the same motion.

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1. See lectures in the chapter UNIZOR.COM → Physics+ 4 All → Laws of Newton
2. See lectures in the chapter UNIZOR.COM → Physics+ 4 All → Laws of Kepler
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