Physics+ N-dimensional Lagrangian: UNIZOR.COM - Physics+ 4 All - Lagrangian

Notes to a video lecture on UNIZOR.COM

Lagrangian
for N-dimensional Systems

In one of the previous lectures we considered a point-mass object oscillating on a spring in an empty one-dimensional space with Cartesian coordinates (one degree of freedom).

In that case we defined a Lagrangian of an object as a difference between its kinetic and potential energies and built a theory equivalent to Newtonian but based on the Euler-Lagrange equation instead of the Newton's Second Law.

In another lecture we discussed a spring pendulum (two degrees of freedom) and used non-Cartesian parameters to define a state of a system - an angle of pendulum from a vertical and a spring's length.
It was possible but rather complicated to use the Newtonian approach, so we applied Lagrangian Mechanics to come up with the differential equations to describe a state of this system.

An important reason for using energies instead of vectors of force and acceleration in that second example was that energies are scalars, while forces and accelerations are vectors.
Manipulations with scalars are simpler, especially dealing with complex systems with more than one degree of freedom.

Obviously, no matter how we solve a problem, the calculated actual physical trajectories of objects in space must be the same.

Before proceeding any further, we strongly recommend to refresh your knowledge about conservative forces, independence of work performed by these forces from a trajectory of an object moved by them, a concept of a field and its potential.
The chapter Laws of Newton of this course is a good source of this information.

Recall that conservative forces are defined as those that depend only on position in space, independent of time. Their work, performed by moving an object from one position to another, does not depend on trajectory or speed along this trajectory, but depends only on the beginning and ending positions of an object.

From the Energy Conservation Law follows that the work performed by a conservative force that moves an object changes the potential energy of this object by the amount of work performed.

Recall that an increment of potential energy
ΔE_pot=E_pot(P₂)−E_pot(P₁)
of an object moved by a conservative force F (a force of a field, a force of a spring etc.) from position P₁ to P₂ along any trajectory equals to the work performed by this conservative force.
ΔE_pot = ∫_[P1~P2]F(P)·dr
where P is a variable position of an object moving along a trajectory from P₁ to P₂,
r=OP is a vector from the origin of coordinates O to a position of an object P as it moves along a trajectory,
multiplication F(P)·dr is a scalar product of two vectors - a vector of force and an infinitesimal vector of increment of a position of an object along its trajectory and
[P₁~P₂] denotes that an integral is taken along a trajectory from P₁ to P₂.

The same conservative force (a force of a field, a force of a spring etc.) can be represented as a vector of gradient of a potential
F = −∇E_pot
where a minus sign '−' indicates that the conservative force (a force of a field, a force of a spring etc.) is always directed towards decreasing of potential.

Lagrangian Mechanics was invented to simplify analysis of complex systems acted upon by conservative forces, like electrostatic, gravitational or spring forces.

We are going to prove that Lagrangian Mechanics of a closed (no external forces) mechanical system with its components acted among themselves with some conservative forces produces differential equations of motion that are equivalent to Newtonian equations, but easier to deal with.

Consider a system that consists of N point-masses with each i^th component Ω_i acted upon by conservative forces (three-dimensional vectors) from all components of this system with a combined force
F_i(r₁,...r_N)
that depends on positions of all components of a system.

The time-dependent Cartesian coordinates of each Ω_i are {x_i(t),y_i(t),z_i(t)}, which we will denote as a vector r_i(t).

The Newton's Second Law of motion for each object in a system is, therefore,
F_i(r₁(t),...r_N(t)) = m_i·r"_i(t)
where each F_i is a vector of three components {F_ix,F_iy,F_iz}, and each component is a function of 3N coordinates of all objects in a system,
mass m_i is a mass of Ω_i and
a symbol " means the second derivative of position vector r_i(t) of object Ω_i by time, that is, its vector of acceleration {x"_i(t),y"_i(t),z"_i(t)}.

In coordinate form the above equation can be written as
F_ix(r₁(t),...r_N(t)) = m_i·x"_i(t)
F_iy(r₁(t),...r_N(t)) = m_i·y"_i(t)
F_iz(r₁(t),...r_N(t)) = m_i·z"_i(t)

In summary, we have 3N differential equations of 2-nd order, three for each component of a system of N objects.

In case of multiple objects in three-dimensional space exerting forces on each other (like all the planets of our Solar system or a nucleus with all its electrons in an atom) the vectors of forces are directed in different directions and the system of differential equations based on Newton's Second law is extremely complex.

Let's approach it differently.
Since all forces F_i are conservative, each one can be represented as
F_i(r₁(t),...r_N(t)) =
= −∇_iU(r₁(t),...r_N(t))
where U(r₁(t),...r_N(t)) is a total potential of an entire system and
∇_i is a vector of its partial derivatives by each coordinate of object Ω_i.
∇_i = {∂/∂x_i,∂/∂y_i,∂/∂z_i}.

Let's shorten for convenience the results above as
F_i = −∇_iU
and rewrite it in (x,y,z) components of Cartesian coordinates
F_ix = −∂U/∂x_i
F_iy = −∂U/∂y_i
F_iz = −∂U/∂z_i

As we see, knowing the potential of a system of objects at each point is sufficient to know all the conservative forces acted on individual objects in this system.

Let's address the m_i·r"_i(t) side of the Newton's Second Law and derive its value from the kinetic energy of an object Ω_i.
We express a vector r"_i(t) in (x,y,z) coordinates as
(x"_i(t),y"_i(t),z"_i(t))

The Newton's Second Law equations for object Ω_i in coordinate form would then be
F_ix = m_i·x"_i(t)
F_iy = m_i·y"_i(t)
F_iz = m_i·z"_i(t)
Using the potential energy, the same equations for object Ω_i would be
−∂U /∂x = m_i·x"_i(t)
−∂U /∂y = m_i·y"_i(t)
−∂U /∂z = m_i·z"_i(t)

Kinetic energy of Ω_i equals to K_i(t)=½m_i·v_i(t)² where v_i is a linear speed along a trajectory
v_i(t)² = r'_i(t)·r'_i(t) =
= x'_i(t)²+y'_i(t)²+z'_i(t)²

Now we can express the components of an acceleration vector r"_i(t) of Ω_i in terms of its kinetic energy
K_i(t)=½m_i·[x'_i(t)²+y'_i(t)²+z'_i(t)²]

Partial derivatives of kinetic energy by each coordinate of velocity vector produces coordinates of an object's momentum
∂K_i /∂x'_i = m·x'_i
∂K_i /∂y'_i = m·y'_i
∂K_i /∂z'_i = m·z'_i
Derivative by time of a momentum gives the right side of the Newton's Second Law d/dt ∂K_i /∂x'_i=d/dt m·x'_i=m·x"_i
d/dt ∂K_i /∂y'_i=d/dt m·y'_i=m·y"_i
d/dt ∂K_i /∂z'_i=d/dt m·z'_i=m·z"_i

We are ready to express the Newton's Second Law in terms of kinetic and potential energy.
From
F_ix = −∂U/∂x_i
and
d/dt ∂K_i /∂x'_i = m·x"_i
follows
−∂U/∂x_i = d/dt ∂K_i /∂x'_i
Similarly,
−∂U /∂y_i = d/dt ∂K_i /∂y'_i
−∂U/∂z_i = d/dt ∂K_i /∂z'_i

The total kinetic energy of a system is a sum of kinetic energies of its components
K = K₁+...+K_N
Since each K_i depends only on a velocity of the i^th object {x'_i(t),y'_i(t),z'_i(t)}, partial derivative of K_i by x'_i(t), y'_i(t) or by z'_i(t) is the same as partial derivatives of an entire kinetic energy of a system K by the same components x'_i(t), y'_i(t) or z'_i(t) of the velocity of the i^th object
∂K_i /∂x'_i = ∂K/∂x'_i
∂K_i /∂y'_i = ∂K/∂y'_i
∂K_i /∂z'_i = ∂K/∂z'_i

Using total kinetic energy of a system, the formulas describing the laws of motion of object Ω_i would be
−∂U/∂x_i = d/dt ∂K/∂x'_i
−∂U/∂y_i = d/dt ∂K/∂y'_i
−∂U/∂z_i = d/dt ∂K/∂z'_i
Now the only participants in these equations are the total kinetic and potential energies of an entire system - just two numbers that depend on positions and velocities of system's components.

To make the theory more elegant, let's introduce a Lagrangian L=K−U that equals to a difference between kinetic and potential energy of this system.

Since kinetic energy of a system K is independent of positions of its components {x_i(t),y_i(t),z_i(t)},
−∂U/∂x_i=∂(K−U)/∂x_i=∂L/∂x_i
and similar with partial derivatives by y_i and z_i.

Since potential energy of a system U is independent on velocities of its components,
∂K/∂x'_i=∂(K−U)/∂x'_i=∂L/∂x'_i
and similar with partial derivatives by y_i and z_i.

Therefore, our equations look even simpler
∂L/∂x_i = d/dt ∂L/∂x'_i
∂L/∂y_i = d/dt ∂L/∂y'_i
∂L/∂z_i = d/dt ∂L/∂z'_i

The equations above are also differential equations of the second order, like withNewton's Second Law.
There are also 3N of these equations (three coordinates for N objects in a system).
But there is only one number, Lagrangian, also called action, a function of all positions and velocities, to deal with for all objects instead of individual forces for each object.
Lagrangian Mechanics allows to deal with complex system in a simpler way.

As the cherry on top, consider the same equations in the form
d/dt ∂L/∂x'_i − ∂L/∂x_i = 0
d/dt ∂L/∂y'_i − ∂L/∂y_i = 0
d/dt ∂L/∂z'_i − ∂L/∂z_i = 0
In these equations Lagrangian L depends on 3N parameters, positions and velocities of N components of our system, which are, in turn, are time-dependent.
So, ultimately, Lagrangian L is a function of time.

On one hand, these are equations that define a motion of a system, that is they define a trajectory of a system changing positions and velocities of its components from one moment in time to another.

On another hand, as we discussed in the Variations chapter of this course, they produce a function of time L(t) that brings to extremum (usually, minimum) the action functional
S = ∫_[t1,t2] L(t)·dt

Therefore, we can say that a mechanical system with only conservative forces present changes its state along an N-dimensional trajectory that minimizes the action functional above.