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The Classical Evolution: Why Lagrangian and Hamiltonian Mechanics Replaced Newtonian Mechanics

PHYSICXION: Lagrangian and Hamiltonian mechanics were introduced to overcome the limitations of Newtonian mechanics
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The Classical Evolution: Why Lagrangian and Hamiltonian Mechanics Replaced Newtonian Mechanics 


Classical mechanics, as formulated by Isaac Newton, laid the foundation for understanding motion, forces, and the physical behavior of objects. However, as physics evolved, the limitations of Newtonian mechanics became apparent, especially for complex systems and modern physics applications. To address these challenges, Lagrangian and Hamiltonian mechanics were introduced. These formulations not only simplified the description of physical systems but also revolutionized the way we understand nature.

In this article, we will explore why Lagrangian and Hamiltonian mechanics emerged as powerful alternatives to Newtonian mechanics.

1. Newtonian Mechanics and Its Limitations

Newtonian mechanics is built upon the second law of motion:
F=ma
This equation works well for simple systems where forces and accelerations are easy to calculate. However, it has notable drawbacks:
  • It relies on vector quantities, which can be cumbersome for systems with multiple interacting components.
  • It requires explicit treatment of constraint forces (e.g., forces acting on a pendulum or a bead on a wire).
  • Newtonian mechanics demands the use of Cartesian coordinates, which may not suit systems with complex geometries or symmetries.
These limitations created a need for a more generalized, elegant, and adaptable framework.

2. The Emergence of Lagrangian Mechanics

In the 18th century, Joseph-Louis Lagrange developed an alternative formulation of classical mechanics based on the Principle of Least Action.

What is the Principle of Least Action?

The principle states that a system evolves in such a way that the action (a quantity integrating the difference between kinetic and potential energy over time) is minimized. This is mathematically expressed as:
S=∫t1t2L dt
where  is the Lagrangian, defined as:
L=T−V
  • TT: Kinetic energy
  • VV: Potential energy

Why Lagrangian Mechanics is Superior

  1. Generalized Coordinates: Lagrangian mechanics allows the use of generalized coordinates (e.g., angles, distances) instead of Cartesian coordinates, making it much easier to describe complex systems.
    • Example: For a pendulum, angular displacement (θ\theta) is more natural than x-y coordinates.
  2. Simplification of Constraints: Constraint forces (e.g., forces keeping an object on a surface) are naturally incorporated into the Lagrangian formulation without solving for them explicitly.
  3. Mathematical Elegance: Instead of vector equations, Lagrangian mechanics reduces the problem to solving a single function: the Lagrangian. The resulting Euler-Lagrange equations describe the motion of the system.

3. The Power of Hamiltonian Mechanics

Later, William Rowan Hamilton reformulated classical mechanics in terms of the Hamiltonian function.

What is the Hamiltonian?

The Hamiltonian H represents the total energy of a system:
H=T+V

Hamiltonian mechanics replaces the second-order Euler-Lagrange equations with first-order differential equations using positions 
qiq_i and momenta pip_i:

dqidt=∂H∂pi,dpidt=−∂H∂qi​

Advantages of Hamiltonian Mechanics

  1. Symmetry and Conservation Laws: Hamiltonian mechanics explicitly connects symmetries to conservation laws through Noether's theorem. For example:
    • Translational symmetry → Conservation of momentum
    • Time symmetry → Conservation of energy
  2. Canonical Variables: By introducing positions and momenta as independent variables, Hamiltonian mechanics simplifies complex systems and provides deeper insight into phase space dynamics.
  3. Modern Physics Applications: Hamiltonian mechanics serves as the foundation for:
    • Quantum mechanics (Hamiltonian operator in Schrödinger’s equation)
    • Statistical mechanics (partition functions and phase space)
    • Field theory (generalized Hamiltonian densities in electromagnetism and particle physics).

4. Applications in Modern Physics

While Newtonian mechanics works well for simple problems, Lagrangian and Hamiltonian mechanics are indispensable in modern physics. Some key applications include:
  • Quantum Mechanics: The Hamiltonian function becomes the Hamiltonian operator, central to Schrödinger's equation.
  • General Relativity: Einstein’s field equations are derived using the variational principles similar to Lagrangian mechanics.
  • Quantum Field Theory: Fields, such as the electromagnetic field, are described using Lagrangian densities.
By replacing forces with energies and action principles, these formulations enable physicists to describe systems ranging from planetary motion to subatomic particles.

5. Symmetry and the Beauty of Nature

One of the most profound advantages of Lagrangian and Hamiltonian mechanics is their connection to symmetry and conservation laws. This relationship is mathematically formalized by Noether's theorem, which states that:

“Every symmetry of a physical system corresponds to a conservation law.”
For example:
  • Time invariance → Conservation of energy
  • Spatial invariance → Conservation of momentum
  • Rotational invariance → Conservation of angular momentum
Such insights are not apparent in Newtonian mechanics but arise naturally in the Lagrangian and Hamiltonian frameworks.


Problem Setup: Planetary Motion in a Central Force Field


Let's take a clear mathematical example of why Hamiltonian mechanics is often more advantageous than Newtonian mechanics when solving astronomical problems, such as describing the motion of a planet orbiting the Sun under the influence of gravity. Here just a simple two-dimensional circular model is considered to describe the utility of Lagrangian and Hamiltonian mechanics. In reality, the scenario is more complex and orbits are not purely circular, so a simplified model is described mathematically. 

Newtonian Mechanics Approach

In Newtonian mechanics, we start with the force equation derived from Newton's second law:
F=ma
For a planet of mass m orbiting a star (e.g., the Sun) of mass M, the gravitational force is:
F=−GMmr2​
Here:
  • GG is the gravitational constant,
  • rr is the distance between the planet and the star.
This force acts as a central force, always pointing toward the star.
To solve for motion, we need to write down Newton's equations of motion in Cartesian coordinates (x,y):
md2xdt2=−GMmr3x,md2ydt2=−GMmr3ym \frac{d^2 x}{dt^2} = -\frac{GMm}{r^3} x, \quad m \frac{d^2 y}{dt^2} = -\frac{GMm}{r^3} y

where
r=x2+y2r = \sqrt{x^2 + y^2}.

These are second-order differential equations, and solving them for x(t) and y(t) can be quite cumbersome, especially when the symmetry of the system is not fully utilized. Moreover, the forces need to be explicitly calculated, which complicates the process.

newtonian and hamiltonian approach of Planetary Motion in a Central Force Field as only illustration

Hamiltonian Mechanics Approach

In Hamiltonian mechanics, we reformulate the problem in terms of energy and generalized coordinates. For a central force problem, it’s easier to switch to polar coordinates (r,θ) instead of Cartesian coordinates.

Step 1: Hamiltonian Formulation

The Hamiltonian H is the total energy of the system, which is the sum of the kinetic energy T and potential energy V:H=T+V
  • The kinetic energy in polar coordinates is:
    T=pr22m+pθ22mr2T = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2mr^2}

    where
    prp_r and pθp_\theta are the generalized momenta conjugate to rr and θ, respectively:

    pr=mr˙,pθ=mr2θ˙.
  • The potential energy due to gravity is:
    V=−GMmr.
Thus, the Hamiltonian becomes:
H(r,pr,pθ)=pr22m+pθ22mr2−GMmr.

Step 2: Hamilton's Equations

Hamilton’s equations of motion are first-order differential equations given by:
dqidt=∂H∂pi,dpidt=−∂H∂qi.
Here, the generalized coordinates are

q1=r 
and
q2=θ, 
with corresponding momenta
pr pθ. 
Let’s derive the equations:
  1. For  (radial coordinate):
drdt=∂H∂pr=prm.\frac{dr}{dt} = \frac{\partial H}{\partial p_r} = \frac{p_r}{m}.
dprdt=−∂H∂r=−∂∂r(pθ22mr2−GMmr).Simplifying:dprdt=pθ2mr3−GMmr2.\frac{dp_r}{dt} = \frac{p_\theta^2}{mr^3} - \frac{GMm}{r^2}.
  1. For θ\theta (angular coordinate):
dθdt=∂H∂pθ=pθmr2.\frac{d\theta}{dt} = \frac{\partial H}{\partial p_\theta} = \frac{p_\theta}{mr^2}.
dpθdt=−∂H∂θ=0.\frac{dp_\theta}{dt} = -\frac{\partial H}{\partial \theta} = 0.
Here,
pθ is conserved because the Hamiltonian does not explicitly depend on
θ. This reflects angular momentum conservation.

Advantages of Hamiltonian Mechanics in This Problem

  1. Symmetry Utilization:
    • Hamiltonian mechanics naturally identifies conserved quantities. Here, the conservation of angular momentum (pθp_\theta) arises directly from the symmetry of the system (rotational invariance).
    • In Newtonian mechanics, we must manually derive this result.
  2. Simplified Equations:
    • Hamilton’s equations are first-order differential equations, unlike Newtonian mechanics, which involves second-order equations. This simplification is significant when solving systems numerically or analytically.
  3. Energy-Centric Approach:
    • The Hamiltonian focuses on energy (a scalar quantity), making it more elegant and adaptable to systems with complex constraints or many degrees of freedom.
  4. Generalized Coordinates:
    • Switching to polar coordinates (r,θr, \theta) is natural in Hamiltonian mechanics, while Newtonian mechanics requires re-expressing forces in polar form, which is tedious.

Conclusion

Lagrangian and Hamiltonian mechanics were introduced to overcome the limitations of Newtonian mechanics and provide a more elegant, generalized, and powerful framework. By focusing on energies, symmetries, and variational principles, these formulations simplify the analysis of complex systems and extend seamlessly into modern physics.

In the case of planetary motion, Hamiltonian mechanics provides a more elegant and efficient approach by leveraging symmetries, conserved quantities, and generalized coordinates. The formulation reduces the problem to first-order differential equations, highlights angular momentum conservation naturally, and simplifies the mathematics of the system.

From the motion of planets to the behavior of particles in quantum mechanics, Lagrangian and Hamiltonian mechanics have become essential tools for understanding the universe.