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Lagrangian formalism prefers simple logic without any details about a system to complicated mechanical equations as in   Quantum Mechanics and newtonian mechanics.

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Lagrangian ‘action’  is just an auxiliary quantity which gets us to original equations of motion.
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Hamiltonian information of phase space entropy  is always conserved, but it is not equivalent to energy because energy is not always conserved.

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Hamiltonian is the method of describing diverse mechanical systems as the conservation of phase space entropy.

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Schrödinger’s old quantum mechanics uses the Hamiltionian  state vector, which is intuitively easier than energy to represent probabilities with the help of a spacelike slice at a particular moment in time.

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In quantum mechanics ,  a “sum of possibilities” arises naturally if we use the path integrals.
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The principle of least action might arise from the path integral formalism of quantum mechanics.

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The probability amplitude for some particle to go from A to B is given by a sum over all possible paths from A to B. The action is stationary along the most probable path because adjacent paths interfer constructively only where the action does not change much.

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The universe has no preference for storing energy as kinetic or  potential.
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The universe tries to distribute energy between kinetic and potential  with minimal difference between the two.

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The universe is a multi level  container (HUP obeying quantum foam) with energy trying to seek the same level everywhere. Life is  just one of the infinite ‘quantum tunnels’   for energy to “level out.”

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Just like water, energy tries to redistribute itself between the kinetic and potential buckets evenly.

Depending on the constraints at a particular time, energy may be sloshing from one bucket to another , trying to “level out.”
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A system evolves through phase space as a function of time.
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Two main forms of energy are potential and kinetic.

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Classical quantities and units are respectively replaced by operators and probabilities in quantum mechanics.

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Lagrangian does not examine the system’s particulars at one point in time in configuration space, it examines the shape of a multidimensional state-space bowl, and asserts that the state of the system will be found at the extremum, the  farthest point from the middle, where T-V (Kinetic Energy minus Potential Energy=Work/consumption) is minimized.

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The Lagrangian is simply a mathematical construct to define the action S, a quantity identical to path or  the work performed by the system.

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Out of many possible paths, a  particle will take the path whose action is hardest to change by moving it a bit.
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Lagrangian is the mathematical definition of a universe. The Lagrangian for a classical harmonic oscillator is exclusively  identical to a universe with a particle that only interacts with this Potential.

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The Lagrangians of Quantum Field consist of interactions that only are allowed in this universe.

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There is no fundamental difference between Newtonian mechanics, Lagrangian mechanics and Hamiltonian mechanics.
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Classical mechanics is more Lagrangian than Hamiltonian.

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Lagrange made dynamical  measurement showing indifference to the positions of the various parts of the system.

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Instead of Newton’s three laws, Lagrange mechanics deals with a single fundamental logic: the principle of least action. Since the laws of physics are not absolute, the same physical phenomenon can be explained by many different fundamental laws.Even this so called ‘fundamendality’ is just a practical assumption without any proof.

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According to Lagrange mechanics, we can calculate the motion by minimizing a certain quantity, called the action, a product of time and energy.

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Lagrangian mechanics is centered around energies. Forces are no longer needed to determine the motion of objects.
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Many fundamental laws of physics can be described by Lagrangian  principle of least action.

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Lagrangian mechanics is a very widely used and mathematically equivalent formulation of Newtonian mechanics.

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Lagrangian principle of stationary action  says that “action” is always stationary (always a maximum or a minimum). Like Newton’s Laws scientists accept it without any proof.

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The principle of stationary action is  called the principle of least action because action is usually a minimum.

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Quantum mechanics explains why action is stationary.

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The lagrangian is a way of making  classical mechanics easier to solve by  emphasizing conservative forces and ignoring constrain forces.

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Instead of saying “here’s which way you accelerate” , Lagrangian mechanics says “you move in such a way that you minimize the action between fixed endpoints.”
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When tragectories and frequencies are known better than the forces ,the Lagrangian  reveals  the nature of forces by lagrange-eular equation.
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Classical Lagrangian mechanics is used for an individual particle with a finite number of degrees of freedom.

Lagrangian coordinate  describes  the motions of particles of a solid or fluid in continuum mechanics.
Lagrangian function is the kinetic energy (energy of motion) minus the potential energy (energy of position).

Lagrangian field theory is the field-theoretic analogue of Lagrangian mechanics.
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