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Why Hamilton's equations are called canonical equations?
The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
What is Hamiltonian function?
The total energy of the system simply described in classical mechanics called as Hamiltonian.
Is momentum hamiltonian operator is hermitian operator?
The hamiltonian operator is the observable corresponding to the total energy of the system. As with all observables it is given by a hermitian or self adjoint operator. This is true whether the hamiltonian is limited to momentum or contains potential.
What are the applications of partial differential equations in computer science?
All the optimization problems in Computer Science have a predecessor analogue in continuous domain and they are generally expressed in the form of either functional differential equation or partial differential equation. A classic example is the Hamiltonian Jacobi Bellman equation which is the precursor of Bellman Ford algorithm in CS.
What are equivalant equations?
Equivalent equations are equations that have the same solution set.
What does the Hamiltonian system refer to?
The Hamiltonian system refers to a dynamical system in classical mechanics that is described using Hamilton's equations of motion. It is a formalism that combines the equations of motion of a system with a specific function called the Hamiltonian, which represents the total energy of the system. It is widely used in physics and engineering to analyze and model the behavior of complex physical systems.
What has the author G A Sardanashvili written?
G. A. Sardanashvili has written: 'Generalized Hamiltonian formalism for field theory' -- subject(s): Mathematics, Constraints (Physics), Manifolds (Mathematics), Hamiltonian systems, Lagrange equations, Field theory (Physics)
How do Lagrangian and Hamiltonian mechanics differ in their approaches to describing the dynamics of a system?
Lagrangian mechanics and Hamiltonian mechanics are two different mathematical formulations used to describe the motion of systems in physics. In Lagrangian mechanics, the system's motion is described using a single function called the Lagrangian, which is a function of the system's coordinates and velocities. The equations of motion are derived from the principle of least action, which states that the actual path taken by a system is the one that minimizes the action integral. On the other hand, Hamiltonian mechanics describes the system's motion using two functions: the Hamiltonian, which is a function of the system's coordinates and momenta, and the Hamiltonian equations of motion. The Hamiltonian is related to the total energy of the system and is used to determine how the system evolves over time. In summary, Lagrangian mechanics focuses on minimizing the action integral to describe the system's motion, while Hamiltonian mechanics uses the Hamiltonian function to determine the system's evolution based on its energy.
What is the relationship between Lagrange and Hamiltonian mechanics in classical physics?
In classical physics, Lagrange and Hamiltonian mechanics are two equivalent formulations used to describe the motion of particles or systems. Both approaches are based on the principle of least action, but they use different mathematical formalisms. Lagrange mechanics uses generalized coordinates and velocities to derive equations of motion, while Hamiltonian mechanics uses generalized coordinates and momenta. Despite their differences, Lagrange and Hamiltonian mechanics are related through a mathematical transformation called the Legendre transformation, which allows one to derive the equations of motion in either formalism from the other.
Why Hamilton's equations are called canonical equations?
The word canonical means "by a general law, rule, principle or criterion". When the Hamiltonian operator is applied to the (average momentum) wave function it gives quantized values. In this sense the Hamilton equations gives the Schrodinger equation discreet values by a general law.
What is the relationship between the Lagrangian and Hamiltonian formulations in classical mechanics?
In classical mechanics, the Lagrangian and Hamiltonian formulations are two different mathematical approaches used to describe the motion of a system. Both formulations are equivalent and can be used interchangeably to solve problems in mechanics. The Lagrangian formulation uses generalized coordinates and velocities to derive the equations of motion, while the Hamiltonian formulation uses generalized coordinates and momenta. The relationship between the two formulations is that they both provide a systematic way to describe the dynamics of a system and can be used to derive the same equations of motion.
What is the proper adjective for Hamilton?
Hamiltonian is the proper adjective for Hamilton. For instance: The Hamiltonian view on the structure of government was much different from that of Jefferson.
How can the Hamiltonian path be reduced to a Hamiltonian cycle?
To reduce a Hamiltonian path to a Hamiltonian cycle, you need to connect the endpoints of the path to create a closed loop. This ensures that every vertex is visited exactly once, forming a cycle.
How can the Hamiltonian cycle be reduced to a Hamiltonian path?
To reduce a Hamiltonian cycle to a Hamiltonian path, you can remove one edge from the cycle. This creates a path that visits every vertex exactly once, but does not form a closed loop like a cycle.
When is the Hamiltonian conserved in a dynamical system?
The Hamiltonian is conserved in a dynamical system when the system is time-invariant, meaning the Hamiltonian function remains constant over time.
What is Hamiltonian function?
The total energy of the system simply described in classical mechanics called as Hamiltonian.
How can the Hamiltonian be derived from the Lagrangian in classical mechanics?
In classical mechanics, the Hamiltonian can be derived from the Lagrangian using a mathematical process called the Legendre transformation. This transformation involves taking the partial derivatives of the Lagrangian with respect to the generalized velocities to obtain the conjugate momenta, which are then used to construct the Hamiltonian function. The Hamiltonian represents the total energy of a system and is a key concept in Hamiltonian mechanics.
