# Why do conservation laws in mechanics lead to symmetry principles?

• neelakash
In summary, the conversation discusses the relationship between the Lagrangian and Hamiltonian principles in mechanics and the conservation laws of linear momentum, angular momentum, and energy. The conversation also mentions the Noether theorem, which states that symmetries lead to conservation laws. The connection between these concepts is not typically explained in purely Newtonian terms and may require further research using references such as the Noether theorem. The conversation also mentions the virtual work method and its connection to Lagrangian mechanics. Finally, it is noted that the conservation laws can be intuitively understood as maintaining symmetry in various physical systems.
neelakash
I do not know the Lagrangian or Hamiltonian principles of mechanics. However,I came to know that conservation of linear momentum,angular momentum and conservaton of energy lead to homogeneity of space,isotropy of space and homogeneity of time.Can anyone show why it is in purely Newtonian terms?I am familiar with virtual work methods...Otherwise if these conservation laws can be proved from the symmetry principles?You may also refer me to some link.

It is the other way around ...
A symmetry leads to a conservation law, this is called the Noether theorem.
It is more general than Newtoniam mechanics.
The virtual work method is the origin of Lagragian mechanics.
So indeed, it should be possible to explain it on this basis, but this is not usual.
Google for Noether, maybe you will find the connections you need.

Intuitively, this theorem is very clear.
Think to a freely spinning object for example: no force break the symmetry around the rotation axis.
If some force would define a preffered position for the spinning object, the angular momentum would not be conserved any more.

Thank you for your help.

## 1. What is conservation and symmetry?

Conservation and symmetry are two concepts in science that are closely related. Conservation refers to the idea that certain physical properties, such as energy and momentum, remain constant in a closed system. Symmetry, on the other hand, refers to the balance and proportion between different parts of an object or system.

## 2. Why is conservation and symmetry important in science?

Conservation and symmetry are important because they help us understand and describe the physical world around us. By studying these concepts, we can make predictions and better understand the behavior of objects and systems. They also play a crucial role in many scientific theories and laws.

## 3. How are conservation and symmetry related?

Conservation and symmetry are related because they both involve the idea of balance and equilibrium. In conservation, the balance is between different physical properties, while in symmetry, the balance is between different parts of an object or system. Both concepts help us understand how the physical world operates.

## 4. What are some examples of conservation and symmetry in nature?

Examples of conservation and symmetry in nature can be found in various phenomena, such as the conservation of energy in a closed system, the symmetry of snowflakes, and the balance of forces in a pendulum. These concepts can also be observed in living organisms, such as the symmetry of a butterfly's wings or the conservation of mass in biological processes.

## 5. How are conservation and symmetry applied in real-world situations?

Conservation and symmetry have practical applications in many fields, such as engineering, physics, and biology. For example, engineers use the principles of conservation to design efficient systems, while physicists use symmetry to describe and predict the behavior of particles. In biology, these concepts are used to study the balance and stability of ecosystems and the conservation of species.

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